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Cornell University Library 
 
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 An elementary treatise on conic sections 
 
 3 1924 031 214 384 
 olin.anx 
 

CONIC SECTIONS 
 
 AND 
 
 ALGEBRAIC GEOMETRY. 
 
Cambriise: 
 
 PRINTEB BY 0. J. OLAT, M.A. 
 AT THE nNIVERSITT PBESS. 
 
AN 
 
 ELEMENTARY TREATISE 
 
 ON 
 
 CONIC SECTIONS 
 
 AKD 
 
 ALGEBRAIC GEOMETRY, 
 
 WITH NUMEKOUS EXAMPLES AND HINTS FOR THEIE SOLUTION 
 
 CF^pecialla liegfgneli fot tjc SSgc of JStgititiew. 
 
 G. HALE PUCKLE, MA., 
 
 ST John's college, Cambridge, head mastee of windbemeeb college. 
 
 FOURTH EDITION, REVISED AND ENLARGED. 
 
 Itontlow : 
 
 MACMILLAN AND CO. 
 
 1870. 
 
 [All Sights reserved.'] 
 
unsve:rssty 
 
 LIBRARY^ 
 
PEEFACE. 
 
 The first edition of this book was published shortly after 
 the appearance of Dr. Salmon's admirable treatise, with the 
 hope that I could write a short and easy work upon a 
 similar plan, without losing the obvious advantages of his 
 harmonious and consecutive arrangement of the subject. 
 Hence the Reduction of the General Equation of the Second 
 Degree will be found to precede the discussion of the pro- 
 perties of the curves, and the Parabola is treated of after 
 the Ellipse andHjrperbola; but I have arranged the chapter 
 on the Reduction of the General Equation, so that a very 
 small part of it will suffice, when the subject is read for 
 the first time. 
 
 My chief object was to write with special reference to 
 those difficulties and misapprehensions, which I had found 
 most common to beginners. In the third edition, published 
 two years ago, I tried, without losing sight of my original 
 purpose, to make the book more suitable to the require- 
 ments of the time. It was almost entirely re-written and 
 re-arranged, and a large amount of new matter was added. 
 The present edition has been carefully revised, and some 
 few additions have been made. 
 
ALGEBRAIC GEOMETRY. 
 
 CHAPTER I. 
 Position of a Point on a Plane — Loci — Equations, 
 
 1. Algebra is applied to Geometry, to investigate pro- 
 blems which concern the Tnagnitudes of lines or areas, or to 
 express the position of points and the form of curves. The 
 Algebraic proofs of the propositions, of the second book of 
 Euclid are examples of the former application, and the use 
 of positive and negative signs with the Trigonometrical ratios 
 has introduced the reader to the latter. It is upon the latter 
 branch of the subject that we are now about to enter; and 
 we prqeeed to explain the system invented by Descartes, and 
 hence called 'the Cartesian system,' in which the positions 
 of points are determined by means of Co-ordinates. 
 
 2. In ordinary Algebra we have been in the habit of 
 considering the symbols + and — as symbols of the reverse 
 operations of addition and subtraction. In Algebraic Geo- 
 metry, as in Trigonometry, we use these symbols to indicate 
 contrariety of position. Thus, in any indefinite straight line 
 XX', let us consider as a fixed point from which distances 
 are to be measured, and let us take two points M, J/', equi- 
 distant from and on opposite sides of it; then, if we 
 li 1 
 
2 POSITION OF A POINT ON A PLANE. 
 
 denote the distance OM hy + a, we shall express the distance 
 
 X' M' A M X 
 
 OM' by — a, i.e. we shall consider lines measured from left 
 to right as positive, and lines measured from right to left as 
 negative. 
 
 The propriety of this convention will best appear from an 
 example. Let us suppose X the east and X' the west of 
 ; then, if a man starting from walk 6 miles to the east 
 and then 2 miles to the west, the magnitude of his walk 
 will be represented by 8, but his position with reference to 
 his starting place by +4, or, in order to find his position, 
 if we consider the distance he has walked east as positive, 
 we must consider the distance he has walked west as 
 negative. Similarly, if he walked 4 miles east and then 6 
 miles west, his position would be expressed by — 2, or two 
 miles to the left of 0, but the magnitude of his walk by + 10. 
 This distinction must be carefully borne in mind, and it 
 must be remembered that the convention established above 
 has reference simply to the position of points on the line 
 XX', with reference to ; hence if we take 0M= + 4 and 
 OM' = — 4, the student must be careful to avoid the error 
 of assuming MM' to be represented by 4—4 or 0, as this 
 would be reasoning about the magnitude of a line upon 
 assumptions which have only been made about the position 
 of its extremities with regard to a fixed point. 
 
 3. A simple example will shew the great advantage of 
 ■ the preceding convention, in rendering generally true formulae 
 that would otherwise be true for a particular case only. Let 
 and A be two fixed points upon the line XX', and M 
 a point moving upon the same line. In order to express the 
 distance of M from the point by means of the distances 
 OA and AM, let 
 
 OM=x, OA=a,AM = x'\ 
 
POSITION OF A POINT ON A PLANE. 3 
 
 then, if the point if is placed to the right of A, we have 
 
 x = a + x' (1) ; 
 
 if it is between and A, 
 
 x = a-x' (2); 
 
 if it is to the left of 0, 
 
 x = x —a (3). 
 
 Here we have supposed x and a?' to be the magnitudes of 
 OM and AM, without regard to sign ; and we see that three 
 formula are necessary to express the distance of M from 0, 
 when we consider all the positions that it can occupy. But, 
 if attention is paid to the signs of x and x', the formula first 
 obtained is equally true for the other two cases. For, if we 
 suppose M to pass from the right to the left of A, x' becomes 
 — x, and formula (1) becomes the same as (2), as long as x 
 is less than a or M lies between and A. If x' is greater 
 than a, Jlf will fall to the left of 0, and x becomes negative; 
 in this case formula (1) becomes 
 
 — x = a — x', 
 
 which is the same as formula (3). 
 
 4. To determine the position of a point on a plane. 
 
 Let us suppose that we know the position of two straight 
 lines Ox and Oy; then, if we draw the w 
 
 parallels PM, FN to these lines from y/ 
 
 any point P, we shall know the po- / / 
 
 sition of that point, if the lengths of ^ ^/ — -^ — ^ 
 
 PM and PiV are given. For example, / 
 
 if PN= a, PM = b are two equations 
 given us, we can determine the position of the point P with 
 regard to the lines Ox and Oi/; for, if we measure OM (= a) 
 along Ox, and ON ( =s J) along Oy, and complete the paral- 
 
 1—2 
 
4 POSITION OF A POINT ON A PLANE. 
 
 lelogram OMPN, P will be the point whose position we 
 wished to determine. 
 
 5. The line PJIf is usually denoted by the letter y, and 
 is called the ordinate* of the point ; OM, which = PA'', is 
 denoted by the letter x, and is called the abscissa of the point; 
 the two lines are called the co-ordinates of P. The Ijnes xx, 
 yy are called the axes of co-ordinates, and their point of 
 intersection is called the origin: xx is called the axis of x, 
 and yy the axis of y. 
 
 The point P is said to be determined when the values of 
 X and y are given, as by the two equations x = a, y==h; as, 
 for example, if it were given that a; = 3 feet, y = 2 feet, we 
 should determine the point of which x and y are the . co- 
 ordinates by measuring 3 feet along Ox and 2 feet along Oy, 
 and completing the parallelogram of which these two lines 
 formed the adjacent sides. The corner of the parallelogram 
 opposite to would be the position of the point. The point 
 whose position is defined by the equations x = a, y = i is 
 commonly spoken of as the point {ab). The axes are said^ 
 to be rectangular or oblique, according as angle y Ox is or is 
 not a right angle, 
 
 6. We have supposed hitherto that x and y, the co- 
 ordinates of the point, are positive 
 quantities, and have measured the 
 distances along the lines Ox and Oy. 
 If a; or y be negative, it will indi- 
 cate, according to the convention 
 established above, that we must 
 measure along Ox' or 0^, in order to 
 find the position of the point. For 
 
 • The lines PM, &c. drawn paraUel to one another from a series of 
 points, were called by Newton 'lines) ordinatim applioats,' and the abscissa! 
 ■OM, &c. were the distances cut offhj these lines from a fixed line as Ox. 
 
POSITION OF A POINT ON A PLANE. 
 
 example, if P, P^, Pj, P, be points, situated in the four 
 angles made by the axes, whose co-ordinates are of the same 
 •magnitude, i.e. 
 
 PM=P,M' = PJd'=P,M=h, 
 
 and OM = OM' = a, these points will be represented by the 
 following equations : 
 
 (x = a, 
 
 x = — a, 
 
 Mr--. 
 
 The point represented by a; = 0, 3/ = is the origin ; by 
 x = 0,y = h\s the point N on the axis of y ; by a: = a, ^ = is 
 the point M on the axis of x, and so on. 
 
 r 7. To find the distance between two points P, E, whose 
 co-ordinates are known with reference to axes inclined to each 
 other at a given angle. 
 
 Let angle yOx = a, and let the co-ordinates of P be 
 PM{=y'), OM{=x'), and of B, 
 RN{=^y"),ON{=^x"); 
 
 draw PQ parallel to Ox, then 
 PQ= ON-OM=x"-x', 
 RQ = BN-PM=y"-y', 
 and ^ PQR = ir — o>; hence 
 
 PR' = PQ' + RQ''-2PQ.RQ cos PQR 
 = (a;" _ x'Y + {y" - yj 4- 2 (a;" - a;') [y" - /) cos «". 
 
 If one of the points as P were the origin, so that x = 0, 
 y = 0, we should have 
 
 Pfi" = x"' + y"' + 2x"y" cos to. 
 
6 THE STRAIGHT LINE 
 
 Cor. These formulae become much more simple when 
 £0 = 90°, or the axes are rectangular; in that case, smce 
 cos ft) = 0, 
 
 PiJ==(x"-a;7 + (y"-.yr, 
 
 or, if P be at the origin, 
 
 8. In using these formulae, attention must be paid to 
 the signs of the co-ordinates. If 
 the point P, for instance, be in 
 the angle xOy', the sign of its 
 ordinate {y) will be negative, and 
 we must write y" +y' instead of 
 y" — y in the formulae ; this may- 
 be seen to agree with the figure, 
 s&PQ will now = B^-^ PM. 
 
 The reader should draw figures, placing P and R iii 
 
 different compartments and in other varieties of position, 
 
 that he may assure himself of the universal truth of the 
 
 expressions obtained for the distance PR. 
 
 Ex. To find the distance between two points whose eo-ordinates are a;=2, 
 y = - 3 and a; = - 5, y=&, the axes being inclined at an angle of 60°. 
 
 Here a;"-a!'=- 5-2 = -7, /'-/=6 + 3=9, 
 
 and cos a= \ : hence, if d be the distance, 
 
 d''=49 + 81-2.7.9.^ = 67, 
 
 d=^y67. 
 
 9. We have said that the student should, by different 
 figures, convince himself of the uuiversal truth of the formula 
 of Art. 7 ; but a careful consideration of a few propositions 
 in this manner will shew him that this is not necessary, and 
 that the formula first obtained will, as in Art. 3, adapt itself 
 to the changes of the figure. Thus, in the fig. of Art. 7, QR 
 is the difference between the actual magnitudes of RN and 
 
JOINING TWO POINTS. 
 
 PM, and must remain so, as long as P and Q are above Ox, 
 that is, as long as both the ordinates are positive. If P falls 
 below Ox, while R remains above it, QR is the sum of PM 
 and RN; and this change is provided for by the change of 
 the sign of PM. Hence, in Algebraic Geometry, we need not 
 examine every modification of the figure, as in Euc. II. 13 ; 
 for any general figure, that we draw consistent with the con- 
 ditions of a problem, will lead us to a result which will be 
 true for all possible cases comprehended in it. 
 
 10. To find the co-ordinates of a point (hk), where the 
 straight line joining two given points (x'y), (x"y") is cut in a 
 given ratio. 
 
 Let P, Q be the two given 
 points (sc'y), ix'y"), R the point 
 {hie) whose co-ordinates are 
 sought, and let 
 
 PR : RQ :: m : n. 
 Draw the ordinates PM, 
 RL, QN, and the line PEF 
 parallel to OT; then 
 
 PR_PE ML 
 RQ~ EF~TN' 
 m _ x' — h' 
 n 
 
 or 
 
 whence 
 
 h = 
 
 Similarly 
 
 h-x" 
 mx" + nx 
 m+n 
 
 m+n 
 
 li m=n, or PQ is bisected in R, 
 
 h = 
 
 x" + x' 
 
 1c = 
 
 y" +V' 
 
 2 ' ' — 2 ' 
 a result which is very frequently of use. 
 verified for different figures, as in Art, 8. 
 
 It should be 
 
8 
 
 POLAR CO-ORDINATES. 
 
 11. Polar Co-ordinates. 
 
 Besides the method of expressing the position of a point, 
 that we have hitherto made use of, 
 there is another which can often be 
 employed with advantage. If a fixed 
 point he given, and a fixed line OA 
 through it, we shall evidently know the 
 position of any point P, if we know the 
 length OP and the angle POA. The 
 line OP is called the radius vector, the 
 
 fixed point is called the pole, the line OA the initial line; 
 and this method is called the method of polar co-ordinates. 
 We shall, for the sake of brevity, call the point whose polar 
 co-ordinates are p and 0, 'the point {p6)! 
 
 12. The sign - is applied to polar co-ordinates on exactly 
 the same principles as those already explained in the case of 
 rectangular co-ordinates. Thus, if p represent any distance 
 measured from towards P, — p represents an equal distance 
 measured from towards P' ; and, if 6 represent any angle 
 measured from A. towards P, —6 will represent an equal 
 angle measured from A towards Q. We shall define the 
 positive direction of p as that part of the line which marks 
 the boundary of the angl^ 6. A few examples will make this 
 clearer. 
 
 Let a be any distance OP, measured from towards P, 6 
 being the angle which OP makes with OA ; then 
 
 Fig. 1. Kg. 2. Mg. 3. 
 
 N-l 
 
TBANSFOBMATION. 
 
 Fig. 4. Fig. 5. 
 
 ..«C 
 
 O —a. x> Ji 
 
 4' 
 
 TT 
 
 2' 
 4 
 
 ^ = 
 d = 
 5 = 
 
 p = a represents P in Fig. 1 ; 
 /> = -« T'ig.2; 
 
 p = — a. 
 p= — a. 
 
 p = a .. 
 
 .Fig. 3; 
 .Fig. 4; 
 ,Fig.5. 
 
 It is important to observe that the direction in which p is 
 measured depends not only on its sign, but also on the value 
 
 of 6 ; thus, when 6= -r- and p = - 
 
 ■ a, p must be measured from 
 
 to P as in fig. 3 ; and when ^ = -t" » P = <^, p must be mea- 
 sured in exactly the same direction. Again, when ^ = 0, p = a, 
 and when 5 = TT, p= — a, p must in both cases be measured 
 from towards A. 
 
 13. To transform polar co-ordinates into rectangular, or 
 rectangular into polar. 
 
 . Suppose P to be the point whose polar co-ordinates 
 OP{=p), and angle PO^ (=6) are ' 
 known. Take the pole as origin 
 of rectangular co-ordinates, OA and 
 a perpendicular through 0, as axes 
 of X and y, and let OMi^ai) and 
 PM(=t/) be the rectangular co-ordi- 
 nates of P. 
 
 m: a. 
 
10 POLAK CO-ORDINATES. 
 
 Hence we have 
 
 1/ = psinff, x = p cos 0, 
 
 , 'j? + f = p\ ta,n.0=^. 
 
 These equations will enable us to transform any equation from 
 polar co-ordinates to rectangular, and vice versa. 
 
 Ex. To transform the equation 
 
 a'pBio? B=2<d/' cos e-b'p cos' e 
 to rectangular co-ordinates. 
 Multiplying by p we have 
 
 ay sin" e = 2a5 V eos 6 - J V "os" 6, 
 or a'y'=iab"x-h"i?, 
 
 the initial line being now the axis of x, and the pole the origin. 
 
 14. To determine the distance between two points whose 
 polar co-ordinates are given. 
 
 Let F,P' be the two points, OP = p, angle POA = 0, 
 OF = p, angle F 0A=& ; then from y 
 
 the triangle POP we have 
 
 PP'" = 0P'^+ OP" -2 OP. OP cos P OP', 
 
 or PP^ = p'= + p' - 2/j'p cos {& - 0) , 
 which gives the required distance. 
 
 15. We shall now shew how the method of co-ordinates 
 may be applied to determine the position of straight lines and 
 the form of curves, and we will begin with a few of the sim- 
 plest cases. 
 
 We have seen that the position of a point P is completely 
 determined by two equations x = a, y = h. Suppose we have 
 one only of these equations, x=a, given us ; then, evidently, 
 if we draw a straight line PM parallel to Oy so that 0M= a, 
 the equation a; = a is satisfied by every point in that line pro- 
 
POSITION OF LINES ON A PLANE. 
 
 11 
 
 duced every way indefinitely. Hence the equation x=a, 
 instead of representing a point, represents a straight line 
 parallel to the axis of y. In like manner the equation y = h 
 represents a straight line parallel to the axis of x. Hence the 
 point (ab) may be considered as the point of intersection of 
 the two straight lines represented by the equations x = a, 
 y = h, for at that point both equations are satisfied. 
 
 Again, the equation a; = is evidently satisfied by every 
 point in the axis of y, and may therefore be said to represent 
 that axis ; and similarly, the equation ?/ = represents the 
 axis of X. The intersection of these straight lines is the point 
 where both the equations x — Q, y = are satisfied, or the 
 origin. 
 
 16. If, instead of two equations to determine x and y, we 
 have a single equation, expressing a relation between them, 
 (as, for instance, 2x + 3y + 9 = 0,) the position of the point P 
 will be indeterminate ; for we 
 may assign any values we 
 please, OM, 0M„ OM^, &c., 
 to x, and the equation will 
 furnish us with corresponding 
 values of y, FM, F^M^, FJM^, 
 &c., where we draw the ordi- 
 nate downwards, if the value 
 which we assign to x gives us 
 a negative value of y ; hence 
 we may obtain any number of 
 
 points P, Pj, Pj, &c,, whose co-ordinates satisfy the equa- 
 tion. We may take the points M, M^, M^, &c. as near as we 
 please, and so get an assemblage of points P, F^, P^, &c. as 
 near to one another as we please. If the points are brought 
 indefinitely near to one another, they will form a continuous 
 line ; hence we may consider the equation to represent the 
 
12 LOCI AND 
 
 line which passes 'through all the points whose co-ordinates 
 satisfy it. 
 
 Ex. Suppose the equation to be y=S-3x; then, the unit of measure 
 being Imown, if we take 0M=.\, we have PM=h, and P is a point in the line , 
 represented; if OM-i=% P^M^ — 1, and Pj is another point; by taking values 
 of SB between 1 and 2 we may determine any number of points between P 
 and Pi. If 0Jl/2=3, P^M^=-\, and must therefore be drawn downwards; 
 if OMz=i:, PiM3=- 4, and must also be drawn downwards ; by takiiig values 
 of X between 2 and 3, or 3 and 4, we may obtain any number of points be- 
 tween P, and Pj or Pj and Pj. If we take a negative value of x, or suppose 
 OMi=- 1, we have PJdi=U, and so on : hence the line represented by the 
 equation wiU be that which passes through P, P,, &c. and aU other points 
 which satisfy the equation. 
 
 17. An equation between x and y may give us more than 
 one value of y for each assumed value of x, or vice versd ; for 
 example, the equation y' = 6a! will give, for every value of x, 
 two equal values of y with opposite signs, so that, for every 
 position of M, we shall have two positions of P, and two lines 
 will be traced out at the same time. 
 
 When a point is restricted by conditions of any kind, to 
 occupy any of a particular series of positions, that series of 
 positions is called the locus of the point, or the loam of the 
 eqvMion that expresses the conditions ; and the equation is 
 called the equation to the locus, or the equation to the curve 
 which passes through all the positions of the point. 
 
 The curve then, or locus, represented by an equation be- 
 tween X and y, is the assemblage of all those points whose co- 
 ordinates satisfy the equation ; hence, if x, y be quantities 
 which, when substituted for x and y, satisfy the equation, the 
 point (aj'y) is a point on the locus. 
 
 18. Eemarks similar to the above (Arts. 15 — 17) will 
 apply to polar co-ordinates ; for, if p and d are connected by 
 an equation, every value of d will, according to the degree of 
 the equation, give one or more values of p, which must be 
 interpreted according to the rules of Art. 12. 
 
EQUATIONS. 
 
 13 
 
 19. The following simple examples will shew the method 
 of representing loci by means of equations. 
 
 (i) To determine the' locus of the equation y=x-i, the axes being rect- 
 angular. 
 
 hence, 
 
 JJf Jc 
 
 Let Ox, Oy be the axes ; now, when !i;=0 in the equation, y=- 4 
 if we take OD — i on the negative part of 
 the axis of y, D will be a point in the locus 
 (Art. 6). Again, when ^=0, x=i; hence, 
 if we take 02*= 4, on the positive part of 
 the axis of x, T will be a point in the locus. 
 Draw the straight line DT, and produce it 
 indefinitely both ways; this will be the - 
 locus of the equation; for take any point 
 P in this Une and let its co-ordinates be 
 OM=x, PM=y; then, evidently, PM=MT^ 
 since the line makes an angle of 45" with 
 Ox, and therefore 
 
 PM=OM-OT, 
 
 or y=x-i. 
 
 Heno6 the co-ordinates of any point in the line DT satisfy the equation, and 
 BT is the locus required. 
 
 (ii) To determine the loci of the equations y=x and y—-x. 
 
 It is easily seen that, whatever be the inclination of the axes, the former 
 equation is satisfied by every point in a straight line bisecting the angle yOx, 
 and the latter by a straight line perpendicular to it, bisecting the angle SOx. 
 
 (iii) To determine the locus of the equation a? + y'=9, the axes being rect- 
 angular. 
 
 This equation asserts (Art. 7) that the distance of the point (icy) from the 
 origin is constant and =3. Hence the locus of the point is a circle whose 
 centre is the origin and radius =3. 
 
 (iv) To determine the locus of the equa- 
 tion, p=p sec 9. 
 
 Let be the pole, OA the initial line. 
 Taie OD=p, and draw DP perpendicular to 
 OA ; then, if P is any point on the line, 
 OP=OD eeaPOD, or, H OP=p, and the 
 angle P0D=9, we have p=p sec for any 
 point on DP ; hence DP is the locus. 
 
14 LOCI AND 
 
 (v) To determine the loci of the equations p=a constant, e=a constant. 
 The first is evidently a circle with the pole for centre. The second is a 
 straight line passing through the pole, and is the initial line, if 6=0. 
 
 20, As examples of determining the equation to a locus 
 from some known property it possesses, we may take the 
 following. 
 
 (i) To find the equatim to the Ucus, of which every ordinate (PM) is 
 greater than the corresponding abscissa (OM) by a given distance (5). 
 
 The condition is evidently expressed by 
 the equation i/=x + b, and, if we take 
 OI>=OT=i, as in the figure, the line XT 
 will plainly be the locus of the equation, for, 
 whatever be the position of P on the line, 
 PM=MT,oy:y=x + h. 
 
 (ii) To find the equation to the locus, of which every ordinate (PM) is a 
 mean proportional between MO and MB, B being a point in Ox, and the axes 
 being rectangular. 
 
 Let 0B=2r and the angle yOxhe supposed =901"; then since PM^=OM.MB, 
 the equation is 
 
 y^=x(2r-x) or y^=2rx-7?. 
 
 21. It is evident then, that the examination of any equa- 
 tion between two unknown quantities must generally give 
 rise to an assemblage of points, or a geometrical locus ; and, 
 conversely, if we can, by knowing some property of a locus 
 or the circumstances of its description, determine a relation 
 between the co-ordinates of any point taken arbitrarily on 
 the locus, that relation is the equation to the locus. 
 
 If a problem relate to the position of a single point, and 
 the data be sufficient to determine the position of that point, 
 the problem is determinate ; but, if one or more of the con- 
 ditions be omitted, the data which remain may be sufficient 
 to determine more than one point, each of which satisfies the 
 
EQUATIONS. 15 
 
 conditions of the problem; the problem is then indetermi- 
 nate. Such problems will in general result in indeterminate 
 equations representing straight lines or curves, upon which 
 the required points are situated. If, for example, the hypp- 
 tenuse and one side of a right-angled triangle be given, the 
 position of the vertex is fixed, for not more than one triangle 
 can be described on the same side of the hypotenuse, with 
 these data ; but, if the hypotenuse only be given, the ver- 
 tex may lie anywhere in the circumference of the semicircle 
 described on the hypotenuse as diameter; the problem of 
 finding the vertex is then indeterminate, and the semicircle 
 is said to be the locus of all the vertices of all right-angled, 
 triangles described on one side of the given hypotenuse. 
 
 22, In the following pages we purpose to investigate 
 those lines only, which are represented by equations of the 
 first and second degree between two variable^ (x and y). An 
 equation of the first degree is an equation that involves no 
 power of either of the variables higher than the first, nor 
 their product ; its most general form is 
 
 Ax + By+C=0. 
 
 An equation of the second degree contains no term in which 
 the sum of the indices of the variables is greater than two ; 
 its most general form is 
 
 dx' + Bxy + Cy^+Dx + Ey+F=0. 
 
16 EXAMPLES I. 
 
 EXAMPLES I. 
 
 1. Find the points whose co-ordinates are (0, 1), (—3, 1), 
 (-5, 6), (-2, -3). 
 
 2. Draw a triangle, the co-ordinates of whose angular points 
 are (0, 0), (2, — 3), ( — 1, 0), and find the co-ordinates of the middle 
 points of its sides. 
 
 3. A straight line cuts the positive part of the axis of ^ at a 
 distance 4, and the negative part of the axis of fc at a distance 3 
 from the origin : find the co-ordinates of the point where the part 
 intercepted by the axes is cut in the ratio of 3 : 1, the smaller seg- 
 ment being adjacent to the axis of x. 
 
 4. There are two points P (7, 8) and Q (4, 4) : find the dis- 
 tance PQ, (i) with rectangular axes, and (ii) with axes inclined 
 at an angle of 60°, 
 
 5. "Work Ex. 4 when P is {-2, 0), Q (-5, - 3). 
 
 6. The co-ordinates of P are x = 2, y = 3, and of Q, x= 3, 
 y= 4; find the co-ordinates of ^, so that PR : RQ :: 3 : 4. 
 
 7. The polar co-ordinates of P are p = 5, 0= 75°, and of 
 §, p = 4, e = 15°i find the distance PQ. 
 
 8. Eind the polar co-ordinates of the points whose rectangular 
 co-ordinates are 
 
 (1) x = j3, (2) x = -^f3, (3) x = -l, 
 
 y= 1; y= i; y= i; 
 
 and draw a figure in each case. 
 
 9. Eind the rectangular co-ordinates of the points whose polar 
 co-ordinates are 
 
 (1) p = 5, (2) p = -5, (3) p= 5 
 
 and draw a figure in each case, 
 
 O'V '- P »'-p 
 
EXAMPLES I. 117 
 
 ' 10. Transform the equations 
 
 a; cos a + y sin a = a, os' + xy + y^ = V, 
 
 from rectangular to polar co-ordinates. 
 
 11. Transform the equation ps = d^ cos 2^ from polar to rect- 
 angular co-ordinates. 
 
 12. A straight line joins the points (2, 3) and ( — 2,-3); 
 find the co-ordinates of the points which divide the line into three 
 equal parts. 
 
 ' 13. If ABC is a triangle, and AB, AG are taken as axes of 
 X and y, find the co-ordinates (i) of the biseetion of BG, (ii) of the 
 point where the perpendicular from A meets BG, and (iii) of the 
 point where the line bisecting the angle BAG meets BG. 
 
 14. Find the co-ordinates of the same points, when AB is the 
 axis of X, and a straight line drawn from A perpendicular to AB 
 the axis of y. 
 
 15. The rectangular co-ordinates of a point & are h and h, 
 and a straight line PS is drawn, making an angle B with the axis 
 of as; shew that the co-ordinates of P are 
 
 a; = A -f (3 cos ^, y = h + p€\vi6, 
 
 where &P = p. 
 
CHAPTER 11. 
 
 General Equation of the first degree. The straight line. 
 
 23. The general equation of the first degree between two 
 variable quantities (x and y) represents a straight line. 
 
 Every equation of the first degree is included in the form 
 
 Ax+By+C=0 {A), 
 
 A, B, and C being quantities which do not involve a; and y ; 
 they represent numbers, are always invariable for any par- 
 ticular equation, and are therefore called constants. Let now 
 (*i.yi)> i^il/il' {^aV^ ^® three points in the line (whatever kind 
 of line it may be) represented by equation {A), and let the 
 abscissiE be in order of magnitude, and therefore, from (-4), 
 the ordinates also in the same order. Then, since the rela- 
 tion among the co-ordinates is true for every point of the 
 line, it is true for these three, and we have 
 
 Ax,+By^+C=0 (1)^ 
 
 Ax,JrBy^+C=Q (2), 
 
 Ax, + By,+ C=Q (3), 
 
 from (2) - (1) A {x, -x,) + B (y, - y^) = 0, 
 
 from (.3) - (1) A ^ - a;,) + 5 (y, - y^) = 0, 
 
 which evidently gives us 
 
GENERAL EQUATION OF THE FIRST DEGREE. 19 
 
 Now, if P, Q, R be the points (a.y,), {x^^, {x^y^ and 
 PLM be drawn parallel to 
 Ox, equation (4) gives us 
 
 MP~ LP' ^^^' 
 
 hence the trianglesPilffi, PiQ 
 are similar (Euc. vi. 6), and the angles BPM, QPL are equal, 
 and therefore Q is on the straight line PR, In like manner 
 it may be shewn that every other point in the line repre- 
 sented by the equation 
 
 Ax + By +0=^0 
 
 is in the same straight line with P and R. The line repre- 
 sented is therefore a straight line, 
 
 24. It has been already stated that A, B, G are fixed 
 quantities for any particular line. They are, therefore, the 
 quantities which distinguish one line from another ; for the 
 same symbols x and y are used, not only in the same line for 
 different points, but also for points in different lines. A little 
 care and practice are sufficient to prevent this apparent am- 
 biguity from causing any confusion, 
 
 For' instance, 
 
 Ax-\^By^- C=0 (1), 
 
 ax + by + c = (2), 
 
 represent (except when one equation is formed from the other 
 and therefore expresses no new relation between x and y) two 
 different straight lines, since A, B, C are different from a,l,c; 
 but X and y, though the same in both, have not the same 
 meaning in both, for in the first case x and y represent the 
 co-ordinates of any point in (1), and in the second the co- 
 ordinates of any point in (2). 
 
 It is manifest that the position of the line does not 
 
 2-2 
 
20 GENERAL EQUATION OF THE FIRST DEGREE. 
 
 depend upoH the absolute magnitude of A, B, C, since, if we 
 multiply or divide the equation by any constant, it will still 
 represent the same line. It is seen, indeed, by dividing the 
 equation by one of the constants A, B, C, that there are in 
 reality only two independent constants involved ; for instance, 
 if we divide by A, the equation becomes 
 
 73 Q 
 
 ■where -r and -j are the two constants that particularize the 
 
 line ; and we shall see hereafter, that two conditions, which 
 determine these constants, are sufficient and necessary to fix 
 the position of a straight line in a plane. 
 
 We shall hereafter, for the sake of brevity, often speak of 
 "the line represented by the equation Ax-\-By-\- 0=0," as 
 "the line {Ax + By + C)." 
 
 25. The converse of the preceding proposition is equally 
 true, that all straight lines are represented by equations of 
 the first degree ; but, if we wish to reason about any par- 
 ticular straight line, with reference to two fixed axes, we must 
 have some data by means of which we may construct it, such 
 as the position (with reference to the axes) of one or more 
 points through which it passes, the angle it makes with 
 either of the axes, its. distance from the origin, &c., &c.; and 
 we shall find that, as we suppose these data to vary, the form 
 of the equation {Ax + Bi/+ C=0) will vary too. We say 
 form, because, as the line itself is supposed to remain the 
 same, and only the means by which we determine its po- 
 sition to alter, it is plain that its equation must remain really 
 the same also. 
 
 We shall now proceed to examine some of the most obr 
 vious of those conditions which fix the position of a line 
 
CONSTANTS OF THE GENERAL. EQUATION DETERMINED. '21 
 
 witli reference to the axes, and find what form the equation 
 "will take in each particular case. We shall first deduce the 
 forms from the general equation (^a; + 5y + (7 = 0), and 
 afterwards yerify our results by independent geometrical con- 
 siderations. 
 
 26. (I) Let DT he the line, and let the lengths DO 
 and OT (which arei called its intercepts 
 on the axes) be the data to determine its 
 equation. It is plain that these are suf- 
 ficient to do so, as there can be only one 
 straight line passing through D and T. 
 
 .(^). 
 
 Let then OD = b, OT=a. 
 
 Now, in the equation 
 
 Ax + Bij-\-G = 
 
 G 
 when x=0,y= — t,, which are the co-ordinates of the point D; 
 
 and 
 
 G 
 wheny = 0, x = —-2> 
 
 and therefore 
 
 T, 
 
 -^^OJD = b, -^^OT=a; 
 
 A 
 
 but equation {A) may be written in the form 
 
 y-^i 
 
 X 
 
 
 A 
 
 + ■ 
 
 G 
 B 
 
 (1), 
 
 and therefore becomes 
 
 ? + -^ = l 
 a 
 
22 CONSTANTS OF THE GENERAL EQUATION 
 
 for these data ; hence, if by dividing out we write any equa- 
 tion of the first degree in the form of (1), the quantities in 
 the d^iominators will be the intercepts on the axes. 
 
 27. (II) Next let the data be the length of the perpen- 
 dicular OE from the origin on i>T, and the angle EOx which 
 it makes with the axis of x. These being known, the point E 
 is evidently known, and, as there can only be one straight 
 line drawn through E at right angles to OE, these data are 
 sufficient to determine the position of DT. 
 
 Let OE=p, angle EOT=a, and i^OT = », the known 
 angle between the axes. 
 
 Then, as before, 
 
 C 
 — "5 = OD, and /. = 
 
 B ' " cos (« — a) * 
 
 -~.= 0T, and.-. =-^, 
 
 A cosa' 
 
 and therefore {A) becomes 
 
 -^+ 2^-=l. 
 
 P P 
 
 cos a cos (q) — a) 
 
 or xcoa a. + y cos (w — o) =p. 
 
 The coefficients of x and y in this equation are called the 
 direction-cosines of the line. It must be carefully remembered 
 that by a is meant the angle EOx which the perpendicular 
 makes with the positive part of the axis of x, i.e. it is the 
 angle through which Ox must be turned towards Oy, in order 
 that Ox may coincide with OE, with x on the same side of 
 as-E' is ; for instance, if the data to determine the position of 
 the line were 01' = p, angle £0a5= 180° + a, E would lie in 
 EO produced, at a distance =^ from 0; but^ will in this 
 case, and always, be a positive quantity, since the positive 
 
OF THE FIRST DEGREE DETERMINED. 23 
 
 direction of OE may be defined as that which marks the 
 boundary of the angle o. 
 
 COE. 1. If w = 90°, or the axes be rectangular, the equa- 
 tion becomes 
 
 X cos Oi + y sin a =p, 
 a very useful form of the equation. 
 
 Cor. 2. If the equation to any line be 
 
 Ax-vBy-irG=0 (1), 
 
 and the equation to the same line be written in the form 
 
 X cos oL + y cos((a — a)=p (2), 
 
 or a;cosa+^sina=p (3), 
 
 according as the axes are oblique or rectangular, then, since 
 equations (2) or (3) are really identical with (1), we see that 
 A, B, and C are proportional to cos a, cos(a> — a), and —p 
 for oblique axes, and to cos a, sin a, and — p for rectangular 
 axes ; and therefore, for the latter case, 
 
 ABC ... 
 
 =-■ — = ~ (4), 
 
 cos a sin a ^ ^ ' 
 
 equations which are often useful. For example, if we wish 
 to write equation (1) in the form (3), we have from (4) 
 
 sin a _ B cos a _ A 
 cos'g + sin'q ^ ^' + .g' 
 
 since p is always a positive quantity. Therefore, from (4) 
 
 A . B 
 
 cos a ^ — , _^ =- , sm a = , , — . 
 
 ^A' + B'' ^A' + B" 
 
24 
 
 CONSTANTS OF THE GENERAL EQUATION 
 
 Hence equation (1) must be written, supposing C a positive 
 quantity, 
 
 A B ... 
 
 05 — 
 
 Hence, any equation may be written in the form of (3), 
 by so adjusting the signs, that the term not involving x ot y 
 may stand, as a positive quantity, as the right-hand member, 
 and then dividing ' the whole equation by the square root of 
 the sum of the squares of the coefficients of a; and y. 
 
 For instance, the equation 
 
 Sa; - % + 7 = 
 
 will be written 
 
 3,4 7 
 
 (6). 
 
 We say that p is always a positive quantity, because we have 
 agreed above always to consider a as the angle which the 
 perpendicular on the line, and not that perpendicular pro-, 
 duced, makes with the positive part of the axis of x. If we 
 were to remove this restriction, p might sometimes be a 
 negative quantity, as we have explained (Art. 12) in the case 
 of polar co-ordinates. For instance, 
 equation (6) represents a line JDT, 
 
 where sin^6>a5 = ^, cos EOx= — , 
 5 5 
 
 7 
 and OE = ^ , but the equation may 
 
 be written 
 
 3 
 
 5^- 
 
 62' = -5' 
 
 where we must take the angle KOx{=l%Qi'' + EOx) whose 
 
 sine = — ^, and cosine : 
 
 and then measure 0E=~ in 
 
 o 
 
. OF THE EIRST DEGREE DETERMINED, 
 
 25 
 
 what is now the negative direction of the perpendicular. 
 This will evidently give us the same line BT; but in the 
 following pages, when we speak of the equation to a line 
 written in the above form, we shall always suppose p to be 
 positive. 
 
 28. (Ill) Again, let the data be the length of 0Z> and the 
 angle DTx. Now, although we can draw an infinite number 
 of straight lines through D, there is only one that makes this 
 particular angle with the axis of x ; for, when we speak of the 
 angle which a straight line make^ with the axis of a;, we always 
 mean the angle which the part of the line above the axis 
 makes with the axis produced in a positive direction, i.e. the 
 angle DTx, not DTO. These data then determine the line. 
 Let OD = h, DTx = a, and DOx = (o, as before. 
 
 Then --2 = 5, 
 
 -9^=0T= ^™ (" ~ ") . I 
 A sin a 
 
 sin (ft) — a) , 
 = „■ „ -0, 
 
 sma 
 
 and therefore 
 
 A 
 
 sin a 
 
 sm (o) — aj 
 but equation {A) may be written 
 
 _ A G 
 
 and therefore the equation becomes for these data 
 
 siUa , , 
 
 ■U = -; -, r .X + 0, 
 
 " Sin (ft) — a) 
 which is often written. 
 
 y = mx + h, 
 the constant m being a short way of writing the constant 
 
 sin a 
 sin (o) — a) ' 
 
26 CONSTANTS OF THE GENERAL EQUATION 
 
 Cob. 1. If 6j = 90°, or the co-ordinates be rectangular, 
 
 m = tan a. 
 
 Also m will be positive only when a is less than w. In the 
 figure the -dotted line has m positive; in Z) Tit is negative. 
 
 Cor. 2. Every equation of the first degree may, by 
 dividing by the coeflSicient of y and transposing, be written 
 in the form 
 
 y = mx + h, 
 
 where, for rectangular axes, m represents tana, and for 
 
 oblique axes, m represents —. — ; — r , and where, for all 
 
 ^ ^ sm (tu — a) 
 
 axes, h represents the intercept of the line on the axis of y, 
 
 and will be positive or negative according as the line cuts 
 
 that axis above or below the origin. The constant m may 
 
 be called the angular coefficient of the line. 
 
 Cor. 3. If the distance OD = 0, or the line pass through 
 the origin, we shall have 
 
 -^=0 
 
 and the equation to the line will be 
 
 sin a 
 ^ sm (a» — a) 
 
 Hence the equation to any straight line passing through the 
 origin is of the form 
 
 y = mx^ 
 
 29. (IV) Next let the data be the point P' through 
 which DT passes, and the angle DTx 
 which it makes with Ox. Let the 
 known co-ordinates of P' be x' and y, 
 and let /.DTx=a, DOx = a>, as be- 
 fore. 
 
OF THE FIRST DEGBEE DETERMINED, 27 
 
 Then, since i^y") is a point on the line 
 
 therefore Aaa + By +0=0, 
 
 and A{x-x)+B{y— y) =0, 
 
 y — y A 
 
 or *' — ^ = ■ 
 
 x-id B' 
 
 but as before, 
 
 -i=OD. and --, = 0T; 
 
 B~^"' """ A 
 and therefore 
 
 A OJD sin a 
 
 £ or sin(Q)-a)' 
 
 hence ^= ■ f " , . 
 
 x—x sin (w — a) 
 
 X —x' y ~'y' 
 sin (w — a) sin a 
 
 Cor. 1, If a) = 90'', or the co-ordinates be rectangular, 
 sin (<a — a) = cos a, 
 
 and the equation may be written 
 
 {x-x) ^ y -y 
 cos a sin a ' 
 
 or more commonly 
 
 y-y' = m{x-x'), 
 where 7» = tana. 
 
 CoK. 2. If the given point be the origin, and therefore 
 x' = 0, y ^ 0, this equation reduces to 
 
 y = mx, 
 as in III. Cor. 3. 
 
'28 > • STRAIGHT LINES. 
 
 30. (V) Next let the data be the co-crrdinates of the 
 two points F aad P" through which the line passes. Let 
 the co-ordinates of P be x, y, and of P", x", y" ; then, since 
 P' and P" are points on the line, 
 
 Ax^-By + G=<^ (1), 
 
 therefore Ax-\-By'+G=^ (2), 
 
 and ^a;"+%"+C=0 (3); 
 
 from (1) - (2) A{x-d)-\-B{y- y') = 0, 
 from (3) - (2) A (x" -x')+B (y"-^ y')=0, 
 whence we have for the required equation 
 
 y-y ^ y -y ^ 
 
 x — d x" — a! ' 
 
 . CoE. If one of the points {oc'y") be the origin, the equa- 
 tion becomes, since x' = 0, y" = 0, 
 
 y-y' _y' 
 
 I — ' ) 
 33 — 0! X 
 
 or, after reduction, 
 
 i-yl. 
 
 — / 1 
 
 X X 
 
 which is therefore the equation to a line passing through the 
 origin and (xy'). 
 
 31. All these equations may be obtained independently 
 from geometrical considerations, instead of being deduced 
 from the general equation of the first degree ; thus 
 
 (I) The intercepts OD = h, OT=a being given. Let 
 PM{=y), OM{=x) be the co-ordinates 
 of any point P in the line ; then the 
 triangles DOT, PMT are always similar, 
 whatever be the position of P, and we 
 have ■ ~W ^ \ ^ 
 
 PM : MT= OB : OT, 
 
or 
 
 EQUATIONS .GEOMETRICALLY OBTAINED. 29 
 
 y b 
 
 whence, as before, 
 
 a — X a 
 
 - + f = 1. 
 
 This equation will be easily remembered, as each variable 
 stands over the intercept on its ' own axis. It is the same, 
 whether the co-ordinates are rectangular or oblique. 
 
 32. (II) Given the perpendicular (p) and the angle (a) 
 
 it makes with the axis of x. Let F 
 
 be any point in DT; OM, PM (cc, y) its 
 
 co-ordinates. Draw MS perpendicular 
 
 to OE; then 
 
 08 ,.. 
 
 cosa = -— (1), 
 
 arid cos (m — a) = cos 8RM 
 
 SB 
 ~EM' 
 
 RE 
 ~ BF' 
 
 SB + BE 
 
 and also 
 and therefore 
 
 ~ BM+BF' 
 
 ^8E 
 
 y 
 
 from (1) 08=x cos a ; from (2) 8E=y cos (w - a) ; 
 ,•. a; cos a 4- ?/ cos (tu — a) = 08 + 8E 
 
 CoE. If yOx\>Q& light angle, we have 
 
 08 . n-pyr SE 
 
 cos a. = , sm a = ccs biUu = — , 
 
 10 y 
 
 -(2); 
 
30 ' STRAIGHT LINES, 
 
 and we obtain the equation 
 
 X cos a + 1/ sma=p, 
 
 S3. (Ill) The length Oi>(=i) and the angle i)ra!( = a) 
 being given. 
 
 Let OM {=x) and PM{=y) be the co-ordinates of any 
 point P in the line ; through 
 D draw DQ parallel to Ox 
 to meet MP produced in Q; 
 then 
 
 PQ _Bm.PDQ 
 DQ sin BPQ' 
 
 QM-PM sin (180" -a) 
 BQ sin(a-o)) ' 
 
 or 
 
 or 
 
 sm a 
 
 sin (a> — a) ' 
 
 , sina . , 
 
 whence y = —. — -. r .x + b, 
 
 " sm(ei) — a) ' 
 
 or y = wa; + 6, 
 
 as before. 
 
 Cor, \iyOx be a right angle, we have 
 
 ^ = tanPi?Q 
 
 = tan (180° -a), 
 
 , . = — tan a, 
 
 X 
 
 y = tan a . a; + 5. 
 
 It will be observed, that, in the figure we have chosen, 
 m is a negative quantity, since sin (a> — a) and tan a are both 
 iiegative. If we take the dotted line, the geometrical con- 
 struction will shew that m is positive in that case. 
 
EQUATIONS GEOMETRICALLY OBTAINED. 31 
 
 34 (IV) The co-ordinates (a;', y') of the point P', and 
 the angle DTx (= a) being given. 
 
 Let P'be any point {xy) on the line, and draw F Q pa- 
 rallel to Ox, to meet the ordinate of P in Q\ 
 
 then |^-^^"^^'« 
 
 or 
 
 FQ sin QPP" 
 
 y-y ^ sin (180° -g) 
 x' — X sin (a — to) 
 
 hence -^ — S = -r—, r, 
 
 x—x sin (q) — a) 
 
 as before (Art. 29). This equation may be written 
 
 x—x 
 
 _ y-y 
 
 sin (a> — a) sin a ' 
 
 FP 
 •where each member of the equation is evidently = —. — ; 
 
 hence we may write the equation thus, 
 !C — x'_y—y' 
 c 8 
 
 ^here « = ^, o = ^i^l^^, l^FP, 
 
 smo) sm tB 
 
 and where we must remember, that s, c are constants for any 
 given line, but Z is a variable quantity, and equals the length 
 of the straight line between the points (jvy) and {x'y'). 
 
 Cob. If the axes are rectangular, 
 
 x' — X _ y—y 
 mnr(a-90") ~ sin(180»-a) ' 
 
 and each member of the equation is equal to FP] hence we 
 may write the equation 
 
 x — x' _ y — i/ _ J 
 c ~ s ~ ' 
 where a = sin «, c = cos a, I =» FP. 
 
32 STRAIGHT LINES, 
 
 The geometrical meaning of this equation should be care- 
 fully noted, as we shall frequently have occasion to use it 
 hereafter. The formula will evidently make I positive or 
 negative, according as the distance is measured from P 
 towards D or T; and, as sin a and sin <u are always positive, 
 the sign of I is always the same as that oiy —y. 
 
 It is also useful to obsei-ve, that, when the co-ordinates 
 are rectangular, this equation connects the rectangular and 
 polar co-ordinates of any point {xy^, if we consider (aj'y') the 
 pole, and the initial line to be drawn through (aj'y) parallel 
 to the axis of x \ for we have 
 
 y=y' -\rsl, a; = a;'+cl; 
 
 or, with the ordinary notation of polar co-ordinates, 
 
 y=y' + psin0, x=x' + pcosd. 
 
 35. (V) The co-ordinates of two points P' {x'y) and 
 F" {x"y") being given. 
 
 Let P be any point (xy) on the line ; and draw P' Q" Q 
 parallel to Ox, to meet the ordij 
 nates of P and P" in Q and Q"; 
 then 
 
 PQ_P^^ 
 ■ Qp-Q'P" 
 
 or 
 
 t It I 
 y-y ^y_iil. 
 oc —x X — a;" 
 
 or 
 
 y-y' ^ y -y 
 
 x—x x' — x' 
 as before. 
 
 This equation may be written in the form 
 y-y' = m{x-s:), ■ 
 
NUMKRIGAL EXAMPLES. 83 
 
 where ■m=—,-, — ^ 
 
 05—05 
 
 = -: — 7 c , if the axes be oblique, 
 
 sm (w — a) ^ ' 
 
 = tan a, if the axes be rectangular, 
 
 which agrees with the result of (IV). 
 
 Cor. Since m =■ —. — -. r , we have 
 
 sm (o) — a) 
 
 m (sin Q) cos a — cos a sin a) = sin a ; 
 
 therefore m (sin m — cos w tan a) = tan a ; 
 
 m sin ft) 
 
 whence tan a = 
 
 \-\- m cos ct) ' 
 
 36. We will now give a few numerical examples on the 
 preceding articles. 
 
 Ex. 1. To find the equation to a straight line ■which cuts off intercepts on 
 the axes of x and y_ equal to 3 and - 6 respectively. 
 Writing a= 3, 6 = - 5 in the equation ol (I) , 
 
 - + '^- = 1 
 
 ■we have, for the required equation, 
 
 3 6 
 
 Ex. 2. The perpendicular from the origin on a straight line =5, and 
 makes an angle of 30" iirith the axis of x; find the equation to the hne, (i) 
 ■when the axes are inclined at an angle of 60°, and (ii) ■when they are rectan- 
 gular. 
 
 "Writing u=60'', 0=30", andf)=5 in the equation of (II), 
 X cos a+y cos (w - a) =p, 
 we have 
 
 V3(ai+y) = 10, 
 
 for the equation when the axes are inclined at an angle of 60°; and it thff 
 axes are rectangular, 
 
 seooso + y sina=iJ, 
 
34 STRAIGHT LINES. 
 
 or J^x+y=10, 
 
 is the equation reguired. 
 
 Ex. 3. To find the equation to a straight line which makes an angle of 
 135° -with the axis of x, and outs off an intercept = - 8 on the axis of y, (i) if 
 the co-ordinates are rectangular, and (ii) if they are inclined at an angle 
 of 45». 
 
 (i) Writing i= -3, and m=tan 135°= - 1 in the equation of (III), 
 
 we have, for the required equation, 
 
 y=-x-Z. 
 (ii) Writing a =135°, w-a=45»- 135°= -90° in the equation of (III), 
 sino 
 
 ^=SE(^r:^-»'+*' 
 
 we have, for the required equation, 
 
 1 
 
 or y=~^.x-%. 
 
 Ex. 4. To find the equation to a straight line which passes through the 
 points whose co-ordinates are 
 
 a;=4, y=-2, Bnda;=-3, y= -5. 
 
 Writing 4 and - 2 for x' and y', and - 3 and - 5 for «" and y" in the 
 equation of (V) 
 
 x-x! ai'-x'^ 
 we have, for the required equation, 
 
 y+2_ -5H-a 
 
 x-i -3-4' 
 or 7sf-3ic-h26=0. 
 
 , Ex. 5. To find the equation to a straight line which passes through thr 
 origin and a point whose co-ordinates are 
 
 a;=B, y=-2. 
 
 Writing 3 and - 2 for st' and y' in the equation of (V) Cor. 
 
 X x" 
 VS have 
 
 3y+2!c=0, 
 for the equation required. 
 
/ 
 
 1 
 
 N 
 
 3 
 
 
 , 
 
 
 Q 
 
 s, 
 
 \ 
 
 D 
 
 
 Q' 
 
 
 \^ , 
 
 M 
 
 
 
 
 rx 
 
 M' 
 
 
 
 
 
 
 ■ NEGATIVE dO-OEBINATES. 35 
 
 37. We have hitherto considered those points in the 
 line only, whose co-ordinates are positive ; but it may be seen, 
 in every case, by noticing the direction, in which the co- 
 ordinates are drawn, that the equation is satisfied by every 
 point in the line, if produced indefinitely both ways. We 
 will take the form of equation given in (III), with rectangular 
 co-ordinates, and suppose 
 the equation to DT to be 
 
 y = moo + h, 
 
 where h, as we have shewn 
 above, is a positive quan- 
 tity and = OD, and m is, 
 for this line, a negative 
 quantity, and = tan a. 
 
 Hence, for the part between D and T, y is equal to the 
 difference of the absolute lengths represented by h and mx, 
 as was shewn in Art. 33 ; but, if we take P on the other side 
 of the axis of y, so that x is negative, the quantity mx 
 becomes positive, and y is now equal to the sum of the 
 absolute lengths represented by mx and h. We may easily 
 see from geometrical considerations that this is the case; for 
 P ^ is the quantity which now represents the absolute length 
 of the quantity mx or tana.D^ (the signs being disre- 
 garded) and QM— h ; therefore PM, or y, which = the sum 
 of P § and QM, = the sum of mx and h. 
 
 Again, if we take P' on the other side of the axis of x, 
 the ordinate will, as in the portion DT, represent the differ- 
 ence of the absolute lengths of h and mx, since x is still 
 positive and m is negative; and here M'Q' = h, and F ^'=the 
 absolute length of mx or of tan a . DQ, and we see, geometri- 
 cally, that the ordinate FM' is the difference of these lines : 
 also it is drawn upwards or in the negative direction, as it 
 evidently shoiild be, since of the two quantities h and mx, 
 
36 
 
 TEACING STRAIGHT LINES. 
 
 whose difference it is, the negative quantity, mx or P' Q', is 
 the greater, 
 
 38. In order to trace the straight line represented by any 
 equation of the first degree, it is only necessary to determine 
 two points, through which the line passes ; the line joining 
 these two points, and produced indefinitely both ways, will be 
 the locus of the equation. The following examples will shew 
 how this may be done most conveniently, (I) "when the line 
 does not pass through the origin, and (II) when the line does 
 pass through the origin. 
 
 Ex. (I) To trace the Unas 
 
 (i) 5sr + 3x+15 = 0, 
 (ii) 23f-5a:+10=0. 
 
 In (i), when a;=0, y= —3, or the line 
 cuts the axis of ^ at a distance OD (= 3) 
 below the origin; when 2^=0, x=—Z, or 
 the line cuts the axis of a; at a distance 
 Or(=5) to the left of the origin. Hence 
 the line BT, prodnoed indefinitely, is the 
 locus of equation (i). 
 
 In (ii), when a;=0, y= -5, and when 
 j/=0, a;=2; 
 hence, if we tale 0Z»'=5, and 0J"=2, 
 D'T' produced indefinitely will be the locus of equation (ii). 
 
 £x. (II) To trace the lines 
 
 (i) 3ai+4y=0, 
 (ii) 2a;-3y=0. 
 
 These lines pass through the origin, since the equations are satisfied by 
 the values a!=0, ^=0. To find another 
 point, we may take any value of one vari- 
 able, and find the corresponding value of 
 the other. Thus in (i) take 0M= - 4, 
 then PM=Z\ hence P is a point in the 
 line, and PO produced indefinitely is the 
 locus of equation (i). 
 
 Again, in (ii) take ,0Jf'=3, then 
 
 hence P' is a point in the line, and P'O produced indefinitely is the locus 
 required. 
 
INTERSECTlOSr OF STRAIGHT LINES. 87 
 
 39. Hie equations to two straight lines being given, to 
 find the co-ordinates of their point of intersection. 
 
 Since the co-ordinates of every point on a line must 
 satisfy its equation, the co-ordinates of the point through 
 which both the lines pass will satisfy both equations, or they 
 will be the values obtained by solving the equations as 
 simultaneous. 
 
 Ex. To find tlie co-ordinates of the point where the two lines, 
 
 3?/ + 4a!-ll = 0, 4y + 3a;-10=0, 
 intersect. 
 
 The values of x and y, obtained by combining these two equations, are 
 x=2, y=l, which are, therefore, the co-ordinates of the point of intersection 
 of the lines. 
 
 40. We remarked (Art. 24) that the variables x and y 
 had not the same meaning in two different equations. If, 
 however, we combine these equations in any manner what- 
 soever, we tacitly introduce the condition, that the values of 
 the variables are the same in each. All equations so obtained 
 are therefore satisfied by the co-ordinates of the points which 
 the loci have in common. It does not follow, that every point 
 in the locus of an equation so obtained is one of the common 
 points of the original loci. We may combine in such a 
 manner as absolutely tp determine these co-ordinates, or we 
 may obtain some relation between them, expressed by an 
 equation, and, as this equation must represent some line, 
 straight or curved, and is satisfied by the co-ordinates of the 
 point or' points 6i intersection, it represents a line passing 
 through those points. 
 
 We will take two simple examples of this, 
 (i) The values x = %y = l, which satisfy the equations 
 Art. 39. Ex, 
 
 8^ + 4:i;- 11 = 0, 4y-l-3jJ- 10 = (1), 
 
38 INTERSECTION OF 
 
 will evidently satisfy the equation 
 
 3?/ + 4*- 11 +43/ + 3a3- 10 = 0, 
 or y + a; - 3 = 0, 
 
 which is therefore the equation to a straight line, passing 
 through the intersection of the two lines represented by 
 equations (1). 
 
 (ii) Let the equations to two straight lines referred to 
 
 rectangular axes be 
 
 1 
 y = mx + h (1), ?/ = ---« + &' (2), 
 
 "lit 
 
 The geometrical meaning of these equations is (Art. 33) 
 that the lines cut off intercepts on the axis of y equal to h 
 and y, and tliat they make angles 6, ff with the axis of x, so 
 
 related that 
 
 tan 0. tan 6' = - 1 (3). 
 
 Now, if 9 and ff be quantities whose value is absolutely 
 known, we may treat (1) and (2) as simultaneous equations, 
 and obtain definite values for x and y, the co-ordinates of the 
 intersection of (1) and (2), in terms of the constants 6, ff, 
 h, y. But, if all we know about 6 and ff is that they are 
 connected by equation (3), it is evident that for every value 
 of 9 there will be a corresponding value of ff, and that each 
 pair will produce a new point of intersection. Suppose then 
 that we combine (1) and (2), so as to eliminate this variable 
 quantity m, (which = tan 9, by Art. 33,) instead of one of the 
 variables x or y. We obtain the equation 
 
 {y-V){y-V)+x'=0 (4). 
 
 Equation (4) is a relation between the co-ordinates of the 
 intersection of (1) and (2) ; and, as it represents a line of some 
 kind, that line must pass through the intersection. But, 
 whatever value 9 and ff have in (1) and (2), we shall, by the 
 same means, obtain equation (4), provided only that the 
 relation (3) exist between them. Hence, (4) passes through 
 
STRAIGHT LINES. 39 
 
 eveiy -poini of intersection -wMch can be generated by the 
 change of 6, and is called the locus of the intersection of (1) 
 and (2). 
 
 41. We may here remark that the student cannot he too 
 careful in considering the. geometrical meaning of ever?/ 
 algebraic step he takes. The very facility with which alge- 
 braic expressions are combined and manipulated is often the 
 most serious drawback to his progress. 
 
 42. In finding the co-ordinates of the points where two 
 loci intersect, we shall sometimes fall upon impossible, and 
 sometimes upon infinite values of the variables. We gather 
 from the former, that the loci do not meet there ; and the 
 latter will often give us some information about the geome- 
 trical position of the loci. For instance, from the equations 
 
 y = mx+b (1), y =m'a; + b' (2), 
 
 we obtain, for the abscissa of the intersection, 
 
 h'-b 
 
 x = ,. 
 
 m — m 
 
 If we suppose h and b' to remain the same, while m' ap- 
 proaches indefinitely near to to in value, the value of x 
 becomes indefinitely great, and we infer from this that the 
 lines become parallel, which is evidently true by Ai't. 28. It 
 is in accordance then with the algebraic result, to say that 
 parallel straight lines meet at infinite/; but it must be clearly 
 understood what this means. 
 
 Suppose DT and I/T 
 to be the two lines, and 
 draw D'L parallel to DT. 
 Then, if we suppose OB' 
 (=&') to remain the same, 
 and the point of intersec- 
 tion F to move along BT 
 to an infinite distance, the 
 
40 INTERSECTION OF 
 
 line B' T tends to coincide with D'L as its limiting position.. 
 The equation of the second degree will furnish similar exam- 
 ples, where it is very important to bear in mind this expla- 
 nation. 
 
 43. If the equations to two straight lines be 
 
 Ax + By+G=Q (1), 
 
 Ax+B'y+G'=0 (2), 
 
 the equation 
 
 {Ais + By+G)+k{A'x + B'y+C') = (3), 
 
 where h is any constant quantity, positive or negative, will 
 represent a straight line passing through the point where (1) 
 and (2) intersect; for it is an equation of the first degree, 
 and therefore represents a straight line ; and the co-ordinates 
 of the point of intersection of (1) and (2) satisfy both those 
 equations, and therefore evidently satisfy (3) ; hence the point 
 is on the line represented by (3). 
 
 Since h is an arbitrary quantity, undetermined by the 
 nature of the problem, equation (.3) will represent a system of 
 lines fulfilling one condition only, viz. that of passing through 
 the intersection of (1) and (2). Any individual of the system, 
 particularized by the fulfilment of a second condition, can 
 have the appropriate equation assigned to it, by giving the 
 proper value to k. For example, if any line of the system 
 passed through a point {x'y'), we should obtain the value of h 
 for that line from the equation 
 
 {Aai + By'+C)+k (A'x' + B'y' + C) = 0, 
 
 and, substituting this value in (3), we should have the equa- 
 tion required. 
 
 Also, generally, if the equations to two curves be 
 8 = 0, 8' = 0, 
 
STRAIGHT LINES. 41 
 
 where 8 and 8' stand for expressions involving so and y, the 
 equation 
 
 8 + JcS' = 
 
 will represent a locus passing through the points of intersec- 
 tion of the two first-mentioned curves. 
 
 Hence, if any equation involve an arbitrary constant h, in 
 such a manner that the equation may be written in the form 
 8+k8' —0, that equation will represent a series of loci 
 which pass through the points of intersection of 
 
 ^ = 0, 5" = 0. 
 
 Ex. 1. The equation 
 
 or (3+4i)2/ + (4 + 3i)K-ll-10i=0, 
 
 will, according as we give different values to Jc, represent a series of straight 
 lines passing through the point (a; =2, y=l), which is the point of intersec- 
 tion of the two lines represented by 
 
 3y + ix-ll=0, %-)-3a!-10 = 0. 
 
 Ex. 2. The equation y=mic + h may be written 
 (y- J) -mx=0, 
 
 and, if m is indeterminate, this will represent a series of straight lines passing 
 through the intersection of the Unes 
 
 y — h=0, x=0, 
 
 or cutting off an intercept =5 from the positive part of the axis of y, as we 
 have already seen (Art. 28). 
 
 Ex. 3. The equation 
 
 y-/-m(a;-a!')=0 
 
 represents, if m is indeterminate, a system of straight lines passing through 
 the intersection of the Unes 
 
 y-y'=0, x-x'=^Q, 
 
 as we saw in Art. 29, 
 
42 -POLAU EQUATION TO 
 
 44. To find the general equation to the straight line, 
 referred to polar co-ordinates. 
 
 Writing x = pcosd, y = psVD.0 (Art. 13), in the equation 
 Ax + Bi/+ C = 0, 
 we liave 
 
 Ap cos6 + Bpshix6+<J=0 
 
 for the general polar equation. 
 
 45. It is easily seen, that, as in the case of rectangular 
 and oblique co-ordinates, the form 
 of the general polar equation to the 
 straight line will vary according to 
 the data which determine the po- 
 sition of any particular line. The 
 following data present the equation 
 under its most convenient form. 
 
 Let DThe the line, and let the perpendicular OE{=p) 
 from the pole, and the angle EOT [= a) which it makes with 
 the initial line, be the data to determine its position. Now if 
 
 Ap cos e + Bp sin 0+ C=0 (1), 
 
 be the equation to JDT, we have, since E{pa) is a point on 
 the line, 
 
 Ap cos a + B^ sin a. + G = (2). 
 
 From (1) and (2) 
 
 p sin 6 ■ 
 
 ■psma 
 
 p cos 6 —p cos a 
 
 A 
 ' B' 
 
 .(3). 
 
 G 
 
 But from (1), when 6 = 0, p = —-j, the point T, 
 and when 0=^90°, p = -j^ B; 
 
THE STEAIGHT LINE. 43 
 
 tence -^= OD, ~^=0T, 
 
 B 'A 
 
 , A 01) 
 
 and ■g = ^=cota; 
 
 and therefore, substituting in (3), 
 
 psin^. sin a— ^Jsin^ a = — pcos 0. cos a+p cos' a; 
 therefore p cos (^ - a) = p, 
 
 or p=^sec(^ — a); 
 
 the polar equation for the above data. 
 
 This equation may be deduced at once from the equation 
 to the straight line in the form 
 
 cc cos a + ?/ sin a =^p, 
 
 by writing p cos 9 for x, and p sin 9 for y, the data being 
 exactly the same in the two cases. 
 
 The same equation may be obtained very simply from 
 geometrical considerations ; for, if P be any point (fi&) in the 
 line, we have 
 
 OP =01!. sec POE, 
 
 or p=psec(9 — a). 
 
 If a = 0, or OE be taken for initial line, the equation is 
 
 p = j3 sec 9. 
 
 46. We will collect here, for the sake of reference, the 
 different forms of the equation to the straight line, which we 
 have investigated. 
 
 - + 1 = 1. Arts. 26 and 31. 
 a 
 
 xooB<, + y cos (« - a) =p1 ^^^ ^7 and 32. 
 X cos 0. + y sm a =p) 
 
u 
 
 EXAMPLES II. 
 
 Arts. 28 and 33. 
 
 y = mx + b 
 y = mx 
 
 c s j 
 
 {y-y')=m[!C-x)\ 
 
 f ft . f 
 
 x — x x' — X 
 
 y=yl. 
 
 X x' 
 
 Apcose + Bpii\ne-\- <7=01 
 p =p sec {9 - a) J 
 
 = constant 
 
 Arts. 29 and 34. 
 
 Arts. 30 and 35. 
 
 Art. 4.5. 
 Art. 19, iii. 
 
 EXAMPLES II. 
 
 1. Deaw the lines whose equations are 
 
 (I) y = 5x+'2, (2) 2/-7 = 5a!+3, (3) 7y-3a! = 0, 
 
 (6) 2a! + 3 = 0. 
 
 (4) 6-a: = 2y, (5) |+^ = 2, 
 
 2. Find the equation to the straight line which passes through 
 the points (2, 5) and (0, - 7). 
 
 3. The co-ordinates of the angular poiuts of a triangle being 
 given, find the equations to the three straight lines, each of which 
 bisects two of the sides. 
 
 4. Two straight lines make each of them an angle of 45° 
 with the axis of x, and their intercepts on the axis of y are 6 and 
 8 j find the equation to the straight line which is equidistant 
 from the two, the axes being rectangular. 
 
 5. Find the equation to a straight line on which the perpen- 
 dicular from the origin = 6, and makes (1) an angle of 45°, and 
 
EXAMPLES II, 45 
 
 (2) an angle of 225° with the axis of x, the axes being rect- 
 angular, 
 
 6. Determine the point of intersection of the two lines 
 (3y-a; = 0) and (2iB + y = l). 
 
 7. Find the equation to the straight line which passes through 
 the point of intersection of the straight lines 
 
 a!-2y-a = 0, x+iy-2a=0, 
 and is parallel to the line Zx + 4:y = 0. 
 
 8. Find the equation to a straight line which is equidistant 
 from the t-yiro lines represented by the equation y = TOa;.+ c±c'. 
 
 9. Find the equation to the straight line that joins the 
 points of intersection of the two pairs of lines 
 
 2x + Zy-4:a = 0, 1 a;+6y-7a = 0, V 
 
 2x + y-a = Q, / Zx-2y + 2a=Q.] 
 
 10. Find the length of the perpendicular from the origin on 
 the line a {x — a) +b {y —h) = Q, and the portion intercepted by 
 the axes, which are rectangular. 
 
 11. The rectangular co-ordinates of two points are 3, and 
 4, 4 respectively; find the equation to a, sti-aight line which 
 bisects the distance between them, and makes an angle of 45° with 
 the axis of x. 
 
 12. Find the equation to a straight line which passes through 
 a given point (ah), and makes equal angles with the axes. 
 
 13. Find the equations to the diagonals of the parallelogram 
 formed by the four lines x-a,x = a', y = h,y = h'. 
 
 14. A straight line, inclined to the axis of x at an angle of 
 150°, cuts the positive axes of rectangular co-ordinates in A and , 
 B; find the equation to a straight Hue bisecting AB and passing 
 through the origin. 
 
 15. Find the equations to the four sides of a square, the co- 
 ordinates of two of its opposite angular points being (2, 3) and 
 (3, 4), the co-ordinates being rectangular. 
 
46 EXAMPLES II. 
 
 16. Fiad tlie distance of the origin of co-ordinates from the 
 line jy + I = 1, the axes being rectangular. 
 
 17. Find the equation to a straight line which passes through 
 the intersection of the lines x = a, x + y + a=0, and through the 
 origin. 
 
 18. The axes of co-ordinates being inclined to each other at 
 an angle of 60°, find the equation to a straight line parallel to the 
 line (x + 2/ = 3as), and a distance from it equal to \aj'5. 
 
 1 9. Shew that the lines y = 2x+Z, y=--Zx + 4:, y = ix+ 5, 
 all pass through one point. 
 
 20. Find the value of m, in order that the line (y = mx + 3) 
 may pass through the intersection of the lines (y = x + l) and 
 {y=^2x+2). 
 
 21. A straight line cuts off intercepts on the axes, the sum 
 of the reciprocals of which is a constant quantity ; shew that all 
 straight lines which fulfil this condition pass through a fixed 
 point. 
 
 22. A straight line slides along axes of x and y, and the dif- 
 ference of the intercepts is always proportional to the area it 
 encloses; shew that the line always passes through a fixed point. 
 
 23. If the distance of a point from the origin = twice its 
 distance from the axis of x, shew that it always lies in one of 
 two straight lines that pass through the origin ; axes rectangular. 
 
 24. Find the cosine of the angle which the line (Ax +By + C) 
 makes with the axis of x, the axes being inclined at an angle 
 of 45°. 
 
 25. If a straight line cuts the (rectangular) axes of x and y 
 at equal distances from the origin, and a straight line be drawn 
 from the origin, dividing it in the ratio m : n, find the tangent of 
 the angle which this latter line makes with the axis of x. 
 
EXAMPLES II. 47 
 
 26. All equilateral triangle, whose side = a, has its vertex at 
 the origin, and its sides equally inclined to the positive directions 
 of rectangular axes ; find the co-ordinates of the angles, and thence 
 of the point bisecting the base. 
 
 27. Shew that the three straight lines which bisect the 
 angles of an equilateral triangle meet in a point, taking two of 
 the sides as axes, 
 
 28. The base of a triangle and the straight line joining the 
 bisection of the base with the vertex being axes, form the equa- 
 tions to the straight lines which join the bisection of the other 
 sides with the opposite angles, and find their co-ordinates of inter- 
 section. 
 
 29. Find the polar co-ordinates of the point of intersection 
 of the lines | p = 2a sec T^ - | U and i p = a sec (O -^]\ , and 
 the angle between them. 
 
 30. Trace the line whose polar equation is 
 
 p=2flscos(^-i-^j. 
 
 31. Shew that the polar" equation to a straight line, passing 
 through the points (pff), {p"ff'), is 
 
 p'p sin {ff -6)+ p"p' sin {&' - 6') + pp" sin {6 - 6") = 0. 
 
 What is the geometrical interpretation of this equation ? 
 
 , 32. Shew that the three straight lines 
 Ax + By+G=Q, A'x + B'y + 0' = 0, A"x + B"y+C" = (i, 
 will meet in a point, if 
 
 A {FG" - ff'C) + B {C'4" - G"A) + C {A'B" - A'B") = 0. 
 
CHAPTER III. 
 
 The straight Line continued. — Angles. — Perpendiculars. — 
 Equations representing straight lines. 
 
 47. To find the angle between two straight lines whose 
 equations are given, the axes being supposed rectangular. 
 
 Let ta,uTPT' = t,a.nd let the equations to DT, D'T' be 
 y == mx + b, 7/ = m'x + h', 
 since any equation may be written in 
 this form ; then (Art. 33) 
 
 m = tan DTx, m' = tan 1/ T'x, ^^■ 
 and tan TPT = tan {DTx - D'Tx), 
 m — rd 
 
 or 
 
 < = . 
 
 1 + m/ni ' 
 which determines the angle TPT. 
 
 CoE. 1. Hence we see that, ii m = m', tan TPT' = 0, or 
 
 the lines are paralleL Also, if m' = , tan TPT' = so , or 
 
 the angle TPT' is a right angle. 
 
 The condition then that the two lines 
 
 y = mx +h, y = m'x + V 
 
 should be parallel, is 
 
 m = 'm, 
 
 and that they should be perpendicular to each other, is 
 
 m — — 
 
 m 
 
ANGLE BETWEEN STRAIGHT LINES. 49 
 
 Cob. ^. If the two lines are 
 
 Ax +By +0=0, Ax + B'y + C = 0, 
 
 the conditions of parallelism and perpendicularity are re- 
 spectively 
 
 A^A_ A^_^ 
 
 B B" B A'' 
 
 hence two straight lines are perpendicular, if the coefficients of 
 X and y are interchanged, and the sign of one of them changed. 
 
 Ex. To find the angle between the lines 
 
 2y + x+l=0, 3y-x-l=6, 
 the axes being rectangular. 
 
 Here m= -^, m'=^, and, if olie the angle between the lines, 
 
 tana= - ^ -==-1, 
 or a=r35». 
 
 48. The condition of parallelism, deduced in the last 
 article, is evident independently, and we have, in fact, as- 
 sumed it in previous articles; for m and m' are the tangents 
 of the angles which the lines make with the axis of x, and 
 those angles are equal, if the lines be parallel. The condition 
 of perpendicularity may be proved independently; for, if we 
 cohsider TFI" a right angle, we have 
 
 m' = tan ]yTx = - cot BTx 
 
 1 
 
 m* 
 
 *49. To find the angle between two straight lines, the axes 
 being oblique. 
 
 Let the lines be as before 
 
 • 'y = tnx + i, y^nix + V; 
 
 i 
 
50 ANGLE BETWEEN STBAIGHT LINES. 
 
 making angles a and a' witli the axis of x; and let the angle 
 between the axes be a. Then (Art. 35, Cor.) 
 
 m sin a> , , m' sin « 
 
 tan a = =— , tan a = : 
 
 ' 1 + m cos 6) ' 1 + m' cos ft) ' 
 
 hence, if /3 be the angle between the lines, 
 tan ^ = tan (a — a") 
 
 TO sin ft) to' sin oj 
 
 1 + TO cos ft) 1 + m' cos as 
 
 mm sm 6) 
 J J. 
 
 (1 + TO cos ft)) (1 + m' cos ft)) 
 
 (to — to') sin ft) 
 1 + (to + to') cos ft) + jhto' " 
 
 Hence the condition of parallelism is as before to =to'; and 
 the condition of perpendicularity is 
 
 1 + (to + to') cos ft) + toto' = 0. 
 
 The condition of perpendicularity is much more complicated, 
 when the axes are oblique, than when they are rectangular, 
 as, indeed, are all formulae in which angles are involved. 
 For this reason we shall choose rectangular axes in the 
 solution of questions concerning angles, and shall not extend 
 the remaining articles of this chapter to the case of oblique 
 co-ordinates. 
 
 50. To find the equation to a straight line, which shall 
 make a given angle with a given straight line (y = mx + b), 
 the axes being rectangular. 
 
 Let a be the given angle, and let tana = <, and let the 
 
ANGLE BETWEEN STEAIGHT LINES. 
 
 equation to the given line (DT) be 
 y = mx + b (1), 
 
 and the equation of the required line 
 
 y = m'x + b' (2), 
 
 where m' is to be found by the con- 
 ditions of the problem. 
 
 Then, since the required line may 
 lie either asPQor PR, we shall have, by Art. 47, 
 
 51 
 
 m —m 
 1 + mm! ' 
 
 or = 
 
 m—m 
 1 + mm! ' 
 
 Hence 
 
 m =: 
 
 m + t 
 
 l + rat' 
 
 and the required equation is 
 
 m + t 
 
 2/ = 
 
 1 +mt 
 
 X 
 
 + h' 
 
 (3), 
 
 where V still remains undetermined, as an infinite number of 
 straight lines may be drawn fulfilling this condition. 
 
 If we add another condition, that the line should pass 
 through a point F(xy), the equation will be (Art. 29, Cor. 1) 
 
 y — y'= id {x — a;') 
 
 9M + < 
 
 1 +mt 
 
 {x -^ x) 
 
 (4), 
 
 and it may be seen from the figure, that there are, generally, 
 two straight lines fulfilling these conditions. 
 
 4-» 
 
62 AKGLE BETWEEN STRAIGHT LINES. 
 
 Cob. 1. If < = 0, or the problem be to find the equation 
 to a straight line which passes through a given point, and is 
 parallel to a given straight line, the equation is 
 
 2^-y = m(a!-a;'.). 
 
 Cor. 2. If * = oo , or the problem be to find the equation 
 to a straight line which passes through . a given point, and is 
 perpendicular to a given line, the equation is 
 
 for, if t becomes = oo , we have 
 
 m + t t - 1 
 
 1+VU 1 _ TO 
 
 Ex. To find the equation to the straight lines which pass through the 
 point (1, 2), and make an angle of 45" with the line 
 
 3a:+43^ + 7=0. 
 
 3 
 
 Here, m= - j , t=»l; hence, according as we take upper or lower signs in 
 
 equation (4), we have 
 
 7y-9;-13=0, y + 7a;-9=0, 
 for the required equations. 
 
 51. The equation obtained in Cor. 1, is the same 
 as that of Art. 34, where its geometrical meaning is ex- 
 plained. 
 
 The geometrical meaning of the equation of Cor. 2, may 
 be seen thus : 
 
 Let DThe the given line (2/ = mx + l), F(x'^') the given 
 
PERPENDICULAR DISTANCE OF A POINT FROM A LINE. 33 
 
 point, through which the line 
 lyT is drawn perpendicular 
 to DT, and let P be any other 
 point (xy) on D'T'; then, if 
 P'B be drawn parallel to Ox, 
 to meet the ordinate PM in 
 B, we have 
 
 ~ = tan PT'x = - cot DTx, 
 jr a 
 
 or, since tan I>Tx = m, 
 
 x — x'' 
 
 1^ 
 m' 
 
 52. To find the length of a straight line drawn from a 
 given point (x'y) to meet the line {Ax + By+ = 0). 
 
 The equation to a straight line through {x'y) is (Art. 34) 
 
 ^^^^y^^i (1), 
 
 c s 
 
 where I is the distance of any point [xy) from the fixed point 
 {oiy'). If we substitute for x and y from (1) in the equation 
 to the given line, the resulting equation will refer to the 
 point where the two lines intersect, and we shall have for the 
 distance (Z) from (»y) to that point, the equation 
 
 ^ (c? + .-r') ■\-B{sl^- y') +G=Q, 
 
 {Ac + Bs)l + Ax + By + C= 0, 
 
 or 
 
 ■. ?=- 
 
 Ax' + By'+G 
 Ac + Bs 
 
 .(2). 
 
 Cor. If the axes are rectangular, and the line (1) is 
 perpendicular to (Ax + By+ C=0), we have (Art. 47) 
 
 • ! = ? (3). 
 
54 PERPENDICULAR DISTANCE OF A POINT FROM A LINE. 
 
 since the two lines make with the axis of x angles whose 
 tangents are, respectively, equal to 
 
 s J A 
 - and — ^. 
 c Is 
 
 But, since s and c are the sine and cosine of the angle 
 which the line (1) makes with the axis of x, 
 
 from which and from (3) we have 
 
 . A ^ B 
 
 WA' + B'' -^/A'JrB'' 
 
 hence from (2) we have for the perpendicular, 
 
 Ad + By'+G 
 
 - 'Ja'+b^ 
 
 (4), 
 
 according as we take c and s with the lower or upper signs. 
 
 As we are now considering the magnitude only of the 
 perpendicular, the algebraic sign with which it is affected is 
 immaterial. We shall see hereafter, that it is sometimes 
 necessary to select the appropriate sign for I; but for this 
 purpose we shall use a geometrical construction, and the 
 equation to the line in a less general form. 
 
 CoE. 1. Since Ax' + By' -^^ O is the only part in (4) 
 that varies with the position of {x'y), it follows that the 
 Ax + By+ C oi any point (xy) varies as the distance of the 
 point from the line Ax + By + G = 0. 
 
 Cor. 2. If the point {x'y') be the origin, x' = 0, y = 0, 
 and we have, for the distance of the line from the origin, 
 
 1 = —^— 
 
AREA OF TRIANGLE. 55 
 
 Ex. To find the length of the perpendicular from the point {x=3, 3/ =5) 
 fon the line (3y- 7a+ 9=0). 
 
 , 3x5-7x3+9 3 
 
 53 \ To find the area of a triangle in terms of the co- 
 ordinates of the angular points, the axes being rectangular. 
 
 Let the three points be {xy), {^y), {x'y"). The 
 equation to the straight line joining {x'y) and {x'y") is 
 (Ajt. 35) 
 
 (2/ -y) {x" - x') -{x~ x') {3,"-y') = 0, 
 
 ?ind the perpendicular upon this from the point {xy) is 
 (Art. 52) 
 
 ■ (y - y) jx" - A ~i^- x') iy" - y) . 
 
 and the distance between (a;'y) and {x'y") is the denominator 
 of this fraction. But this perpendicular multiplied by this 
 distance is double of the area of the triangle. Hence 
 
 ±\{(^-y) (:^"-A-{x-x) {y"-y')\ 
 
 is the area of the triangle, the upper or lower sign being 
 used, according as the expression is positive or negative. 
 
 If the three points {xy), {x'y), {x'y") are in the same 
 straight line, the area becomes = 0, which gives, as it ought, 
 the equation of Art. 35. This gives an easy way of remem- 
 bering the formula. 
 
 For the case of oblique axes see Chap. IV. Ex. 8. 
 
 54. To find the length of the perpendicular from a 
 point (x'y') on the line (a; cos a +^ sin a =p), the axes being 
 rectangular. 
 
 ^ From Salmon's Conic Sectiont. 
 
56 PERPENDICULAR DISTANCE OF A POINT FROM A LINE. 
 
 Let Z)rbe the line 
 
 X cos a + y sin a=p, 
 
 and P the point (xy). Draw a 
 line R8 through P parallel to BT, 
 and draw PQ, OJ/S" perpendicular 
 to the lines i)Tand US; then the 
 equation to B8 must be 
 
 X cos a + ^ sin a = OH 
 
 =p suppose ; 
 
 but since («'/) is a point in P8, we have 
 ic' cos a + 3/' sin a =y . 
 Tiio-yj PQ = HE =p'-p, 
 
 .: PQ = x cos a. +y' sin a—p, 
 and is known. 
 
 If the point {x'y) be on the other side of the line, as F, 
 the length of the perpendicular P'^ will evidently be =p—p', 
 or =^ — a;' cos a— y sin a. Hence, if the equa.tion to a line 
 be a cos a + ^ sin a =p, where ^ is a positive quantity, the 
 length of the perpendicular from (xy')is 
 
 ± (x cos a + y' sin a — p), 
 
 the lower sign being used, when, {x'y) is on the origin side of 
 the line. 
 
 55. Thus we see that the a;cosa + 2/sina— j) of any 
 point {xy) is negative or positive, according as (xy) is or is 
 not on the origin side of the line, and vanishes when {xy) is 
 on the line. Hence when we have once adopted a certain 
 name for a line, such as a; cos a + 2/ sin a —jp, we may Say 
 that the line has a positive and a negative, side ; the 
 positive side being that, for any, point on which the ex- 
 
PERPENDICULAR DISTANCE OF A POINT FROM A LINE. 57 
 
 pression called the name of tlie line is positive, so that the 
 perpendicular on that side is written with the same sign 
 as the adopted name. The origin side is negative in Art. 
 54 ; if we called the line p—x cos a — y sin a, the origin 
 side would be positive. The sign of the origin side, then, 
 is the sign of the expression which we have called the name 
 of the line, when x and y each = 0, 
 
 Cor. 1. Similarly, as in Art. 52, Cor. 1, we see that the 
 Ax+By+ C of any point {xy) changes sign, as the point 
 crosses the line Ax + J3y+ C=0; so that the origin side is 
 the positive or negative side of that line, according as C is 
 positive or negative. 
 
 Cor. 2, It is easy to see that the results of Arts. 52 and 
 54 diifer in appearance only ; for, if the equation 
 
 Ax + By+ (7=0 
 
 is written in the form a; cos a+^ sina— ^== 0, it becomes 
 (Art. 27, Cor. 2) 
 
 A ^ G _ 
 
 'Ja' + b' V^^ + 5'"^ '^a'+b^ 
 
 and the pei-pendicular on this line, obtained by Art. 54, coin- 
 cides with the result of Art. 52. 
 
 Ex. If the equation to a line be 
 
 the expression Zy~ix+Z is positive on the origin side, and negative' on the 
 other, -and the expressions 
 
 3/ -4a;' +3 -3y'+4x'-3 
 5 ■' -5 
 
 represent perpen^ieulars froin points (xff) on the origin side and the other 
 side, respectively. 
 
58 STEAIGHT LINES PASSING THROUGH THE 
 
 56. The reasoning of the preceding article supposes 
 that the equation can be written in the form 
 
 a; cos a + ^ sin a — j3 = 0, 
 
 where p is & positive quantity, and a is defined by Art. 27, 
 Cor. 2; but, when the line passes through the origin, this 
 definition of a evidently fails, and the equation Ax + By = 0, 
 written in the form required, is either 
 
 A B . 
 
 or — ,- -x , v = 0. 
 
 The length of the perpendicular from {x'y) is still 
 
 , , / . N Ax' + By 
 
 + [x cos a. + 11 sm a), or + , ; 
 
 and we require to know, on which side of the line the ex- 
 pression adopted as its name is positive. If the equation be 
 written in the form y — mx = 0, it is evident that, for any 
 point above the line, y — mx is positive ; for y will then have 
 a larger positive or a smaller negative value, than it has on 
 the line for the same abscissa. Similarly, for any point 
 ielow the line, y—mx is negative, where we consider the 
 positive part of the axis of y as above the line. Hence we 
 obtain the following rule. Write the coeflScient of ^ positive; 
 then the positive part of the axis of y is on the positive 
 side of the line. 
 
 Ez. If the equation to the line be 
 
 the ezpreBBion 2y-ix is positive for polats above the line, and negative for 
 points below it ; and the expressions 
 
 Sy'-iscf -3y'+4a!' 
 
 "~5 ' 5 • 
 
 represent perpendiculars from points (a;'?/') above and below the line, respec- 
 tively. 
 
MTERSECTION OF TWO STRAIGHT LINES. 59 
 
 *57. It has already been shewn (Art. 43) that the equa- 
 tion 
 
 Ax-\- B;/ + + Jc {A'x + JB't/+ C) =0 (1) 
 
 is the equation to a straight line passing through the inter- 
 section of the lines 
 
 Ax + By+C=0, A'x + B'y + G' = 0... (2). 
 
 Now equation (1) admits of a very simple geometrical inter- 
 pretation, if the lines (2) be written _/+ 
 in the form 
 
 a; cos a +^ sina— ^ = (3), 
 
 a; cos /S -1- y sin y8 - g' = (4); 
 
 for let EK and ER be the lines (3) 
 and (4), and suppose the origin some- 
 where in the angle opposite to KER, so that the positive 
 and negative sides of the lines are as in the figure. Then, if 
 we take any point, as P, {x'y), the perpendiculars PK and 
 PR will be represented (Art. 54) by the expressions 
 
 OS cos OL + y' sin a —p, x cos /3 + 3/ sin ^ — q, 
 
 and the equation 
 
 a;' cos a + 2/' sin a —p — A; (a;' cos /3 4- 2/' sin /3 — g-) = (5) 
 
 , asserts that 
 
 PK _k 
 PR~1' 
 
 and hence the locus of the point P, or of the equation 
 
 X cos 01 + y sin a— p — k(x cos ^ + ysin^ — q) = (6) 
 
 is a straight line passing through the intersection of (3) and 
 (4), and such that, if perpendiculars be dropped from any 
 point in it upon the lines (3) and (4), those perpehdicu,lars' 
 will be to one another in the ratio of k to 1. Hence also the 
 sines of the angles which (6) makes with (3) and (4) are in 
 the ratio of Ic to 1. 
 
60 
 
 STRAIGHT LINES PASSING THROUGH THE 
 
 *58. It -will now be seen, why we have (Art. 54-) con- 
 sidered so carefully the signs of the perpendiculars, or rather, 
 investigated a method by which 
 we may always be able to write 
 "the expressions for them, so that 
 those expressions shall represent 
 positive quantities; for if the 
 lines in question be BT and 
 D'T', it is evident that the rea- 
 soning of Article 54 will give us, 
 for the point P (x'y'), if its per- 
 pendiculars on BTsiJid JD'T are 
 as k to 1, the equation 
 
 x' cos a + y sina—p _k 
 
 x' cos yS + 2/' sin /3 — g' 1 ' 
 
 or xcosa. + ysma—p — k{xcos^ + ysm^ — q) =0...(7), 
 
 as the equation to FR ; whereas the point Q would, under 
 the same circumstances, give us the equation 
 x cos a + y sin a —p _ h 
 — (a' cos y8 + 2/' sin j8 — 2') 1 ' 
 or a; cos a + y sin a — p + k{x cos /3 + 2/ sin jS — g) = 0. . . (8), 
 as the equation to QR. 
 
 CoE. If A = 1, or the perpendiculars are equal, equations 
 (7) and (8) will represent the straight lines which bisect the 
 angles URT and D'RD. These lines are easily seen to be 
 at. right angles, by the condition of Art. 47, Cor. 
 
 *o9. In order to select the proper equation for any par- 
 ticular line passing through R; where the ratio of the perpen- 
 diculars or sines is giyen, the following rule, which may be 
 readily deduced from Art. 58, is universally true. 
 
 If any point in the required line is on the origin side of 
 both or neither of the given lines, the equation, is of the form 
 
INTERSECTION OF TWO STEAIGHT LINES. 
 
 61 
 
 (7); if it is on the origin side of one only, the equation is 
 of the form (8). 
 
 The equations to the lines, if given in any form, may (Art. 
 27, Cor. 2) be reduced to the form here used, and the above 
 test applied. Arts. 54 — 59 are equally true for oblique axes, 
 if (Art. 32) we use the equation x cos a. + y cos (a — a) = p; 
 but in that case the. reduction of any equation to the re- 
 quired form is more complicated. 
 
 * 60. Equations (6) and (7) (Arts. 57, 58) usually present 
 some difficulty to the student, because we use the same sym- 
 bols, X and y, to represent the co-ordinates of the point P 
 and the co-ordinates of any point on the given lines. In 
 these articles, we have endeavoured to avoid this difficulty, 
 by first calling the co-ordinates of P x and y', considering 
 P as one fixed point. When we have obtained relations 
 (5) between these co-ordinates, by means of the conditions 
 of the problem, we may evidently (Art. 21) write x and 
 y for x' and y, in the equation to the locus of P, with- 
 out any fear of confusion. We shall hereafter frequently 
 speak of the perpendicular from the point (xy) on the line 
 X cos a -t- ^ sin a —p = 0, bearing in mind this explanation. 
 
 *Bx. To find the equations (axes rectangular) to the straight lines which 
 bisect the supplementary angles between the two lines, 
 
 y-^/3a:-5=0, ,y%-a!+ 6^/3=0. 
 
 let DR, jys be the lines, -which will evidently lie as hi the figure ; then, 
 comparing their equations with the equation xeoBa+y wn a=p, 
 
 and -writing them in that form, we have 
 (Art. 27, Cor. 2) 
 
 for DiJ, 
 
 ioxJyR 
 
 
 Hence we have for the equation to the 
 bisector PS, any point of which is on the 
 origin side of both or neither of the gi-ren 
 lines, 
 
62 EQUATIONS REPRESENTING 
 
 or 2y-ix=nJi-'iZ. 
 
 For the bisector EQ„ any point of which is on the origin side of one only. 
 
 'i 
 
 rifi 
 
 or 2a!+23^= -23-11;^. 
 
 61. Equations representing straight lines. 
 
 We shall now notice a few particular cases, where equa- 
 tions of a degree higher than the first may fee interpreted by 
 means of the preceding articles. 
 
 If we have the equations to two straight lines 
 Ax + By+0=Q, A'x + £'i/+O' = 0, 
 it is evident that the equation of the second degree 
 
 {Ax + By+C) . {A'x + B'y + C) = 0, 
 will represent both the lines, for the co-ordinates of any 
 point in either of the lines, substituted in the equation, will 
 make one of the factors vanish, and the equation will be 
 satisfied. Similarly, if we multiply together the equations 
 to n straight lines, we shall obtain an equation of the mth 
 degree, which represents them all. Conversely, if any equa- 
 tion of the nth degree can be separated into n factors of the 
 first degree, it represents n straight lines. 
 
 62. To find the condition that an equation of the second- 
 degree should represent two straight lines. 
 
 Let the equation of the second degree, ia its, most general 
 form, 
 
 Ase' + Bxy+Of + Dx + Ey + F^O (1), 
 
 be written as a quadratic in y, 
 
 Cf + (Bx + E)y + Ax'' + I)x + F=^0; 
 
STRAIGHT LINES. 63 
 
 Solving this equation, we have 
 
 _ -{Bx + E)±J{Bx+I!Y-4!C(Ax' + I)x + F) 
 
 The expression under the root becomes 
 
 {E'-iAC)x^ + 2{BE-2GB)x + E'-'^CF, 
 
 and, in order that equation (1) may break up into two equa- 
 tions of the first degree, it is sufficient and necessary that 
 this expression should be a perfect square, that is with respect 
 to x, for the constants in the root maybe surds or imaginaiy; 
 hence the condition is 
 
 {BE-2GBY = [B'-iAG) {B'-iGF). 
 
 If the quantity under the root do not contain x at all, the 
 equation will represent two parallel straight lines ; for, in 
 this case, the coefficient of x in the right hand member of the 
 equation, which is (Art. 28, Cor. 2) the angular coefficient 
 of the lines, is not affected by the alteration of the sign of 
 the radical. These lines are imaginary, or the locus im- 
 possible, when the quantity under the root is negative. 
 
 Ex. 1. The equation 
 
 may be ■written 
 
 y^-3(x-l)y + 2{x-iy==0. 
 
 Solving for y, we have the equations 
 
 y-2x+2=0, y-x+l=0, 
 
 which are the equations required, since the original equation is obtained by 
 multiplying these two equations together. 
 
 Ex. 2. The equation 
 
 y^+2xy+Zx'' + Gx+2y-h3=0, 
 when solved becomes 
 
 y+X:^■lJ=^/'^{x+l)=0, 
 
64 - EQUATIONS EEPRESENTING 
 
 which represents two imaginaxy straight lines, which may be said to intersect 
 in the only point for which the equation is satisfied, viz, 
 
 y+x + l=0, !i! + l = 0, ora!=-l, y=0. 
 
 Ex. 3. The equation 
 
 4x'+8xy + 4y^+Sx+3y=0 
 represents the two parallel lines, 
 
 y+x=0, Ay + 4x + d = 0. 
 
 63. The equation 
 may be written 
 
 Solving this as a quadratic in ^, we obtain 
 
 2Cy = -{B-jB'-4,AC)x, 2Cy = -{B+jB'-4A.C)x. 
 
 which represent two straight lines through the origin, possible 
 and different, possible and coincident, or imaginary, according 
 as£'^-4^C> = <0. 
 
 Similarly every homogeneous equation of the nth degree 
 may be written in the form 
 
 ^•(i)"-A,(r- ^..©.A-o, 
 
 and will have n roots, possible or impossible, 
 ^ = m, ^ = m'&c., 
 
 which will give n straight lines, possible or imaginary,; passi 
 ing through the origin. 
 
STEAIGHT LINES. 65 
 
 64. The equation 
 
 ■will give two solutions of the form a; = constant, and will 
 therefore represent two straight lines, possible or imaginary, 
 parallel to the axis of y. In like manner the equation 
 
 represents two straight lines parallel to the axis of x. 
 
 Similarly any equation of a higher degi-ee, which involves 
 only one of the variables, will represent a series of straight 
 lines, possible or imaginary, parallel to one of the axes. 
 
 65. If an equation of the second order can be written in 
 the form 
 
 {Ax + By+ Cy + {ax+bi/ + c)'= 0, 
 
 it can be satisfied by those values only of x and y, which 
 make 
 
 Ax + By+ C=0, and ax-\-hy + c = 0; 
 
 for otherwise we should have the sum of two positive quan- 
 tities equal to zero. Hence the locus is a point, or may be 
 considered as two imaginary lines 
 
 AxJrBy+ C+ V-1 (aa; + 5y + c)=0, 
 which intersect in a real point. 
 
 66. If an equation of the second order can be written in 
 
 the form 
 
 {Ax + By+Cf+{ax + ly + cf= - P, 
 
 where P is positive, it can be satisfied by no real values of 
 X and y, since the sum of the two quantities on the left-hand 
 side can in no case be negative ; the locus is therefore entirely 
 imaginary. 
 
 5 
 
66 EQUATIONS EEPRESENTING 
 
 *67. To find the angle between the straight lines repre- 
 sented by the equation 
 
 Ax^ + Bxy + Cy* = (1), 
 
 the axes being rectangular. 
 
 If the lines are y = nx and y = filx, and 6 the angle be- 
 tween them, then (Art. 47) 
 
 tan^=^^£^; 
 
 also (Art. 63) /* and fi are the roots of the equation 
 
 Cm^ + Bm+A = (2), 
 
 where m is written for - ; hence 
 
 X 
 
 -(/*+/*') = ^, A*/*' = ^ (3); 
 
 therefore (/* — /t')" = (^ + /*')" — 4;i4/*' 
 
 , . , , A+G 
 
 and l + /i/i= — -^ — ; 
 
 therefore tan = ^ — , 7=-^ . 
 
 A+G 
 
 Cor. If instead of (1) the equation representing two 
 straight lines were 
 
 Ax'' + Bxy+Gf + I)x + Ey + F=0 (4), 
 
 it could be written in the form 
 
 {y-,ix-^){y-fi'x-^') = (5), 
 
 or fifi'x''-{fJ.+ij;)xy+f + Px+ Qy + R = (6), 
 
 where P, Q, and M are constants. Hence, if (4) and (6) are 
 in reality the same equation, 
 
 fl/i ^t + fi' _ 1 
 
 A~ £ ~G' 
 
STRAIGfHT LINES. 67 
 
 which equations for /* and jj! are the same as (3) ; henca 
 fi and /i' are as before the roots of 
 
 Cm^ + Bm + A = Q, 
 
 and the straight lines represented by (4) are parallel to those 
 represented by (1), and contain the same angle. 
 
 Cob. 2. The straight lines represented by (1) are coin- 
 cident, and those represented by (4) are parallel, if 
 
 5" -4^(7=0, 
 
 in which case 
 
 As^ + Bayy + Ojf: 
 
 is a perfect square. They are at right angles, iiA+C=Q. 
 
 *68. To find the equation to the straight lines which bisect 
 the angles between the straight lines 
 
 Ax'+Bxy+Cy'^O.. (1), 
 
 the axes being rectangular. 
 
 Let the lines (1) be y — /ta; = 0, y — ij,'x = 0, where fi and 
 fi' are the roots of the equation Cm" + Bm + -4 = 0. Then 
 the equations to the bisectors are (Art. 58, Cor.) 
 
 VnV Vl+Z* ' Vl+yu,'' Vl+A*" ' 
 or, when expressed as one locus, 
 
 (]f-f^T (y^A)'' _o 
 
 or, simplifying and dividing by /*' — ^, 
 
 f(ji+fi')+2xy (!-/*/*')-«''(/* + /*') =0, 
 or, (Art. 67) 
 
 a!'-2,^^xy-y'=0 (2). 
 
 5—2 
 
68 EXAMPLES III. 
 
 The condition (Art. 63) that the lines (2) should be real 
 is {A—CY+B'>0; hence the bisectors are always real, 
 whether the original lines are real or imaginary. They are 
 at right angles by Art. 67, Cor. 2. 
 
 EXAMPLES III. 
 
 1. Find the equation to the straight lines which pass 
 through the point (1, 3), and make an angle of 30° with the line 
 (2y — a; + 1 = 0) ; axes being rectangular. 
 
 2. Draw the lines represented by the equation 
 
 {2y,-x + c) (3y + £B-c) = 0, 
 
 and determine (1) where they intersect, and (2) at what angle; the 
 axes being rectangular. 
 
 3. Find the equation to a straight line which passes through 
 the point (c, 0), and makes an angle of 45° with the line 
 (J)x—ay = ah) ; axes being rectangular. 
 
 4. Find the equation to a straight line which is perpendicular 
 to the line (8y + Sa: — 3 = 0), and cuts the axis of y at a distance 
 = 8 from the origin ; axes being rectangular. 
 
 5. Find the cosine of the angle between the lines 
 
 {y-ix+9> = 0) and (y-6a; + 9 = 0)j 
 axes being rectangular. 
 
 6. Find the angle between the lines {iy + 3a; + 5 = 0) and 
 (4a5 - 3y + 6 = 0), ; axes being rectangular. 
 
 *7. Find the equations to the straight lines which pass through 
 the intersection of the lines {y-'2x + 4), {y = Zx+ 6), and bisect 
 the supplementary angles between them ; axes being rectangular. 
 
EXAMPLES III. 69 
 
 8. "What is the geometrical signification of the equations 
 x' + y'-O, xy^Qi 
 
 *9. Find the equations to the straight lines -which bisect the 
 angles between the lines (l2x + 5t/ = 8) and (3a:-4y = 3); axes 
 being rectangular. 
 
 10. Shew that the lines represented by the equation 
 
 6y''^-xy~2x + y-x'-l = 0, • 
 
 are inclined to one another at an angle of 45° ; axes being reot 
 angular. 
 
 1 1 . The equation 2y' -3xi/-2x'-Si/+6x = 0, represents two 
 straight lines at right angles ; axes being rectangular. 
 
 12. The equation y" — 2xy sec + of = 0, represents two straight 
 lines inclined to one another at an angle 6; axes being rect- 
 angular. 
 
 13. "What is the inclination of the co-ordinate axes, when 
 the lines represented by y^ — x" = Q, are perpendicular to one 
 {mother ? 
 
 *14. The equations to two straight lines are 
 
 x + 3y-a=0 (1), y-x + a = (2); 
 
 find the equations to the straight lines which pass through the 
 intersections of (1) and (2), so that the ratio of the sines of the 
 inclination of each to (1) and (2) may be as 1 : ,J5. 
 
 15. "What must be the inclination of the axes in order that 
 the lines (xy —3y — 2x + 6=0) may include an angle of 1 35° ? 
 
 16. Find the equations to the two straight lines which pass 
 through the origiuj and divide into three equal parts the distance 
 between the points in -which the axes of co-ordinates are intersected 
 by the line (x + y= 1). 
 
 17. Find the distance of the point of intersection of the lines 
 {3x + 2y + i = 0), (2:? + 5^-1-8 = 0), from the line (y = 5a; + 6)j the 
 axes being rectangular. 
 
70 EXAMPLES III. 
 
 18. Eind tie perpendicular distance between the lines 
 
 Ax + By + G = 0, Ax+By + G' = 0; 
 the axes being rectangular. 
 
 19. On -which sides of the line 3iB — 2y=l do the points (1, 2), 
 (3,-4) lie 1 Shew that they lie on the same side of the line y = 3ar, 
 
 20. The perpendicular from a point {x'y') on the line 3a!+3=4y, 
 is -^ (4y — Zx' — 3). On which side of the line does {xy") lie ? 
 
 21. Shew that the equation {Zx + y) (2a! — 3y + 8) = 4, repre- 
 sents a locus which lies entirely in two of the angles formed by the 
 lines (3a; + y = 0) and (2a; - 3y + 8 = 0). 
 
 22. Determine the loci represented by the polar equations 
 
 (i) 6-a = 0. (ii) sin(e-a) = 0. 
 
 (iii) (p — a) (p' — a" — ap tan ^ sin ^) = 0. 
 
 23. Find the angle between the lines 
 
 c d 
 
 - = cos ^ — 2 sin ^, - = cos ^ + 3 sin 9. 
 
 P P 
 
 24. Interpret the equations : 
 
 (i) sin 2^ = 0, (ii) cos 35 = 0, (iii) e' + a^' -l-)35 + y = 0. 
 
 25. Find the area of the triangles whose angular points are 
 (i) (0,6), (a, 0), (a, 6); (ii) (a, 2a), (2a, 3a), {3a, - ia) ; 
 
 (iii) (0, 0), {x, y) (a/, yy 
 
 26. Find the area of the triangle contained by the three 
 straight lines 
 
 3x+iy = 12, 4a;+3y=12, x + y = 3. 
 
CHAPTER IV. 
 Trcmsforination of Co-ordinates. 
 
 69. When the position of a point or the equation to a 
 curve is given, with reference to any particular system of co- 
 ordinates, it is frequently necessary to find that position or 
 equation with regard to some other system. "We have already 
 shewn how to pass from rectangular to polar co-oi'dinates, and 
 the converse, and we shall now shew how we may perform 
 other transformations, such as altering the origin, passing 
 from rectangular to oblique co-ordinates, and others of the 
 same nature. 
 
 70. To transfer the origin of co-ordinates to a point 
 (x'y') without altering the direction of the axes. 
 
 Let O'X, O'y be the new axes, respectively parallel to Ox 
 and Oi/, the old ones ; let the co-ordi- 
 nates of any point P, referred to the 
 old axes, be x, y, and, when referred 
 to the new, X, Y\ 
 
 OB = x', O'R^'y', and 
 FM= PM'+ O'B, 0M= O'M' + OR, 
 
 OT y= T+y', x = X+x'. 
 
 These formulae are true for rectangular and oblique axes. 
 
 Hence, to find what the equation to any locus becomes, 
 when the origin is transferred to a point {x'y'), the new axes 
 
72 
 
 TEANSFOEMATION OF CO-OEDINATES. 
 
 remaining parallel to the old, we must write oo+x' for x, and 
 y +y' for 2^. 
 
 Ex. To find what the equation 
 
 y'+iy-4x+8=0 
 becomes, when the origin ia transferred to a point whose co-ordinates are 
 a!=l, y=-2. 
 Writing a + 1 for x, and y-2ior y, we obtain the equation 
 y^=he. 
 
 71. To jind what the co-ordinates of a point become, if 
 the axes, being rectangular, are turned through a given angle 
 (a), the origin remaining the same. 
 
 Let Ox, Oy, OM, PMhe the old axes and co-ordinates of 
 any point P, OX, OY, OM', PM' 
 the new ones ; let angle XOx = a, 
 and let the old co-ordinates be x, y, 
 the new X, Y ; then, if M'R, M'S 
 be drawn parallel to Ox, Oy re- 
 spectively, we have 
 
 0M= 08- EM', 
 
 PM=PB + M'S, 
 
 or, since angle BPM' = angle XOx = a, 
 
 X = Xcos a — Fsin a, y= Fcos a -f- Xsin a ; 
 
 hence, to find what the equation to any locus becomes, when 
 referred to the new axes, we must write 
 
 X cos a —y sin a for x, and y cos a + a; sin a for y. 
 
 The student should remember this figure, and be able to 
 write the formulsB readily from it. 
 
TEANSFOEMATION OF CO-ORDINATES. 
 
 73 
 
 Ex. To find what the equation x^-y^ = a^ becomes, when the axes are 
 moved through an angle of 45°. Here, 
 
 1 
 
 sma=coso= 
 
 V2' 
 
 and we must write 
 
 -^(a;-!/) for x, and -j= {x+y) for y; 
 
 hence, the equation becomes 
 
 {x-yf-(x+yf=2d?, 
 or 2a^ + o^=0. 
 
 JXC X, 
 
 72. To find what the co-ordinates of a point become, 
 when the axes are changed from one oblique system to another, 
 the origin remaining the same. 
 
 Using the same notation as in Art. 71 and a similar figure, 
 we have 
 
 PM=M'S + FB 
 
 sm (xcy) sm [xy) 
 
 orysin (i/x) =Xsxn{Xx) + Fsin( Fa;), 
 where by sin (xi/) we mean the 
 sine of the angle which the axis of 
 X makes with the axis of y, and 
 by sin (Xx) the sine of the angle which the axis of X makes 
 with the axis of x ; and so with the rest. 
 
 Similarly, OM=OS+M'R 
 
 sm (xy) sm {xy) 
 
 or X sin (xy) = Xsin (Xy) + Tsin {Yy). 
 
 These formulae include the particular cases, where we 
 wish to pass from rectangular to oblique, or from oblique to 
 rectangular axes. The signs are those appropriate to the 
 
74 
 
 TEANSFOEMATION OF CO-OEDINATES. 
 
 figure, where [xy), {Xy), {Yy) are all measured on the same 
 side of Oy, and {yx), [Xx), ( Yx) on the same side of Ox. It 
 must be remembered, that, in speaking of the angles made 
 hy the axes, we mean the angles made by their positive 
 directions, and that {xy), {yx) are the same angle. 
 
 It is very rarely necessary to make a transformation 
 such as the above ; and for those which actually occur, it is 
 generally easier to obtain from the figure the formula re- 
 quired, than to adapt those which we have obtained above. 
 We will give a simple case which is often useful. 
 
 73. To transform an equation from a rectangular to an oblique system, the 
 axis of X remaining the same, an^ the new axis of y being inclined at an angle 
 u to the axis of x. 
 
 tTsing the same notation as before, and a 
 similar figure, we have 
 
 OM=OM'+ M'M== OM' + PM' cos PM'M, 
 
 or a!=Z+ Fcoaw; 
 
 PM=PM' am PM'M, or y=rsinw. 
 
 Similarly, if the transformation were from 
 the axes Ox, OY to Ox, Oy, we should have 
 
 X=x — y cotu, Y=y coaeou, 
 
 74. If we wish to transfer the origin to a point {x'y'), 
 and then change the direction of the axes, we have only to 
 add X and y to the formulae of Arts. 71 — 73. Thus, if the 
 changes of Arts. 70, 71 have taken place, 
 
 a; = a;'-l-Xcosa- Fsina, y=y' + Fcosa + Xsina. 
 
 75. The student must bear in mind that we make no 
 change in the locus, by changing the origin or axes. The 
 assemblage of points represented by the new equation is pre- 
 cisely the same as that represented by the old, but the man- 
 ner in which the axes are placed with regard to them is 
 
TEANSFOKMA.TION OF CO-OBDINATES, 75 
 
 changed, and therefore the equation, which expresses this 
 relative position, is not the same as before. 
 
 We may notice also that the degree of an equation cannot 
 be altered by transformation. For let Aafy^ represent any 
 term of an equation of the n* degree, where p + 2 = n: then, 
 since (Art. 74.) the old co-ordinates in terms of the new are 
 expressions of the first degree only, we should have 
 
 Ax'y'i = A (ax' + ly' + c)^ {did + ey +/)*, 
 
 and no term resulting from this multiplication can be of a 
 higher degree than ^ + g' or n. Hence, the equation cannot 
 be raised. It, consequently, cannot be depressed; for, if that 
 were possible, we might, by re-transforming it, raise it, which 
 has been proved to be impossible.' 
 
 76. In the case of polar co-ordinates, if we wish to turn 
 the initial line through an angle a, we must write ^ -I- a for 
 6 in the equation, since the new 6 is less than the old 6 
 by the angle a. 
 
 EXAMPLES IV. 
 
 1.. Tbansfokm the origin to a point (a h) in the equation 
 a;' + 2/' - 2aa; - 26y + o" + 6' - c" = 0. 
 
 2. Transform the equation 
 
 x" cos' a — y^ sin" a = a* 
 by turning the (rectangular) axes through an angle a. 
 
 3. Transform the equations x + y = o and x' — y' = 0, by turn- 
 ing the (rectangular) axes through an angle of 45°. 
 
76 EXAMPLES IV. 
 
 4. If both systems be rectangular, and the equation to the 
 old axis of y referred to the new axes is x-y = 0, the old and 
 new axes are incliaed to one another at an angle of 45". 
 
 5. Transform the equation y' + iay cot a — iax = 0, from a 
 rectangular system to an oblique system inclined at an angle a, 
 retaining the same origin and axis of x. 
 
 6. Transform the equation Sa;" - 5xy + 2y^ = 4, from axes in- 
 clined at an angle of 60", to the right lines which bisect the 
 angles between the axes. 
 
 7. Transform the same equation to rectangular axes, retain- 
 ing the old axis of x. 
 
 8. Shew by the transformation of Art. 73, that, when the 
 axes are inclined at an angle a, the expression of Art. 53 for the 
 area of a triangle must be multiplied by sin u. 
 
 9. If the origin and axis of x are taken as the pole and 
 the initial line, and the angle between the axes is w, shew that 
 we may transform from Cartesian to polar co-ordinates by the 
 formulae 
 
 sin (ti) — 6) sin 5 
 
CHAPTER V. 
 
 Geometrical Applications. 
 
 77. We shall now give a few examples of the applica- 
 tion of the formulse we have obtained to the solution of 
 geometrical problems. In attempting to solve these pro- 
 blems algebraically, the student will find that much depends 
 upon a judicious selection of the origin and axes, and the 
 application of the proper equations and formulae. He should 
 in every case consider the problem well, before he attempts 
 the solution, and form a definite plan, before he begins. He 
 may very possibly not be able to carry out his original 
 scheme, but his attempts to do so will probably suggest 
 some method by which he may solve the problem ; and he 
 will, at any rate, avoid a practice very common to begin- 
 ners, of working without any definite aim, and consequently 
 introducing and combining equations and formulae that only 
 serve to embarrass him, without in any way aiding him in 
 the solution. 
 
 78. To shew that the perpendiculars drawn from the vertices, on the oppo- 
 site sides of a triangle meet in a point. 
 
 Let ABC be the triangle, CD, BE, 4i?the perpendioularB, and assume Ax, 
 Ay as rectangular axes; let the co-ordinates 
 of C be AJ}=3f, CD^i/, and let AB=x". 
 Now the proposition is proved, if we can 
 shew that the abscissa of the point where AP 
 and BE intersect is =!if, for they will then 
 
 evidently intersect in CJ>. In order to shew 
 
 this we must find their eq^uations, which -^ -® J? J? 
 
 we shall obtain (Art. 50, Cor. 2) by observing that they each pass through 
 
78 
 
 GEOMETBICAL APPLICATIONS. 
 
 a given point, and are perpendicular to a given line. We must then first find 
 the equations to the lines AC, JBC, to -wluoh they are perpendicular. 
 
 Since AC passes through the origin and the point C (»V0> ^^b equation is 
 (Art. 30, Cor.) 
 
 2/=^^.. 
 
 •(1); 
 
 and since ££ passes through £ (cb'O), and is perpendicular to (1), its equation 
 is (Art. 50, Cor. 2) 
 
 y=-'^,(!^-«"} (2). 
 
 Also, since BO passes through the points B (a!"0) and C {x'y'), its equation is 
 
 ^^i^^'"-"")-"* (^); 
 
 and since AF passes through the origin (0, 0), and is perpendicular to (3), its 
 equation is 
 
 y= — ^'^ W- 
 
 At the point where (2) and (4) intersect, their ordinates must be identical; 
 hence, equating their values, we must have, at that point, 
 
 whence, at the point of intersection, a!=a;', which proves the proposition. 
 
 The student may exercise himself by solving this problem with DC as axis 
 of y instead of Ay: then, assuming DC, DA, DB to be known, he mayexpress 
 the equations to ^C and CB in terms of the portions of the axes they cut off 
 (Art. 26). It will then remain to prove that AF and BE intersect in a point 
 whose abscissa =0. 
 
 79. To shew that the three perpendiculars through the middle points of the 
 sides of a triangle meet in a point. 
 
 Using the same axes and notation as in the last problem, we must find the 
 co-ordinates of the three middle points M, 
 M', M"; we can then find the equations , 
 to the two perpendiculars MP, M"P, 
 from the condition that they each pass 
 through a given point, and are perpendi- 
 cular to a given line; if we then shew 
 that the abscissa of their point of inter- 
 section =4 ilf', we shall have proved that 
 they intersect in the perpendicular from M'. 
 
 M' 
 
 :B SCr 
 
PROBLEMS ON STRAIGHT LINES. 
 
 79 
 
 The eo-ordinates of M" are evidently ^ > 5^ i also (Art. 10) the co-ordinates 
 
 Of M, the point of bisection of the line contained between the points B (x"0) 
 and (x'y), are 
 
 K'+ir" t/ 
 
 1/ 
 As before, the equation to ^<7 is ^=^1! and the equation to M"P -which 
 
 passes through M" \-n „], and is perpendicular to AC, is 
 
 y' ^ ( a''^ 
 
 Also, as before, the equation to BO is 
 
 y=;^,(»'-»=") (2); 
 
 — 5 — ^ ), and is 
 perpendicular to (2), is 
 
 y-i=-~Y-v—^) ('^- 
 
 At the point where (1) and (3) intersect, their ordinates must be identical ; 
 hence, equating their values, we have 
 
 3!+x''\ 
 
 a;'/ a/N k'-k" / ^+x''\ 
 
 k" 
 
 which gives x— -^ , as the abscissa of the point of intersection ; but this ab- 
 
 seissa belongs to some point in the perpendicular from M', which proves the 
 proposition. 
 
 80. In the figure of Euc. i. 47, if KH and FG he produced to meet in M, 
 and MA produced to meet BQ in T, shew that MT is perpendicular to BC. 
 
 Take AB, reproduced indefinitely, as axes of x and y, and denote the 
 sides of the triangle opposite to A, B, C by 
 a,i,c] then the co-ordinates of M are a;= - 6, 
 y= -c, and the equation to MA, which passes 
 through the origin and M, is (Art. 30, Cor.) 
 
 e 
 V=^x; 
 
 also the line BO cuts off from the axes of x 
 and y intercepts =c and b respectively; hence 
 (Art. 26) its equation is 
 
 
 h 
 
 y 
 
 a: b 
 
80 
 
 GEOMETEICAL APPLICATIONS. 
 
 a: V f 
 - + v=l. 
 
 or y=--x + b. 
 
 a" - • c 
 
 which, hy Art. 47, represents a line .perpendicular to ilf^. 
 
 81. A straight line is drawn parallel to the base of a triangle, and its 
 extremities joined transversely to the base ; find the locus of the intersection of 
 the joining lines. 
 
 Let ABC be the triangle, CB the 
 base, and ED parallel to it. Take AB, 
 AC as axes, and let AB=h, AC=c, 
 AE=e, AD=d; then the equations to 
 EB and CD are 
 
 b'^ e 
 
 ...(1), 
 also (Euo. VI. 2), 
 
 a c 
 
 h 
 c 
 
 ■■(2), 
 d 
 
 from (3) 
 
 whence 
 or 
 
 (3), 
 
 from n.)e=p-; from (2) e?=-^; 
 ^ ' b-x ^ ' c-y 
 
 b _ ex {b-x) 
 c ~ by {c-y}' 
 
 cW - jy -bc{cx-by)=0, 
 
 {ex— by) {ex+by-bc)=0 
 
 .(4). 
 
 Now (4) represents two straight lines, 
 b c 
 
 x 11 , 
 c 
 
 The former of these is a straight line passing through the origin and the 
 
 b c 
 
 bisection of BO, for the co-ordinates of that point, x=^, 3?== , satisfy the 
 
 equation ; and it is evidently the locus required. The latter represents the 
 base BC, which is not the locus of the intersections of EB and CD. 
 
 82. The result of Art. 81 should be noticed. As is fre- 
 quently the case in the solution of algebraical questions, we 
 have arrived at two results, one which we were seeking, and 
 another which does not satisfy the geometrical conditions of 
 the problem. As shewn in Art. 40, the values of x and y 
 which satisfy equations (1) and (2) with the condition (3), 
 will satisfy (4) ; and therefore (4) represents a locus, which 
 
PROBLEMS ON STRAIGHT LINES. 
 
 81 
 
 passes through all the points where (1) and (2) intersect, as 
 d and e vary. It does not follow that every point, whose co- 
 . ordinates satisfy equation (4), should be one of these points 
 of intersection. 
 
 83. The sides containing a given angle are in a given ratio, and the vertex 
 is fixed; supposing the extremity of one of the sides to move in a given straight 
 line, to find the locus of the extremity of the other. 
 
 Let OA, OB be the two sides, ■where 
 pB=n.OA, and let angle AOB = a. Let A 
 move on the straight line AD; it is required 
 to find the locus of B. Take OD (=p), the 
 perpendicular from on AD, as initial line, 
 and let OB=p, angle BOD = 6; then 
 
 OA = OD BeoAOD=p sec (9- a), 
 
 and OB=n.OA, or p=np sec (9- a), which 
 is therefore the polar equation to the locus 
 of P. It represents (Art. 45) a straight 
 line at a distance =np from 0, the inclination of this distance to OD being a. 
 
 84. A straight line is drawn from a point 0, cutting two other straight 
 lines KA, KB in the points A, B, and in the straight line OAB a point P is 
 taken, so that OA, OP, OB are in harmonic progression ; find the locus ofE. 
 
 The condition to be satisfied is 
 
 OP~~OA^ OB' 
 
 Draw any initial line Ox, and let the 
 equations to KA , KB be 
 
 -=ocos9 + 5 sinS, -=a' co&B+V sm.9, 
 
 P P 
 
 since the general equation of Art. 44 may, 
 by dividing and transposing, be vraitten in 
 this form. Then, if the angle BOx=0, 
 
 :prr=<^ cos 9 + 6 sin 9, ---==a' cos 9 + 6' sin 9, 
 OA UB 
 
 2 
 
 therefore ^ = (a + a*) cos S + (6 + 6') sin B, 
 
 and, writing p for OP, we have the polar equation to the locus of P. It is 
 
 6 
 
82 EXAMPLES V. 
 
 (Art. 43) the equation to a straight line passing through the intersection of 
 KA and KB; for it is equivalent to 
 
 ( — a cos 9- 6 sin 51 + ( -- o' cos^- 5' sin j =0. 
 
 EXAMPLES V. 
 
 1. In the fig. of Euc. I. 5, ii BG, CF meet in H, shew that 
 AH bisects the angle BAG. 
 
 3. In the fig. of Euc. i. 47, shew that AL, BK, FG intersect 
 in one point, and that, if BG and GH be joined, the lines will be 
 parallel. 
 
 3. Prove algebraically Euc. vi. 2. 
 
 4. Let Ax bisect any straight line GB in ; draw CB, cut- 
 ting Ax in B and AD produced in E ; join DB and let it meet AG 
 produced in F, and join FF ; FE shall be parallel to BG. 
 
 5. In two given straight lines, drawn from a point 0, take 
 points F, Q in. one, and P", <^ in the other, so that OP, OQ, OF', 
 OQ are in harmonical progression; shew that the locus of the 
 intersections of PQ and FQ is a straight line bisecting the angle 
 between the given lines. 
 
 6. Shew that the locus of a point, the algebraic sum of whose 
 distances from the sides of a polygon is constant, is a straight line. 
 
 7. Taking the requisite data to fix a parallelogram in a plane, 
 by equations to its sides, prove that the diagonals bisect each 
 other. 
 
 8. MANP is. a parallelogram, having a given angle at A, and 
 also its perimeter a given quantity : shew that the locus of F for 
 all such parallelograms is a straight line. 
 
 9. The three bisectors of the sides of a triangle, drawn from 
 the opposite angles, meet in a point. 
 
EXAMPLES V. 83 
 
 10. The hypotenuse of a i-ight-angled triangle is made to 
 slide between two perpendicular straight lines ; find the locus of 
 the right angle. 
 
 11. Given the base and the difference of the squares of the 
 sides of a triangle j find the locus of the vertex. 
 
 12. Given the base and the sum of the sides of a triangle ; 
 if the perpendicular froni the vertex on the base be produced 
 through the vertex, till its whole length equals one of the sides, 
 shew that the locus of the extremity of the perpendicular is a 
 straight line. 
 
 13.' It Ax, Ay be two straight lines, and through any point F 
 there be drawn straight lines PP^Q^, PP^Q^, &c. meeting Ax in 
 /"j, Pg, (kc, and Ay in Q^, Q^, &o., then, if AP ^, AP^, &c. are in 
 harmonical progression, so also are AQ^, AQ^, &c. 
 
 14. AB, AG are two straight lines given in position; a 
 straight line DE meets them in D, E, respectively, so that 
 AD + AE is a constant length ; also DE is divided in the point 
 P, so that DP bears a constant ratio to EP; the locus of P is 
 a straight line. 
 
 15. If the angles of the triangle ABO are given, and A is 
 fixed, while B moves along a fixed str?iight line, shew that the 
 locus of (7 is a straight line. 
 
 6-2 
 
*CHAPTER VI. 
 The Straight Line with Abridged Notation. 
 
 85. We have shewn (Art. 43), that the equation 
 
 Ax + By + G + h{A'x+B'y+C')==0 (1) 
 
 can by varying k be made to represent any straight line 
 passing through the intersection of the lines 
 
 Ax + By+G =0, A'x + B'y+C' = (2). 
 
 Let the symbols L and Jf stand for 
 
 Ax + By + G and A'x + B'y + G'; 
 
 then equations (2) are written L = 0, M=0, and equation 
 (1) is written L + kM= 0. The lines i = 0, M= are 
 usually called 'the line {L)' 'the line {M),' and their point 
 of intersection is called ' the point [L, M).' 
 
 Instead of the equation L + kM= 0, it is often more con- 
 venient to use the symmetrical form IL + mM= 0. This will 
 obviously make no real difference in the equation, since we 
 
 have simply substituted the arbitrary ratio -j for the arbi- 
 trary constant k. 
 
 86. Since (Art. 47) the equations to parallel straight 
 lines may be written so as to differ only in their constant 
 term, any straight line parallel to {L) can be written in the 
 
 ^i i 
 
ABRIDGED NOTATIOK. 85 
 
 • form L + c = 0, where c is a constant. The lines (L + c), 
 {L — c) are parallel to and equidistant from (L). 
 
 If any number of lines [L), [M), (N), &c. be parallel, 
 then the equation 
 
 lL+mM+nN+&c. = (1) 
 
 wUl represent a straight line parallel to them ; for {L), {M), 
 (N), &c. will be of the forms 
 
 Ax + By+C=0, Ax + By+G'=0, Ax + Bi/ + C" = 0, &c. 
 and equation (1) becomes 
 
 Ax+By+ =- ; -f^ =0, 
 
 t + ?M + W + &C. 
 
 which represents a straight line parallel to the given lines. 
 
 Again, if (i), {M), (N), &c. pass through one point, (1) 
 wUl generally represent a straight line passing through that 
 point ; for the coTordinatess of the point make L = Q, M= 0, 
 lf=0 simultaneously, and therefore satisfy (1). We say 
 generally, because I, m, n, might be such as to make the 
 left-hand member of (1) vanish identically. 
 
 87. The following example -will shew how equations to lines may be 
 found in this system of Abridged Notation. 
 
 If {L), (31), {N) be the sides of a triangle ABO, opposite to A, B, C, and 
 points D, E, F he taken in (L), (M), {N), respectively, to shew that the sides 
 of the triangle DEF may be represented by 
 
 L + mM+^=0, Jf+»iV+- = 0, N+lL+—=0, 
 I m n 
 
 where I, m, n are constants. 
 
 We may write the equations 
 
 to OF, L+mM=0, to AD, Jf+niV=0, to BE, N+lL=0; 
 
 and we have for the points 
 
 L=0 ) M=0 ) iV=0 ) 
 
 ' M+nN=0 i • "' N+lL=0 } ' ' L+mM=Q \ * 
 
86 ABRIDGED NOTATION. 
 
 Hence the equation to DE is, since it passes through D, of the form 
 
 M+nN-i-TcL=0 (1); 
 
 and, since it passes through E, it is of the form 
 
 N+lL-i-VM=0 (2); 
 
 and, since these forms are identical, we have 
 
 le _ 1 _» 
 l~V~l' 
 
 whence Ic^ln, V=-, and (1) and (2) each become 
 
 M 
 N+lL + -=0. 
 n 
 
 Otherwise, after writing equation (1), we may proceed thus. Since (1) 
 passes through E, the values M — 0, N+lL=0 satisfy the equation, and 
 therefore, substituting for N and M, we have, as before, 
 
 nlL-iL=0 or h=ln. 
 
 Similarly it may be shewn, that the equations to the other sides may be 
 written as stated above. 
 
 88. If the equatioa to a straight line be written in the 
 form 
 
 p — x cos a —y sin a = 0, 
 
 so that (Art. 55) the origin is on the positive side of the line, 
 it is usual to employ one of the Greek letters for its abbre- 
 viated form a = 0, so that all points, for which the expression 
 a is positive, are on the origin side of the line. Let /S = 0, in 
 like manner, be written for 
 
 g — a; cos /3 — y sin /8 = 0. 
 
 Then we have shewn (Art. 54) that the a and /8 of any 
 point are its distances from the lines (a) and (/8). We have 
 shewn also (Art. 58) that the equations 
 
 are capable of a simple geometrical interpretation, and that 
 they represent two straight lines drawn through the inter- 
 
HARMONIC PENCIL. 87 
 
 section of (a) and (j3), so that the perpendiculars dropped 
 from any point of either of them upon the lines (a), (^), are 
 to one another as fe : 1. 
 
 Particular attention must be paid to the position of the 
 'origin, and the rules laid down in Art. 59 for determining 
 the sign of k. The positive and negative signs in the figure 
 of Art. 58 are now reversed ; but, exactly as in that artic^, 
 for all lines which lie in the same angle as the origin and 
 in the angle vertically opposite, the form is a — le^ = 0; for 
 lines which lie in the other two angles the form is a.4-kP = Q. 
 The student must be careful to bear in mind, that these 
 remarks apply to the particular form of equation only, which 
 we denote by (a), (/S), &c. Many problems occur, as above 
 (Art. 87), where it is not necessary to introduce any limita- 
 tion as to the forms of the equations or the position of the 
 origin. 
 
 89. We have shewn in Art. 58, Cor., that, when k=l, 
 equations (7) and (8) of tJjat article represent the bisectors 
 of the supplementary imgles between the lines; hence the 
 equations to the bissectors of the angles between (a) and 
 (,S) are 
 
 a-/3 = 0, a + /3 = 0. 
 
 A litH« consideration will shew that the lines (a — kj3), 
 Qe9. - /S) are equally inclined to the line (a — /3) ; as are 
 also (a + i/3), ikx + ^) to the line (a + ^). 
 
 90. The properties of the Harmonic Pencil will serve 
 to illustrate this part of the subject. 
 
 Dbf. Any four straight lines meeting in a point are 
 called a pencil of four lines, and a straight line drawn across 
 the pencil is called a transversal. A pencilis called har- 
 monic if it divides any transversal hxtrmonically, that is, so 
 
88 
 
 ABRIDGED NOTATION, 
 
 that the whole line is to one extreme segment, as the other 
 extreme segment is to the middle part. The four points 
 ■where a traDsversal meets the pencil are called a Range. 
 
 Let OA, 00, OB, 
 OD be any pencil, 
 and AGBD a trans- 
 versal ; then we shall 
 shew that the ratio 
 
 AD BD . 
 
 -j-Ty -^ -m^ IS constant, 
 A G BO ' 
 
 in whatever way AI) 
 
 be drawn across the 
 
 system. If we denote 
 
 the angles ^50,^ GO, 
 
 ADO, hjB, G,D,\ie 
 
 have 
 
 AD_AD AO 
 
 AG~ AO'AG 
 
 BD 
 BO' 
 
 BD BO 
 BO' BO 
 
 ^ sm A OD sin (7 
 sinD ' sin AOG' 
 
 sin BOD sInC 
 
 sinD ' sin BO C 
 
 ■ ^^ BD sin AOD sin AOG . . 
 
 '''AG ' BC~sinBOD ' smBOG ^ ^' 
 
 which is a ratio independent of the position of AD, and is 
 called the anharmonic ratio of the pencil. When the right- 
 hand member of equation (1) = 1, the transversal is harmoni- 
 cally divided, and the pencil is harmonic. We then have 
 
 AD:AG = BD:BG.. 
 
 •(2), 
 
 and, if we consider AD, AB, ^ C as the first, second, and 
 third quantities, respectively, equation (2) asserts that the 
 first is to the third as the difference between the first and 
 second is to the difference between the second and third, and 
 
HARMONIC PENCIi. 8& 
 
 'the quantities are therefore in harmonica! progression. The 
 lines KO, KA, KP, KB, in the figure of Art. 84, form an har- 
 monic pencil. 
 
 91. The lines 
 
 form an harmonic pencil. 
 
 Let OA, OB, be the lines, a = 0, /8 = 0, then, if we 
 suppose the origin somewhere in the angle A OB, the lines 
 a-k^ = 0, a + k^ = will (Art. 88) lie as 00 and OD 
 respectively. Then we have 
 
 sin A 00 _ J _ smAOD 
 sin BOG ~ sin BOD' 
 
 therefore, from Art. 90 (1), 
 
 AD : AO = BD : BC. 
 
 92. Art.' 91 is equally true, if the lines are 
 
 ■ L = 0, M=0, L-kM=0, L + kM=0; 
 
 for (Art, 27, Cor. 2) we have L = \a, if = fi^, where \ and /i 
 are constants; hence the equations 
 
 L -kM= and i 4- kM = 0, 
 
 may be written 
 
 Xa—kfij3=0 and Xa.+k/jt,j3=0, or a.—k'^=0 and a+A'/8=0, 
 
 where k' is written for -—■ . These lines therefore form an 
 
 A. 
 
 harmonic pencil with (a) and (/8), that is with (L) and [M). 
 The student must be careful not to assume that (L — M), 
 {L + M) bisect the angles between (i) and {M) ; they do, 
 however, form an harmonic pencil with them. 
 
90 ABEIDGED NOTATIOK. 
 
 93. If (L), (M), (N) be three straight lines, and we can 
 find three constants I, m, n, such that 
 
 lL + mM+nN=0 (1), 
 
 identically, then all the three lines [L), {M), {N),pass through 
 one point, or are parallel. 
 
 For, since (1) is true identically, we have L = — j M— ^ N 
 
 identically, and therefore i= is the equation to a straight 
 line passing through the intersection of [M) and {N), if they 
 meet, or (Art. 86) parallel to them, if they are parallel. 
 
 Ex. (1) The three straight lines that bisect the angles of a triangle meet 
 in a point. 
 
 Taking the origin of co-ordinates within the triangle, let o=0, ;8=0, 7=0 
 he the equations to the three sides; then the equations to the straight lines 
 bisecting the angles are 
 
 o-J8=0, /S-7=0, 7-0=0. 
 Here 2=m=n=l, and the lines therefore meet in a point. 
 
 Ex. (2) If, through the angular points of a triangle, there be drawn any 
 three straight lines meeting in a point, then three straight lines, drawn 
 through the same angles, equally inclined to the bisectors of the angles, will 
 also meet in a point. 
 
 The first three lines may be represented by 
 
 la-mp=0, m)S-»7=0, ji7-Zo=0, 
 
 putting -J- for h &c. 
 
 Then (Art. 89), the equations to the other three will be 
 ma-ip=0, »j3-m7=0, ly-na=0, 
 
 and multiplying these by — ^ , — , and j- , respectively, they become 
 
 « 5=0. 5-5:=o, 5:.« 0. 
 
 I m m n n I 
 
 and therefore we see that these also pass ttrough one point. 
 
TEILINEAB CO-ORDINATES. 91 
 
 Ex. (3) The straight lines joining the angular points of a triangle with 
 the middle points of the opposite sides intersect 
 in one point. Let ABQ be the triangle, and' the ^ 
 
 origin as before within it: let o=0, /S=0, 7=0 
 be the equations to £0,CA, and AB. Then if I> 
 be the middle point of BO, the equation to ADi& 
 
 em CAD 
 . ^ em CAD CD , siaBAD BD or CD 
 
 ^^' ie^-=zp''"'^-^s:b-== ad ' 
 
 therefore sin C^ J ^ sing 
 
 BinBAD BinB' 
 
 and the equation to AD becomes 
 
 /Ssin5-7sinC=0. 
 
 Similarly the equations to the other two lines are 
 
 7 sin(7-osinjl=0, a sin^l -/3 sin£=0, 
 
 and these three lines evidently pass through one point. 
 
 Ex. (4) It may be shewn in the same manner, that, if AD be perpen- 
 dicular to BC, its equation is 
 
 /SoosJ8-7CosC^=0, 
 
 and that those of the other perpendiculars are 
 
 7 cos C-o cos 4=0, o cos 4-/3 cos 5=0, 
 
 and these three pass through the same point. 
 
 94. TriUnear Co-ordinates. 
 
 The distances a, yS, 7 of a point from the three sides of 
 a triangle, formed by the lines (a), (/S), (7), are called the 
 Trilinear Go-ovdinates of the point, and the triangle is called 
 the Triangle qf reference. By paying attention to the signs 
 of a, A 7, the position of a point may be defined by these 
 distances, and the properties of lines investigated by means 
 similar to those employed in the Cartesian System. In this 
 system it is usual to consider a as positive, when P is on the 
 same side oi BG as A, and as negative when on the other 
 
&2 TRILINEAB CO-ORDINATES. 
 
 side, so that the trilinear co-ordinates of a point within the 
 triangle of reference are all positive. It will be seen that this 
 is equivalent to considering the origin within the triangle, 
 and the symbols a, &c. to stand iov p — x cos a — y sin a, &c. 
 
 95. We proceed to shew the relation that must exist 
 between the trilinear co-ordinates of 
 a point. Let AB G be. the triangle 
 of reference, the lengths of whose 
 sides are a, h, c. Take any point P 
 within the triangle, and join it with 
 the angular points; then the dis- 
 tances of P from BG, GA, AB are a, yS, 7, the trilinear co- 
 ordinates of the point, and the areas of the triangles PBG, 
 PGA, PAB are 
 
 acf. iyS <Tf 
 
 "2' T' ¥' 
 
 Hence, if A denote the area ABG, we have 
 aa + S/S -f- C7 = 2A, 
 
 as a constant relation that must always subsist between the 
 
 quantities a, ^8, y. If the point be taken outside the triangle, 
 
 the same relation will be seen to hold, by giving the proper 
 
 sign to a, yS, 7. It is clear that any point on BG gives 
 
 a = 0, which is therefore the equation to BG; similarly /3 = 0, 
 
 7 = are the equations to GA and AB. The point whose 
 
 trilinear co-ordinates are a, /8, 7 may be called the point 
 
 2A 
 (0/87) : thus the point A is the point — , 0, 0. 
 
 Cb 
 
 If the ratios of the co-ordinates of any point be given, 
 the actual values of the co-ordinates may be found by the 
 above relation, as will be seen in the Examples below. In 
 practice, however, we rarely require the absolute values of 
 the co-ordinates. Equations are usually made homogeneous 
 
TRILINEAR CO-ORDINATES. 93 
 
 by the method of Art. 96, and a homogeneous equation in a, 
 A 7 can always be written so as to involve the ratios only of 
 these quantities. Thus we may write 
 
 a'-3a=;8 + 7' = 0, or("^y-3(^y.^+l = 0. 
 
 Ex. To find the trilinear co-ordinatea of the points of intersection in the 
 Examples to Ait. 93. 
 
 In Ex. 1 we have at the point of intersection a=j3=7; 
 
 therefore aa._lp_cy_ aa + ip + cy _ 
 
 a b c o+i+c ' 
 
 therefore a=p=y= 
 
 a+b + c' 
 
 In Ex. 2 we have in like manner 
 
 aa _ 6/3 _ C7 _ aa + 5;8 + C7 _ 
 al ~ im, cn~ al + bm + cn' 
 
 ,, , o ;3 7 2A 
 
 therefore ■==- = - = — ; — = . 
 
 I m n al + bm+cn 
 
 The method is exactly the same for Ex. 3 and 4. 
 
 96. By means of the relation proved above (Art. 95) any 
 equation in trUinear co-ordinates can be made homogeneous, 
 
 by multiplying each term by ^-r raised to a suitable 
 
 power, since this quantity = 1. Thus, the equation to a 
 straight line parallel to the line la. + «w/3 + ny = 0, 
 
 is (Art. 86) l(iL + m^ + ny + k = 0, 
 
 or as an equation homogeneous in a, yS, j, 
 
 2A {la + m^ + ny) +k(aa + h^ + cy) = 0. 
 
 Again, the equation a' + h^ + k^O becomes 
 
 4AV + 2AA/3 {aa + i^ + cy)+k (aa + h^ + <yff = 0. 
 
94 - TEILINEAE CO-ORDINATES. 
 
 97. The equation to any locus can be transformed from 
 Cartesian to Trilinear co-ordinates, without altering the de- 
 gree of the equation. For (Ai-t. 75) the axes may be made to 
 coincide with two sides of the triangle of reference, GB, OA, 
 without raising or depressing the equation ; then, if x, y and 
 a, /S be the co-ordinates of any point F in the two sys- 
 tems, respectively, and the angle 
 
 A GB = 6), we have a; = /8 cosec to, 
 
 y = a cosec 60 ; hence any equation 
 
 in X and y may be written as an 
 
 equation of the same degree in a, 
 
 /3, and can be made homogeneous 
 
 in a, /3, 7, by the method of Art. ^ 
 
 96. Hence the Cartesian equation to any straight line can 
 
 be transformed into an equation of the form 
 
 la. + wi/S -f M7 = 0, 
 
 that is to say, the general equation of the first degree in tri- 
 linear co-ordinates can be made to represent any straight 
 line, by giving suitable values to I, m, n. 
 
 98. Conversely, with one exception, every equation of 
 the form 
 
 la + w/3 + ny = 0, 
 
 will represent a straight line ; for it may be written 
 
 n n c 
 
 fl a\ /m J\ „ 2A . 
 
 or a-l-l /8 + — = 0; 
 
 \n cj \n cj c 
 
 and by writing a = a; sin a, ^ = ysma), as in Art. 97, we 
 have an equation of the first degree in x and y, with GB, GA 
 for axes. This will always represent a straight line, unless 
 
 lamb, J 
 
 -=-, —=- OT I : m : n = a : : c. 
 n c n c 
 
TEILINEAB CO-OEDINATESi 95 
 
 Cob. If the lines (i), {M), (if), are the sides of the 
 triangle of reference, then, since (Art. 92) we may write 
 
 IXa. + m/i^ + nvy = for IL + mM+ nN= 0, 
 
 it follows that any straight line can be represented by the 
 latter equation, and that it will always represent a straight 
 line, unless IX : m/i : nv = a -.h -.c, 
 
 99. We shall now examine the meaning of the equation 
 aa + 6;8 + C7 = 0, or asin^ + y3sinJ5 + 7sin = 0, 
 since these two are equivalent. 
 
 We saw (Art. 26) that the line {Ax-^By+ C=0) cuts 
 
 O 
 off intercepts on the axes — -j . — r > hence, if A and B 
 
 become very small, the intercepts on the axes are very 
 great. Let A and B each become indefinitely small, then the 
 intercepts become indefinitely great, and the line is altogether 
 at an infinite distance from the origin. The equation may 
 now be written, 
 
 0.a; + 0.2/+ C=0, 
 
 which cannot be satisfied by any finite values of x and y, 
 but may be satisfied by infinite values, since the product 
 X 00 may be finite. We may express all this shortly by 
 saying that the equation (7 = 0, that is, a constant = 0, 
 represents a straight line sittiated altogether at an infinite 
 distance from the origin. The direction of the line is wholly 
 undetermined; and it must be clearly understood that the 
 equation, impossible in itself and representing nothing, de- 
 rives its meaning from the possible equation of which it is 
 the limiting form. Similarly it would be absurd to say (Ap- 
 pendix IV.) that the equation (7=0 gave two infinite values 
 of X, or that the equation Bx + G=0 had one infinite and 
 one finite root; but both these statements are intelligible, 
 
96 TRILINEAR CO-ORDINATES. 
 
 if these equations are the limiting forms of Ax'+Bx + G=0, 
 where A and B in the former and A in the latter are in- 
 definitely small. 
 
 In the same way the equation^ 
 
 aa, + b^ + cy = 
 
 is in itself impossible, since we have proved that 
 
 aa + bfi+cy 
 
 is a constant quantity, and cannot = ; but the equation 
 
 la. + »w/8 + ny = 0, 
 
 when the ratios I : m in approach indefinitely near to 
 a -.b : c, will represent a straight line altogether at an infinite 
 distance from the triangle of reference; for when this equa- 
 tion is put in the form Ax + By + C = 0, as in Art. 98, the 
 values of A and B become in this case indefinitely small, and 
 the line, therefore, infinitely distant from the origin. 
 
 100. To find the condition that the two straight lines 
 
 l(X -{■ m^ + ny = (1), 
 
 l'oL + rn^ + n'y = (2), 
 
 should he parallel. 
 
 Suppose a third straight line 
 
 \a-|-/t;S + j/7=0 (3) 
 
 to pass through the point of intersection of (1) and (2). 
 Then for this point (1), (2), (3) are true simultaneously; and 
 a relation between the constants may be found, by solving 
 
 (1) and (2) for - and — , and substituting the values so 
 
 found in (3). Now suppose X, /*, v to bicome a,b,c; then 
 the line (3), and therefore the point of intersection of (1) 
 and (2), has moved off to an infinite distance, and the re- 
 
TRILINEAE CO-OEDINATES; 
 
 97 
 
 lation obtained is the condition of parallelism. It ■will be 
 found to be 
 
 (win — nin) a + {nt — «'Q S + (Im' — Tm) c = 0. 
 
 101. Tofind the equation to a straight line in the form 
 a-a'_/9-j8' 7-7' 
 
 .= Z, 
 
 •(1), 
 
 p q r 
 
 where (a'/Sy) is a fixed point pn the Jine, and I is the dis- 
 tance between (a/87) and {a'^'y), changing sign as in Art. 34: 
 It will be seen from the annexed figure, that, if 0, ^, 1^ be the 
 
 angles which the straight line PQR makes with BC, CA, 
 AB, then 
 
 ol — tj. = EG = l.ws.Q', /. a — a'= — Zsinft 
 
 r Similarly /? - /8' = Z sin 0, 7 - 7' = Z sin i/r, and the equa- 
 tion to the line PQB is 
 
 ^-/S'_7-7'^^ 
 
 a-a' 
 
 — sin d sin ^ sin i|r 
 
 .(2). 
 
 ' ■ The student can satisfy himself, by drawing a variety of 
 figures, that the quantities a — a', p — ^', 7 — 7' cannot all 
 have the samp sign, so that p, q, r cannot be all of -the same 
 sign. They, are always equal to sin 6, sin^, sini^ in irrmgni-^ 
 
 7 
 
98 TRILINEAR CO-ORDI'NATES: 
 
 tude, and the proper signs may easily be found in particular 
 cases'. 
 
 102. To find the equation to a straight line which passes 
 through the points (a'/3V). ,(*"^"7")- 
 
 Let the equation to the line be 
 
 ■la. +mP + m7 = (1); 
 
 then lo! + m^' + ni = (2), 
 
 and Za"+»Mj8"+»»7"=0 (3). 
 
 Eliminating n and m successively from (2) and: (3), we 
 
 have 
 
 I m 'n 
 
 p<y — P7 ya. —y a. a/d —a. p 
 and the required equation is 
 
 (/3 V - y8"7') « + (7'«" - 7"a') ^ + («'/S" - *"/3') 7 = 0. 
 
 103. To find the angle hetween the straight lines 
 
 a-.Jc^=0, a-i'/3 = 0. 
 
 If (a) and (/3) be taken as the axes of x and y respec- 
 tively, it is shewn in Art. 97, that the Cartesian co-ordinates 
 of any point (ay) in either line are in the same ratio as the 
 perpendiculars, a and A upon the lines; hence the equations 
 to the lines referred to these axes are 
 'y = hx, y^lcx; 
 
 and by Art. 49, if a and ^ contain an angle "(7, the angle 
 between the lines is found by the equation 
 
 ^^""P ~ 1 + (k + k') cos C + kk''' 
 
 ^ It -will be observed that one of the segments, QR,SP, PQ (as QJl iii the 
 figure) always subtends the supplement of one of the angles of the triangle ; 
 and it will be found, that, according as that angle is A, S, or C, the single 
 sign will belong to the first, second, or third member of the equation. It will 
 be + or - , according as we choose the positive direction- of I. 
 
TEILINEAE CO-ORDINATES. 
 
 99 
 
 aad the condition of perpendicularity is 
 
 l + {k + k')cosG+kM = 0. 
 
 Ex. 5^0 find the angle between the straight lines 
 
 la + m.p+ny=0, l'a + m'p + n'y=0. 
 
 From the relation aa+i/S+C7=2A, we may, by substituting for 7, write 
 the equations 
 
 (fc - na) a + (fflic - nJ) (3 + 2nA = 0, 
 
 (I'c - n'a) a + (m'c -. n'b) p +, 2»A = ; 
 
 and the equations to lines parallel to these through O, are found by omitting 
 the term 2nA in each. The angle (0) between these is the angle required, 
 ^nd the value of tan (1^) is found to be 
 
 { (fc - «a) (m'e - n'b) - (mc - rib) (I'c - n'a) } sin C 
 
 (I — na) l}'c - n'a) + (mc - rib) (m'c - n'b) - { (<c - na) [m'c - n'b) + (mc - nb) [I'c - n'a) ] ooa i 
 
 which becomes, after a reduction whiph requires care but presents no 
 difficulty, 
 
 (mn' - m'n) sin 4 + (nl' -n't) Bin. S+ {Im' - I'm) sin C 
 W + mm! + rem' - (m»' + m'n) cos X - (ni' + n'C) cosB- (Im' + I'm) C03 C' ' 
 
 The lines are parallel, if the numerator of this fraction vanishes, and per- 
 pendicular, if the denominator vanishes. The former result agrees with 
 Art. 100. 
 
 104. The two following examples are good illustrations 
 of the methods of this chapter. The latter deserves especial 
 notice. 
 
 Ex. 1. Let A,B,F,D\>e any 
 four points in a plane ; then three 
 pairs of straight lines can be drawn, 
 so that each pair includes all four 
 points. Let these pairs be BC, 
 AC; BE, FE; BD, AF. The 
 figure is now called a complete qaeA- 
 rUateral. 
 
 Let ABO be the triangle of re- 
 ference; then we may write the 
 equation io BD 
 
 la-ny=Q (1),, 
 
100 TRILINEAB CO-ORDINATES. 
 
 since it passes through (o, y), and any value may he given to the arbitrary 
 
 ratio - . Similarly the equation to AFmay be written 
 
 m/3-n7=0 (2). 
 
 Hence the equation to FE since it passes through F, the intersection of 
 (a) and (mp - ny), may bp -written 
 
 mp-my + }ca=0 ,., (3), 
 
 and since it passes through J), the intersection of (j3) and (la-Tiry), when 
 p=0 in (S), la=ny : hence, substituting for j8 and y, we have 
 
 ; — Za + Ao=0 or Jc = l, 
 
 and the equation to FE is 
 
 la + mfi-ny=0 (4). 
 
 Similarly the equation to CG, which passes through C (a, jS) and G 
 {la. - "nr/, mp - my), is 
 
 la-mp=0 , (5), 
 
 and the equation to CE, which passes through C (a, p) and E [la + j»j3 - ny, y), 
 is la + mp=0 (6). 
 
 Hence (Art. 91) (X5, CA, CG, OE form an harmonic pencil, for their 
 equations are 
 
 a=0, . |S=0, la-mp=0, Zo + m^=0. 
 
 We leave it to the student to prove that EB, EF, EG, EG apd GF, GD, 
 GC, GE form harmonic pencils. We might with equal propriety have used 
 the abbreviations L, M, N instead of the trilinear co-ordinates a, p, y, 
 
 Ex. 2. If fherfi 6e two triangles ABC, abc, such that the intersectitms of 
 the corresponding sides lie in u, straight line, then the straight lines joining 
 the corresponding angles will meet in a point, and conversely. 
 
 Let P, Q, S, lying in one _p q jj 
 
 straight line, be the intersec- ~ 
 tions of the corresponding sides. 
 
 Take A EC as the triangle of \ ^^^ ■4i- 
 
 reference, and let thp equation 
 toPQRhB £\^ 
 
 la + mp+ my=0. ..{PQR). * ' *?■"■* '* 
 
 Then the equation to be, since 
 
 it passes through the intersec- \ / / a 
 
 tion of PQR andr^C is of the yi/ 
 
 form V 
 
 la+mp+n/y+ha-0, 
 or Z'a + mj3-h»y=0 .» {be). 
 
TRILINEAR CO-ORDINATES. 101 
 
 Similarly the equations to ea and ah are 
 
 Za+»i'/S+»7=0 (ca), 
 
 la+m^+n'y=0 (ai). 
 
 From equations (ic), (co) -we obtain by subtraction 
 
 (J-2')a-(»i-m')i8=0 ((7c), 
 
 which the>-efore represents a straight line passing through c, the intersection 
 of be and ca. But it also represents a straight line passing through the 
 intersection of (a) and (§), i. e. through C. Hence it is the equation to Ce. 
 Similarly the equations to Aa, Bb are 
 
 (m-m')j3-(»-»')7=0 (Aa), 
 
 and (n-n')y-{l-l')a=0 (Bb). 
 
 and therefore (Art. 93) Aa, Bb, Cc meet in a point. 
 
 Conversely, suppose Aa, Bb, Ce to meet in 0, and let the eqnatidn to 
 PQi, the straight line joining the intersections of BO, be and CA, ca, be 
 
 Za + m/3 + nv=0 (PQ). 
 
 Then the equations to be and ca will be (6c) and (ca) as above. We shall 
 shew that the equation to ab is {ab). As above, the equation to COc is 
 (Cc) ; and therefore at the point we have 
 
 (l-l')a={m— m') ;8 = (n - »') 7 suppose. 
 
 Hence the equations to AOa, BOb will be (Aa) and (Bb). Now the line 
 represented by the equation (06) passes through 6, the intersection of BOb 
 and 5c; for it is obtained from the- equations (Bb) and (be) by subtraction; 
 also it passes through a, the intersection of A Oa and ca • for it is obtained 
 from the equations (Aa) and (ca) by addition. Hence the equation (ab) 
 represents a6. But the equation (ah) is evidently Satisfied when y=0 and 
 Za+mj3+n7=0, i. e. the line ab passes through the intersection of AB and 
 PQ, ; or AA, ab intersect in PQ. The triangles ABC, dbc are said fo be 
 Iwmologotis ; PQR is called the axis and the centre of koTnology. 
 
 We have tliEOUghout this chapter, for the sake of sim- 
 plicity, used the equation of Art. 32 in the form 
 
 X cos a +^ sin a —p = 0, 
 
 but the form for oblique axes, 
 
 a5 cos a +^ cos (01 — a) — ^ = 0, 
 
 would (Art. 59) have been equally applicable.. 
 
102 EXAMPLES VI. 
 
 EXAMPLES VI. 
 
 The triangle ABG is supposed to be the triangle of reference. 
 
 1 . Find the equation to a- straight line through the vertex A 
 of a triangle, parallel to the base BC. 
 
 2. Find the equation to the straight line joining the mijldle 
 points o{AB, AG. 
 
 3. Find the equation to the straight line passing through the 
 point (a'jS'y), and parallel to (^a + mj3 + ny = 0). 
 
 4. Find the equation to the straight line joining the feet of 
 the perpendiculars from A and B on BC and GA respectively. 
 
 5i If the lines represented l>y the equations 
 
 {a - b'j{a - bp) + {b-a)y = 0, 
 
 (b-b') (a-aP) + {a-b) y = 0. 
 
 intersect in the line a — a'jS = 0, shew that the following relation 
 holds amongst the constants : 
 
 {a' + b')(a + b} = 2{a'b' + ab). 
 
 6. Find the co-ordinates of the point of intersection of the 
 lines la + myS + Wy = and I' a + m'^ + n'y = 0. 
 
 7. Find the condition that the three points 
 
 (a'^yl {a'^y"), (a"'/3"'/') 
 may lie on one straight line. 
 
 8. Shew that the straight line, drawn through A parallel to 
 BG, and the bisector of BO froni A, form an harmonic pencil 
 with AB and AG. 
 
.EXAMPLES VI. 103 
 
 9. Find tHe trilinear co-ordinates of tie point in which the 
 straight lines meet, which are drawn from the angles of the 
 triangle ABC, to bisect Izhe opposite sides. 
 
 10. In Art. 87 find the condition that AD, BJE, OF may , 
 pass through one point. 
 
 11. Find the equation to a straight line bisecting BG &% right 
 angles. 
 
 12. The straight lines bisecting the three sides of a triangle 
 at right angles meet in a point. 
 
 13. Form the equation to a perpendicular to BGfroia G. 
 
 14. Find the perpendicular distance of the point {a'^'y') 
 from the line (la + m^ + ny = 0). 
 
 15. The straight line joining the middle points of the- sides 
 of a triangle is parallel to the base. 
 
 16i Find the condition that the straight liae> 
 la + mj3 + ny=0, 
 should be parallel to the bisector of the angle A of the triangle of 
 reference. 
 
 17. If the line la + m^ + ny = 0, is drawn across the triangle 
 of reference, shew that I, m, n cannot all have the same sig!&. If 
 it meets AB, AG, not produced, in M, K, find the lengths of 
 AM, AN. 
 
 18. From the angles A, B, G of any triangle are drawn three 
 straight liaes AA', BB', GG', bisecting the angles; through A, B, C 
 are drawn three straight lines perpendicular to A A', BB^, GG', to 
 meet BG, CA, AB, produced, in G, H, K y G, H, K are in one 
 
 - straight line. 
 
 19. ABG is a triangle, D and E are points within the- tri- 
 angle, such that the angle ABE=GBD, and BGD = AGE; shew 
 that BAD = GAE. 
 
 20. Interpret the equations 
 
 a+^ + 7=0, i8+7-a = 0, + 7-/8 = 0, + ^-7=0. 
 
104 JSXAMPLES VI. 
 
 21. The four angles of a quadrilateral A BCD are bisected by 
 four straight lines; the bisectors of A, B meet in E, of B, (7 in F, 
 of C, D in G, of A, J) in II. Prove that the directions of EG and 
 Fff pass through the intersection of the directions of AJD, BG, and 
 AB, CD respectively. 
 
 22. Erom the angles of a triangle ABG straight lines ate 
 drawn through a given point within the triangle, to meet the 
 opposite sides in E, F, G; FG, GE, EF are produced to meet BC, 
 CA, AB, in. P, Q,R; shew that P, Q, R lie in one straight line. 
 
 2.3. If two similar triangles have their homologous sides 
 patellel, the straight lines which join the equal angles meet in a 
 point. 
 
 24. Shew that the result of Ex. 32, page 47, may be obtained 
 by means of Art. 93. 
 
 25. Shew that the trilinear co-ordinates of the centre pf the 
 circle circumscribed about the triangle of reference, are given by 
 the equations 
 
 g _ P _ y _ ^^ 
 
 cos A cos B cos 4 A ' 
 
 26. If be the centre of the circle circumscribed about the 
 triangle of reference, and.if J^O, £0, 00 be produced to meet the 
 opposite sides in A', B', 0', shew that three of the four straight 
 lines represented by the equations 
 
 a sec ^ J= yS sec 5 ± y sec (7 = 
 
 are the sides of th6 triangle A'B'O'. Shew that, if BO, B'O' meet 
 in P', OA, CA' in Q; AB, A'B' in R; P, Q, R will lie on the 
 fourth straight line. 
 
CHAPTER VIl 
 
 The Circle. 
 
 105. We have seen (Art. 61), that the general equation 
 of the second degree 
 
 Ax' + Bxy +. (7/ r\-Dx + Ey + F= 0, , 
 
 may sometimes represent two straight lines. Before ex- 
 amining generally all the loci represented by, it, we shall 
 shew that certain particular forms of it are capable of being 
 interpreted, and will represent circles. We shall afterwards 
 see that the circle is a particular case of a class of curves 
 represented by the general equation; and that the forms 
 which we interpret are particular cases of one of the classes 
 of equations into which we shall divide it. We adopt this 
 plan on account of the simplicity of the circle, and because 
 the reader is already familiar with its principal properties; 
 geometrically treated, 
 
 106, To find the equation to a circle whose centre and 
 radius are given, the co-ordinates being rectangular. 
 
 - If C (ab) be the centre of the circle, P any pomt {ay) oii 
 the circumference, and CB be drawn 
 parallel to Ox to meet the ordinate of 
 P in B, we have 
 
 CB' + PB'=OP\ 
 or (a;-a)^+(y-&r = »•^ 
 
 •^heJe' r = the radius of the circle. 
 
106 THE CIRCLE. 
 
 This follows directly from Art. 7 ; and in fact the equa- 
 tion only asserts, that the distance between the points {ah) 
 and {xy) is constant and equal to r. 
 
 If the co-ordinates be oblique, and inclined to one another 
 at an angle = a, we have, since angle GRP now = 180° — w, 
 
 (x — of + {y — Vf + 2 [x-a) [y — l)cos(o=r'; 
 
 but we shall seldom have occasion to use this equation. 
 
 Cor. Expanding the general equation to the circle re- 
 ferred to rectangular axes, we have 
 
 i)? + y*^2ax-2ly + «" + V - r^= 0; 
 
 and hence it appears that the general equation to the circle 
 is of the form 
 
 x^ + f+Dai + Ey + F=Q, 
 
 D, E, i'' being any constants^ The equatioil 
 
 Aa? + Af+ Dx + Ey + F= 0, 
 
 may be reduced to^ this form by dividing by A, and is there- 
 fore the most general form that the equation can assume, 
 when the co-ordinates are rectangular. 
 
 107. Hence if we can reduce an equation to the form 
 a?+y^JpDx + Ey + F=Q, 
 
 we may always, interpret it ; fbr,^ adding. ^ + "t- to both 
 sides of the equation, we have 
 
 I. If ^> + -J- — ' ^ is positive, this is the equation to a 
 
X- 
 
 FOEMS OF EQTXATIOK. 107 
 
 circle, the co-ordinates of whose centre are 
 
 x= — -^,y = — -^, and whose radius =(-7- +— — Jfj . 
 
 II. If the quantity -^ + -7 — J?' be = 0, the equation 
 
 may be considered to represent a circle with an infinitely 
 small radius, or (Art. 65). two imaginary straight lines 
 
 which intersect in the only real point for which the equation 
 is satisfied, namely the point ( — „- , — "S" ) • 
 
 III. If -7- + -7 — -^ is negative, there are (Art. 66) no 
 
 values of x and y that can satisfy the equation, and the circle 
 is imaginary. 
 
 Ex. 1. The equation x^+y'-ix + iy + l^O, may be -written 
 
 (i«:-l)='+(y+2)2=4, 
 
 which represents a circle, the co-ordinates of whose centre are x=l,y=-2, 
 and whose radins =2. 
 
 Ex. 2. The equation ar'+/ + 2a!-% + 10=0,maybe written 
 (x+l)'' + {y-S)''=0, 
 a circle, the co-ordinates of whose centre are x=-l, y=3, and whose 
 radius —0. 
 
 Ex.3. a?+y^+2x+ey+li=0, or {x+iy+(2/+S)''=-t, represents an 
 imagiuary circle. 
 
 108. If in the equation (a; - 0)'+ (y - hy=r', a=0, &=0, 
 or the centre of the circle' be origin,, the equation becomes 
 x' + i/' = r\ 
 
 If a = r, J = 0, or a diameter be chosen'as axis of x, and 
 its extremity as originy the equation becomes 
 
 a^-2rx + f = 0; 
 
108 THE CIBCLK 
 
 and similarly, if the axis of y be a diameter, and the origin 
 at its' extreinity, the equation is 
 
 109. It may be observed here, that in the circle, as well 
 as every other curve, if the origin is on the curve, there tyill 
 be no term which does not involve either x ov y; for the 
 equation must be satisfied by the values sc = 0, y = 0, which 
 cannot be the case, if there be a term, which does not vanish 
 when X and y vanish. 
 
 110. If we expand the equation to the circle (Art. 106), 
 referred to oblique axes, we obtain 
 
 x^-\-'jf-\-^ cos «*. a;^ — 2 (« + J cos m) a; — 2 (S + a cos w) y 
 
 + a''4-Z'' + 2a6cosw-r' = (1). 
 
 Hence, if the inclination of the axes be a,' the general 
 equation to the circle is .. 
 
 a^ -Vy^ ■\-'lLcQ%io.xy ^Dx \ Ey + F=^Q (2), 
 
 where D, E, F are constants. 
 
 COE. In order then that the general equation 
 
 Aa^ + Bxy+Cy'''\-Dx-\-Ey + F=0. 
 
 may represent a circle, we niust have, 
 
 A = C, B = 2 A cos ft), 
 
 where « is the ahgle between the axes; for then, by dividing 
 by A, the equation can be reduced to the form of (2). 
 
 . Ex. ^o determine the inclination of the co-ordinate axes, in order that 
 tiie equation ' ■ ■ 
 
 may represent a circle, and (0 find the magnitude, of its radios. 
 
POLAR EQUATION. 103 
 
 Comparing the equatjon with equation (1), we have the equations 
 2eosw=-l, a^+H' + iab coao)-r^=0, 
 2 {a+bqoBu)=h=2(b + aBOBa), 
 from which we ohtain 
 
 a=^Tr, a=o=r=k. 
 o 
 
 111. The e(juation 
 
 a;» + y' = r' 
 
 will give us a well-known property of the circle; for it may be 
 obtained by eliminating k, by means of 
 multiplication, from the equations 
 i/ = k{x-r) (1), 
 
 3, = _l(a, + ^) (2), ""^ 
 
 where A is a constant and perfectly arbitrary; 
 
 But these equations evidently represent straight lines 
 which 
 
 (i) pass through the extremities of the diameter QR, which 
 is the axis of x ; for (Art. 43) equation (1) represents a line 
 passing through the intersection of the lines y = 0, a; — r = 0, 
 which is the point B ; and (2) passes through the intersec- 
 tion oi y = Q, x-i-r = 0, which is the point Q. 
 
 (ii) intersect in the circle, sin;ce by eliminating k between 
 them we have the equation to the circle ; and - 
 
 (iii) are at right angles to one another by Art. 47 ; and 
 they represent all lines which fulfil these three conditions!. 
 Hence we see that the locus of the vertices of all right-angled 
 triangles on QR as base, is the semicircle QPR, 
 
 112. To find the equation to the circle referred to polar 
 'co-ordinatesi 
 
110 . THK CIECLE. 
 
 Let'Oa; be the initial line, the pole.; let the co-ordinates 
 of the centre (7 be the known quan- 
 tities p, ff, and of any point JP in the 
 circumference, p, 6 ; then 
 
 co''+po''-2co.po.cospoc= cr, 
 
 or p"' + p''-2p'pcos{d-e')=r\ 
 which is the polar equation required. 
 
 It will be seen that this is the formula of Art. 14, and only 
 asserts that the distance between the points, whose polar 
 co-ordinates are p, 6, and p, 6', is constant and equal to r. 
 
 Cor. 1. The two values of p which may 'be found from 
 the equation 
 
 p" - 2p' cos {0~e')p + p" - r' = 0, 
 
 are the two distances from the .pole of the points P, P', 
 where the radius vector, which makes an angle with Ox, 
 cuts the circle. The product of the roots of this equation 
 (Appendix) = p" — r', a quantity which does not change for 
 different values of 0. Hence the rectangle OP. OP' is constant 
 foj.- all positions of OP. When the roots are equal, or the 
 line touches the circle, as OP, we have 
 
 p"-r''=^OE^. 
 
 Hence QP.OP' = 0R\ as in Euc. Iil. 35, 36. 
 
 Cor. 2. If p =0, or the centre be pole, the equation 
 becomes p = r. 
 
 Cor. 3. If p =r, and 0' = 0, or a diameter be the initial 
 line, and one .extremity of it the pole, the equation becomes 
 p = 2r cos 0. 
 
 The reader will do well to verify by geometrical figures 
 the results obtained here and in Art. 108. 
 
 113. In th€t j^rpgoiug articles we have assumed one only 
 of the well-known geometrical properties of the cjrcle, viz. that 
 
INTERSECTION OF STRAIGHT LINE AND CIRCLE. Ill 
 
 the distance from the centre to the circumference' is constant, 
 and from this property we have deduced the equation. Mo&t 
 of the following articles will admit of being proved in a very 
 simple manner by those properties of the circle with which 
 the reader is .familiar; but we .prefer to deduce our proofe 
 from the equation alone, because this method is theisamB as 
 .that which we shall use in the case of other curves; and it 
 is desirable' that the student should perceive, that all the 
 properties of the circle may be obtained from its equation, 
 without any previous acquaintance with the curve. It will, 
 however, be an exercise very profitable to the student, if he 
 endeavour to deduce the equations of the following articles 
 from any of the properties of tangents, &c. which he may find 
 in Euclid. 
 
 114. To find the length of a straight line-dravm .from a 
 ,point (x'y') to meet the. circle. 
 
 Let PQES be a line whose equation is 
 
 x-x' ^y-y' _.j^ 
 
 c s 
 drawn, from the point P {x'y') to cut the circle 
 X- -^-y^ — r^ 
 
 •(1). 
 (2). 
 
112 THE CIBCLE. 
 
 Now, if we substitute for x and y from (1) in (2), the 
 result will be a quadratic in /, the two roots of which will be 
 the distances PQ, PR of the point {x'y') from the points 
 where PS cuts the circle. As we have drawn the figure, the 
 two values of I are (Art. 34, Cor.) negative ; if P were to lie 
 between Q and R, one root would be positive and the other 
 negative : if P were to lie in RS, both roots would be positive. 
 Making the substitution, we have 
 
 {cl + xy+{sl + yy = T% 
 
 therefore, since s° + c" = 1, 
 
 V + 2 {ex' + sy')l + x" + y'' -r" = (3), 
 
 or F+Ql + R=Q, 
 
 an equation which will always give two values for I ; hence 
 every straight line meets the circle in two real, coincident, or 
 imaginary points, according as the roots of (3) are real and 
 unequal, real and equal, or imaginary. 
 
 115. If the points of section Q, R, become coincident, by 
 Q remaining fixed, till R, moving along the curve, approaches 
 indefinitely near to Q, the line will be a tangent according to 
 Euclid's definition ; for it is of indefinite length, and meets 
 the curve in one point only. We shall, however, find it con- 
 venient to adopt language similar to that used in Algebra, 
 and to speak of two coincident points, just as we speak of 
 two equal roots, and do not call them one root. We shall 
 then take the following as our definition of a tangent, since it 
 is found more convenient than Euclid's, when we treat curves 
 by Algebraic methods, r . 
 
 Def. If two points be taken on a curve, and a chord 
 driawn. through. them; .then, if the firsj; point remains fixed, 
 
INTERSECTION OF .STRAIGHT LINE AND CIRCLE. 113 
 
 ■while the second, moving along the curve, approaches indefi- 
 nitely near to the first, the chord in its limiting position is 
 called the tangent to the curve at the first point, 
 
 116. We shall have occasion to consider the following 
 particular forms which equation (3) of Art. 114 may assume. 
 
 If ^ = 0, the point (a;'^') is on the circle, and one value 
 of I becomes = 0. 
 
 If i? = and Q = 0, both values of I become = 0, and the 
 line passes through two coincident points of the circle and is 
 a tangent. 
 
 If Q = 0, the roots of the equation are (Appendix) equal 
 and of opposite signs, and {x'y') is therefore the middle point 
 of the chord. 
 
 117. To find the equation to a straight line touching the 
 circle at a point (x'y'). 
 
 Let PS (fig. Art. 114) be the line 
 
 "^nA^y^l^^l (1), 
 
 C 8 
 
 cutting the circle 
 
 !^ + f = r' (2), 
 
 in the point Q {x'y). Then for the distances {I) between 
 (x'y') and the points of section of the line and drcle, we 
 obtain as in Art. 114, the equation 
 
 P+ 2 (ex' + sy') I + a;" + y=- r' = 0, 
 
 or Z»-l-2(ca!' + sy)Z=0 (3), 
 
 since [x'y) is on the circle, and therefore 
 
 8 
 
114 f THE CIRCLE. 
 
 Equation (3) gives us l = Q, (as it should, for {xy) coiii-r 
 cides with one of the points of section), and also 
 
 i H- 2 {ex + SI/') = 0, 
 
 the value of 7 in -which is the distance QE. Bu,t-, if we sup- 
 pose the point B to move up to Q, this distance vanishes, 
 and the line becomes a tangent at Q [xy') ; and we have, as 
 the condition' that (1) should he tangent, 
 
 ex -^-sy =0 (4). 
 
 Eliminating c and s by means of this equation and the 
 equation to the line, we have 
 
 {x-x')x ~{y-y')y' = (5.), 
 
 or xx + yy — {x' + y'^) = 0, 
 
 whence xx + yy' = r', 
 
 which is the equation to the tangent at the point {x'y'). 
 
 We leave the reader to obtain this equation geometrically, 
 by means of known properties of the tangent. For example, 
 he will find it easy to shew, from Euc. iii. 16, Cor. that 
 
 x — x 
 
 = -^ 
 
 y-y X 
 which gives us equation (5) of this article. 
 
 118. If we transfer the origin to any point (—a, —5), so 
 that the co-ordinates of the centre are a and h, we must 
 write in the equation to the tangent, by Art. 70, 
 
 , x-a, X- a,y-l,y'--l for x, x, y, y, 
 
 respectively, and the equation becomes 
 
 {x - a) (a;' - «).+ {y-.V)4:y' -h) = r'. 
 
THE TANGENT. 115 
 
 119. If the problem be to find the equation to a tangent 
 which makes a given angle with the axis of x, we must elimi- 
 nate x and y, instead of c and s, from equation (1). 
 
 We have from (4), 
 
 s «'_ 
 
 c~ y" 
 
 s' + c' x'^ + y" 1 r' 
 
 whence — ^— = j^ , or ^ = —,2 „ 
 
 from which and from (4), we have 
 
 y =± re, x' = + rs; 
 
 whence from (1) 
 
 cy—sx = (cy — sx') — + r {c' + s') = ±r; 
 
 or, if - = OT, where m is the trigonometrical tangent of the 
 c 
 
 angle which the tangent makes with the axis, we have 
 y = mx + rVl + m". 
 
 120. We may obtain the above result as follows. Let 
 
 y = mx + b, x' + y^ = r', 
 represent a straight line and a circle; then, if we find values 
 of X and y which satisfy both these equations, these will be 
 the co-ordinates of the points, where line and circle intersect. 
 Eliminating y between them, we obtain 
 
 x' + (mx + by = A 
 (1 + m") x^+ 2bmx + J' - r' = 0, 
 
 the roots of which equation are the abscissae of the points of 
 intersection. Ifthe hne is a tangent, these roots are equal, 
 and we have (Appendix) 
 
 ■ whence '5' = r'{l + m"),. 
 
 8—2 
 
116 THE CIRCLE. 
 
 ■which is the condition of tangency obtained above. Hence, 
 as before, the equation to the, tangent, which makes with the 
 axis of X an angle = tan~'m, is 
 
 y — mx + rVl + iri', 
 
 the double sign referring to the two tangents at the ex- 
 tremities of any diameter, which are parallel. 
 
 It may occur to the student, that we have here used a 
 necessary, but not a sufficient condition of tangency, since the 
 abscissae of the points of intersection would be equal, if the 
 line were a chord parallel to the axis oi y. But in this case 
 the equation to the line would be of the form tc = a constant, 
 and we could not obtain our result by the elimination 
 of y. 
 
 Cor. 1. Hence the condition that any line {y = mx + h) 
 should touch the circle (»" +y^ = /•'), is 
 
 If this condition, be written 
 
 h ^ 
 (l+rf)*"*"' 
 
 it asserts (Art. 52) that the perpendicular from the centre 
 (0, 0) on the line y = mx + & is equal to the radius. Hence, 
 more generally, the condition that the line 
 
 Ax + By+G=0 (1) 
 
 should touch the circle 
 
 {x~af+{y-hr = r^ (2) 
 
 is that the perpendicular from the point {ah) on (1) should 
 be equal to the radius of (2). That is (Art. 52) 
 
 Aa + Bb+0 
 
 + r- =T. 
 
 - {A* + B')^ 
 
THE TAIJGENT. 117 
 
 Cor. 2. Hence the equation to the tangent, in terms 
 of its distance from the centre, and the angle a which that 
 distance makes with the axis of x, is 
 
 X cos a + y sin a. = r. 
 
 121. To determine the equations to the tangents drawn to 
 a circle from any point (x'y'). 
 
 Let the equation to the tangent he 
 i/ — mx= +'"Vl+»i*; 
 then, since it passes through {x'y'), the co-ordinates of that 
 point satisfy the equation, and we have 
 
 (i/'-mxY=r''{l+m'), 
 or {x''-T^m!'-2x'y'm + y"-r' = (1), 
 
 a quadratic to determine the two values of m, in the equa- 
 tions to the two tangents which can be drawn to the circle 
 from {x'y). If /i and fi be the two roots, the required 
 equations will be (Art. 29, Cor. 1) 
 
 y-y'=lju{x-a!), y-y' = tL{x-x) (2). 
 
 Cob. Solving equation (1), we have 
 xy + r Va;'* + y'^ — r' 
 
 whence we see that the values of m are real, if x'^ + y'^>r', 
 or the point is outside the circle; they are equal, when 
 x^ + y"^ = r^, or the point is on the circle; and they are 
 imaginary, when a?"' + y' < r", or the point is within the 
 circle. Hence we say, that from any point {x'y'), there can 
 be drawn two real, coincident, or imaginary tangents to the 
 circle. "We do not attach any geometrical meaning to the 
 term 'imaginary tangent.' We simply mean to say, that, 
 even when /t and /i' are imaginary, equations (2) can be 
 formed, which satisfy the conditions of tangency, and are 
 satisfied by the co-ordinates of the point {x'y). In Art. 62, 
 
118 THE CIRCLE. 
 
 Ex. 2, we have a numerical example of imaginary lines wliich 
 pass through a real point. 
 
 The change in the sign of the expression, x' + y^ — r", 
 should be remarked, and compared with Art. 55. 
 
 122. The straight line drawn through any point in a 
 curve, perpendicular to the tangent at that point, is called 
 the Normal. 
 
 In the case of the circle, if the point be (x'y), the equa- 
 tion to the normal is 
 
 y—y' — m [x — x), 
 
 where m must (Art. 47) have a value, the reciprocal of the 
 value found in the case of the tangent, and must be of a 
 
 different sign; hence, (Art. 119) m=—,, and the equa- 
 tion is 
 
 y-y =|^(a'-a'). 
 
 which, after reduction, becomes 
 
 xy -yx = 0, 
 the equation to a straight line passing through the origin ; 
 hence every normal in the circle passes through the centre, 
 as is proved in Euc. iii. 19. 
 
 123. To find tlte equation to the chord joining the points 
 of contact of two tangents from any exterrial point (x'y). 
 
 Let P' (x'y) be the external point, and let PP" be the 
 line whose equation is required. 
 
 Now the equation to P'P", the 
 tangent at P"(x'y"), is (Art. 118) \ ^ ^, 
 
 XX 4 yy = T, 
 and, since this line passes through 
 P' {x'y'), the co-ordinates of P" satisfy 
 the equation ; hence 
 
 aix' + y'y" = r'; 
 
CHOEB OF COlirTACT. 119 
 
 hence the values x = x", y = y" satisfy the equation 
 
 XX + yy = r", 
 
 or F" {x'y") \s a point on the line it represents; and by 
 exactly similar reasoning, P, the other point of contact, is on 
 this same straight, line ; hetoce 
 
 XX ■\-yy =»• 
 
 is the equation to the chord joining the points of contact of 
 tangents drawn from the .point [x'y) ; for it is the equation 
 to some straight line, and both P and P" have been proved 
 to lie in it. 
 
 Cor. Henee, to draw tangents to the circle from any 
 external point (icy), we have the two equations 
 
 xx' + yy' = r°, aj" + y" = r•^ 
 
 to determine the co-ordinates at the points of contact. These 
 equations will always give two points, real or imaginary, 
 corresponding to the points of intersection of the line and 
 circle. . 
 
 124. A chord of a circle is drawn through a fixed point 
 (x'y), and tangents are drawn at the points where it cuts the 
 circle; to find the equation to the locus of the intersection of 
 these tangents, when the chord is turned about the point (x'y'). 
 
 Let P" be the point {x'y), P'QR the chord, and let 
 the tangents at Q and R intersect in 
 P" {x'y") ; it is required to find the 
 locus of P", as the chord turns about 
 P'. Considering PR as the chord join- 
 ing the points of contact of tangents 
 drawn from the point {x'y"), its equa- 
 ^tion is (Art. 123) 
 
 xx"+yy" = r\ (1); 
 
120 THE CIRCLE. 
 
 but, since it passes through F{xy') we have 
 
 x'a^' +y'y" = r' (2); 
 
 but (oj'y) is any point in the required locus, and its co- 
 ordinates satisfy the equation 
 
 xx +yy' = r^ (3); 
 
 hence the co-ordinates of every point in the locus satisfy 
 this equation, which is therefore the equation required. The 
 locus is therefore a straight line. It is* evidently perpen- 
 dicular to the line [xy' —yx = 0), i.e. to the line joining the 
 centre to the point {xy). 
 
 125. The reasoning of the preceding. article is very often 
 perplexing to the beginner. The difficulty commonly arises 
 from the use of the co-ordinates of P"{x'y") as constants in 
 equation (1), and afterwards as variables in equation (3). 
 We should bear in mind that, although P" is a moveable 
 point, we do not examine its position during its motion, butj 
 taking it in any one of its several positions, we obtain a rela- 
 tion between its co-ordinates, while in that position.. The 
 relation so obtained is equally true for all positions,, and is 
 therefore the equation to the locus of the point. Thus, equa- 
 tion (2) is a relation obtained between the co-ordinates of 
 a certain point P". Equation (3) declares this relation to 
 be true for all points determined by the same law as P", 
 
 126. The line 
 
 xx +yy' = r*, 
 
 is called the polar of the point {x'y') with regard to the 
 circle 
 
 x' + f = r', 
 
 and the point (x'y') is called the pole of the line. These 
 
 > This statement must be omitted, when we refer to this proof in the case 
 of other CRryeSj 
 
POLE AND POLAR, 
 
 121 
 
 terms must not be supposed to have any such meaning, as 
 they have in Art. 11. As the equation to the polar is one 
 of the greatest importance, and will be frequently used in 
 the following pages, we will define its exact meaning in all 
 cases. 
 
 (1) The position of the point P in Art. 124 is not sub- 
 ject to any limitation; hence, wherever the point i^y) may 
 be, the equation xx -\-yy' = r^ represents the locus of the 
 intersection of tangents drawn at the extremities of chords 
 which all pass through {x'y). 
 
 (2) If the point be without the circle, this locus is 
 (Art. 123) identical with the chord joining the points of 
 contact of tangents drawn from {x'y). 
 
 (3) If the point be on the circle, the locus is also 
 (Art. 117) the tangent at the point {x'y). 
 
 In the following figures, JP is the point {x'y'), and BQ is 
 the line {xx' +yy' = r'). 
 
 Let F be within the circle; then, if ST, S'T' are chords 
 
 drawn through P, and the tangents at thfir extremities meet 
 in E and Q, BQ is the locus, and the polar has the inter- 
 pretation (1) only. 
 
122 
 
 THE CIRCLE. 
 
 Let P be without the circle ; then the polar has the inter- 
 pretations (1) and (2) ; that is, ^^ is the chord of contact of 
 tangents drawn from P, and also, if any chord P8T be drawn, 
 the tangents at S and T intersect in R Q. 
 
 Let Pbe on the circle; then the polar is the tangent at 
 P. It has also the interpretation (1) : for, if any number of 
 chords be drawn from P, as PS, PS', the tangents at the 
 extremities of those chords intersect in the line PQ. This is 
 evident, since PQ is itself one of the tangents in every case. 
 It has also the interpretation (2), for the chord of contact 
 now coincides with the tangent. 
 
 127. We saw (Art. 121) that from any point {x'y) there 
 could be drawn two real, coincident, or imaginary tangents 
 to the circle ; and we shewed in what sense we might say 
 that imaginary lines paaaed through real points. In a simi- 
 lar sense we may say that real lines pass through imaginary 
 points. Thus the equation to the polar of {x'y') may be 
 said to pass through the two real, coincident, or imaginary 
 points of tangency, according as {x'y') is without, on, or 
 within the circle. We attach no geometrical meaning to the 
 
POtE AND POLAR. l23 
 
 tei-in ' imaginary point ;' we simply mean that the imaginary 
 values of x and y, obtained from the equations 
 
 xx' + yy' = r', x' + y'=r' (1), 
 
 satisfy the real equation xx' + yy' = r^, which is- obviausly 
 trae. We will examine this a little more closely, that we 
 may see the conditions, under which two imaginary points 
 may lie upon and determine a real straight line, as it is evi- 
 dent that not every two such points wiU do so. Since the 
 imaginary roots of a quadratic equation assume (Appendix) 
 the form a+ySV— 1, a— /SV — 1, it follows that the two 
 values of a, obtained by eliminating y between equations (1), 
 may be assumed to be 
 
 a;, = a + /3V^, a-, = a - /3 V^^ ; 
 
 and the corresponding values of y will be in like manner 
 
 ^j = 7'+Sv — .1, ^, = 7 — Sv — 1. 
 
 The equation to the straight line through (x^y^, {x^y^ 
 will be (Art. 30) . : 
 
 a;-a-/3V~l~-2^V~l' 
 or {y-i)^={x-a)h, 
 
 which is a real straight line. Thus we see that a pair of 
 imaginary points, such as the above, will always determine 
 a real straight line ; also the middle point between them is a 
 real point on the line, whose co-ordinates are 
 
 «f 4- a5„ y. +'.yj 
 
 a; = -iy-^ = a, y^-'-^^ = y. 
 
 Any real straight line will contain any number of ima- 
 ginary points, since any imagjhary value of one variable will 
 
124 
 
 THE CIRCLE. 
 
 give us a corresponding imaginary value of the otlier; but 
 an imaginary straight line will pass through one real point 
 only, namely that for which (Art. 65) the real and the ima- 
 ginary parts of the equation vanish simultaneously. 
 
 128, We saw (Art. 124) that the polar of a point (a'y) 
 is perpendicular to the line joining {x'y") with the centre. 
 Also, if P be the point {x'y'), QB the polar {xx' +yy' = 1^), 
 and CLE the perpendicular from C, we have (Art. 52) 
 
 CB= j; 
 
 {x''+y'^)i' 
 
 but CF={x" + y")K CL = r. 
 
 .-. CR.CP=CU; 
 hence we have an easy geo- 
 metrical construction for the 
 polar ; for, if in CP or CP pro- 
 duced (according as 0P> or 
 < GL) we take a poiiit B, so 
 that 
 
 CB : CL = CL : CP, 
 
 and through B draw BQ perpendicular to CR, BQ will be 
 the polar of P. If 8P is. perpendicular to OR, 8P is the 
 polar of.fi. This is a particular case of the property proved 
 in the next article. 
 
 129. If any point (x'y) be taken on the polar of the 
 point (x"y"), then the polar of (x'y') shall pass through 
 
 ■(x"y")- 
 
 For the equation to the polar of {x"y") is 
 xx" + yy"'=r^, 
 a.nd the condition that (x'y") should lie on this line, is 
 a;x" + y'y"==r^; 
 
POLE AND POLAR. 
 
 125 
 
 but this is also the condition to be fulfilled in order that the 
 polar of {x'y), whose equation is 
 
 xx' + yy' = r* 
 should pass through the point {x"y"). 
 
 Coil. Any straight line (Ax +By+C = 0) can be written 
 in the form 
 
 Ar' Br' , ^ 
 
 which is of the form xx' + yy' - r" = 0, and is therefore the 
 
 Ar^ Br' 
 polar of a point j^, — yj-; hence the preceding propo- 
 
 G 
 
 a 
 
 sition may be thus enunciated : If points he taken in any 
 straight line, the polars of these points will all pass through 
 a fixed point. This fixed point is, from above, the pole of 
 the straight line. 
 
 130. The polar of the intersection of two straight lines is 
 the straight line which joins their poles. 
 
 Let AP, BP be the two lines 
 intersecting in P, and let A', B' 
 be their poles. Then, since the 
 point P is taken upon AP, the 
 polar of A', therefore the polar 
 of P passes through A'. Simi- 
 larly the polar of P passes 
 through B' ; therefore A'B' is 
 the polar of P. Hence, if any number of straight lines meet 
 in a point, their poles lie in one straight line. 
 
 *COE. If there be two triangles ABC, cibc, such that A 
 is the pole of he, B of ca, and G of ab; then a is the pole, 
 of BG, b of CA, and c of AB. Such triangles are called 
 conjugate. 
 
126 THE CIRCLE. 
 
 Also, if ABO be a triangle, such that A is the pole of 
 BC, and B o{ CA; then G is the pole of AB. The tri- 
 angle ABC is said to be $elf-conjugate. 
 
 *Ex. A triangle and its conjugate are homologous; that is, the intersec- 
 tions of the corresponding sides lie in one straight line, and the straight lines 
 joining the corresponding angles meet in a point. 
 
 Let ABC, dbc be the triangles (fig. Art. 104, Ex. 2), and let A, B,C, 
 be (x!y'), (x"y"), (x"'y'") '< tlien, since Aa passes through the intersection of 
 the polars of {xf'y"), (x"'y'''), its equation is (Art. 43) of the form 
 
 (xx" +yy"- r') + k {xx'" + yy'" - r") = ; 
 
 and, determining Jc by the condition that Aa passes through («'/), we have for 
 the equation to .4 a 
 
 (x'x'" + i/y'" - r=) (ax" + yf - r=) - (xV +y'y'' - O K" + VV'" - »•') = 0. 
 Similarly for Bh and Cc we have 
 (« V + yY - r=) {xx'" + yyf" - r"^ - {x"x!" + y"f - r^) (aa/ + yj-' - r=) = 0, 
 (!>!"*"'+2/'y"-0 (a»'+yy'-r«) - {<i!ix!"+y'f' -r^) {xx"+yjj"-,r-^)=0; 
 and by Art. 93 these three lines meet in a point. 
 
 The converse follows from Art. 104, Ex. 2 ; but either of the propositions 
 may be proved from the other, as follows. Since A is the pole of 6c, and o 
 is the pole of BG, 
 
 therefore P, the intersection of BC, be, is the pole of .4 a ; 
 
 similarly Q, CA, ca, Bb,, 
 
 and R, AB, ab, Cc. 
 
 If the three poles P, Q, B lie in one straight line, the three polars Aa, 
 Bb, Cc win (Art. 129, Cor.) meet in a point; and, conversely, if the three 
 polars Aa, Bb, Cc meet in a point, the three pples P, Q, R will (Art. 130) lie 
 in one straight line. 
 
 131. If in the equation to the tangent, 
 
 ■we write x, x for a;, x and y', y for y, y\ respectively, the equa- 
 tion remains unchanged. A careful examination of the proofs 
 of Arts. 123— 127, 129, 130, will convince the student that 
 they all depend upon this property of the equation, and that, 
 if the equation to the tangent of any curve possesses this 
 
INTERSECTION OF TWO CIRCLES. 127 
 
 property, all the above theories of poles and polars are true 
 for that curve. It is very important to see this clearly, as 
 we shall not repeat the proofs. 
 
 132, To find the locus of the middle points of any system 
 of parallel chords in a circle. 
 
 Let. '^^^^Lul^l 
 
 c s 
 
 be the equation to any one of the chords of the system, (ic'y') 
 its middle point ; then, for its intersection with the circle, 
 (af + y^ = »•*), we have, as in Art. 114, 
 
 Z' + 2 {ex' 4- sy') l + x'^ + y'" - r' = 0, 
 
 I being, the distance from the middle point {x'y') to the points 
 of section of line and circle : hence the roots of this equation 
 are equal and of opposite signs, and therefore (Appendix) we 
 have 
 
 ex 4- sy = 0, 
 
 a relation which is true for the co-ordinates of every middle 
 point, since c and s are the same for every chord of the 
 system, and therefore the equation 
 
 ca; + s^ = 
 
 is the equation to the locus of the middle points, and evidently 
 represents a straight line through the centre and perpen- 
 dicular to the chords, a result agreeing with Euc. iii. 3. 
 
 133. The equations to two circles are given ; to find the 
 equation of the straight line which joins their points of inter- 
 section. 
 
 Let the equations to the two circles be 
 
 a? + y^ + Dx + Ey + F=0 (1), 
 
 , £c' + / + X>'a: + jS"yrfi^'=0 (2); 
 
128 THE CIRCLE. 
 
 then, by Art. 43, if h be any constant, 
 s,?+f+Dx+Ey+F+k{x''^f + D'x + E'y + F)=^0....{2i) 
 
 is the equation to some locus passing through the points 
 where (1) and (2) intersect, since it is satisfied by those values 
 of a; and y which satisfy (1) and (2). The equation may be 
 written 
 
 {l+k)x'+{l+h)f+{B-\-kD')x + {E+hE')y + F+1cF'=0, 
 
 which evidently represents a circle (Art. 107) since it can be 
 reduced to the form 
 
 x^ + f+Px-\-Qy + R = (i, 
 by dividing by 1 + ^. 
 
 If ;fc = — 1, or (2) be simply subtracted from (1), the 
 equation becomes 
 
 iP - D') x+ {E - E') y + F -F = 0, 
 
 which is therefore the equation to a straight line passing 
 through the intersections of (1) and (2). 
 
 Cor. Equation (3) may be made to represent any parti- 
 cular circle which passes through the intersection of (1) and 
 (2). For suppose that it passes through some other point 
 {x'y) ; then, if the values as' and y' are substituted for x and 
 y in (3), we shall have an equation to determine the value of 
 k in the circle which we wish to represent. The circle is 
 now completely determined, as we should expect, since only 
 one circle can be described through three points, i.e. about 
 a triangle. 
 
 *134. Let the equation to any circle (Art. 106, Cor.), 
 referred to rectangular axes, be 
 
 jc' + 2r'+i?a! + % + F=0 (1); 
 
INTERSECTION OP TWO CIRCLES. 129 
 
 then (Art. 107) it can be written in the form 
 
 (a;-a)^+(2/-&)'-7^=0 (2), 
 
 where the point {ah) is the centre, and r the radius. Let S 
 denote the left-hand member of (1) and (2) ; then, since 
 [x — aY+{y — hy is the square of the distance of any point 
 {xy) in the plane from the centre {ah), it is evident (Euc. 
 1. 47) that 8 is the square of the length of the tangent drawn 
 from (xy) to the circle. It is easily seen that 8 is positive, 
 zero, or negative, according as {xy) is without, on, or within 
 the circle. 
 
 *135. We have seen, by Art. 133, that, if 8 and 8' are 
 symbols standing for the expressions 
 
 x^ + y'' + Dx + Ey + F; a?+y'' + D'x + E'y + F', 
 
 so that 
 
 8=0 (1), 8' = (2) 
 
 are the equations to two circles, then the equation 
 
 8-8' = (3) 
 
 is the equation to a straight line, and that, if the circles 
 intersect, the points of intersection will lie on (3). But (3) is a 
 real line, whether the circles intersect or not, and in the latter 
 case (Art. 127) is satisfied by the imaginary values of the co- 
 ordinates of intersection. In either case it possesses the fol- 
 lowing important property with regard to the circles; if, from 
 any point of it, straight lines he drawn to touch hoth circles, the 
 lengths of those lines are equal; for the squares of the lengths 
 of the tangents to (1) and (2) from any point {xy) on (3) are 
 (Art. 134) equal to >S^ and 8' respectively, and (3) asserts 
 that these are equal. The line (3) is called the Radical Axis 
 of the two circles. 
 
 9 
 
130 THE CIRCLE. 
 
 *136. The three radical axes belonging to three given 
 circles meet in a point. 
 
 Let the equations to the three circles be 
 ^=0, ;8" = 0, S" = 0; 
 
 then the equations to the radical axes of these circles taken 
 two and two together, are 
 
 8-8' = 0, 8'-S" = 0, 8" -8=0. 
 
 At the point where the straight lines represented by the 
 first two of these equations intersect, we have 
 
 S-8' = 0, 8'-8" = 0, 
 
 hence 8" -8=0; 
 
 that is, the third straight line passes through the intersection 
 of the first two. 
 
 *137. If in the equation to one of the circles, 
 
 {x-aY+(7/-hy-r'' = 0, 
 
 ■we suppose r to vanish, the circle is reduced to the point 
 (ab) ; hence, if we subtract this from the other equation, the 
 radical axis is now a straight line, such that the tangents 
 from any point of it to the circle are equal in length to the 
 distance of that point from a given point (ab). 
 
 If both radii vanish, and both circles become points, the 
 radical axis becomes a straight line, every point in which 
 is equally distant from two given points. 
 
 The student will find it easy to prove, that the radical 
 axis is perpendicular to the line joining the centres of the 
 two circles, and is consequently the common tangent, when 
 the circles touch. 
 
 *138. If there be two unequal circles which do not inter- 
 sect, there will evidently be two points, on the indefinite 
 
EXAMPLES VII. 131 
 
 gtraight line joining their centres, from each of which a pair 
 • of tangents common to the two circles may be drawn. The 
 -points will lie, one between the centres, and the other ex- 
 terior to the smaller circle. They are called the Cmtres of 
 Similitude^. 
 
 EXAMPLES VII. 
 
 1. To find the centre and radius of the circle 
 
 2. Investigate the line or lines represented by the equation 
 
 x^ + xy^ — scV — xr'-ri/' + r^ = 0. 
 
 3. Find the common chord of two circles 
 
 {x-iY+{y-2y=6. {x-2y + (^-3y=8. 
 
 4. To find the equation to a straight hue which passes 
 through the centres of the two circles 
 
 x'+2x + i/' = 0, y'+2y + x' = 0. 
 
 5. To find the equation to a circle having for its diameter the 
 straight line joining the points of intersection of the line, y = mx, 
 and the circle, y^ = 2rx — a?. 
 
 6. Find the equation to the circle, the diameter of which is 
 the common chord of the circles 
 
 7. What is represented by the equation 
 
 a;(a!-2) + y(y-4) + 8 = 0? 
 
 1 The properties of these poiutB are discussed at le^gth in Sahnon's Conic 
 Seciiom. 
 
 9-2 
 
132 EXAMPLES VII. 
 
 8. Find a relation between the coefficients of the equation 
 
 in order that (1) the axis of x, and (2) the axis of y, may be 
 tangents to the circle. 
 
 9. To find the inclination to the axis of x of 'the tangents 
 drawn from any point (icV) to the circle whose equation is 
 
 (a;-a)V(2/-6y-r'''=0. 
 
 10. To find the relation between the quantities a, h, r, in 
 
 order that the line - + y = 1 may touch the circle a? + ^ = 7^. 
 ah 
 
 11. To find the. equation to a circle, the centre of which 
 is at the origin of co-ordinates, and which is touched by the line 
 
 y = 2x + ^. 
 
 12. To find the intercepts on the axes of co-ordinates of 
 the tangent to a circle {of + 'i^ = r^), drawn parallel to a given 
 straight line, (a; cos a + y sin a=p). 
 
 13. If 2a', 2a" be the inclination of two radii of a circle, 
 a? + y^ = r^, to the axis of x, to find the equation to the chord 
 joining the extremities of the radii. 
 
 14. If the pole always lie on a line 
 
 a* b ' 
 
 and the equation to the circle is oi? + y^ = r^, the equation to the 
 polar is of the form 
 
 {ax — 7^+k (fiy - r^) = 0, 
 
 where h is any constant. 
 
 15. If the pole of a straight line with regard to the circle 
 a? + y' = T^ lie on the circle x' + y^= ir', the polar will touch the 
 circle 
 
EXAMPLES VII. 133 
 
 16. Find the equation to the circle which has each of the 
 
 1 2 
 
 co-ordinates' of the centre = — k , and its radius = —j= , the axes 
 
 being inclined at an angle of 60°. 
 
 17. Prove that the circles 
 
 have only one common tangent, and find its equation. 
 
 18. Find the locus of the middle points of chords drawn from 
 the extremity of the diameter of any circle. 
 
 19. Shew that the polar of the point (scV) with regard to the 
 circle (a; — a)' + (y — bf = r^ is 
 
 (x-a) (x' -a)+{y-b) {y'-h) = r\ 
 
 20. Find the locus of the vertices of all triangles which have 
 a given base, and a given vertical angle. 
 
 21. Prove Euc. in. 31, from the resulting equation. 
 
 22. Tangents are drawn to a circle x' + i/^ = r', at two points 
 (a/y'), {x"y"); to find the distance of a point ihh) from a straight 
 line passing through the centre and the intersection of the two 
 tangents. 
 
 23. To find the equations to straight lines touching a circle 
 
 a? + y'=\0, 
 at points, the common abscissa of which is unity. 
 
 24. Fiad the equation to a straight line touching the' cii'cle 
 
 {x-af + l^-hY = ,^, 
 and parallel to a given line y = mx + o. 
 
 25. To find the equation to the straight line passing through 
 the origin of co-ordinates, and touching the circle 
 
 •x'' + y''-3x + iy=0. 
 
134 EXAMPLES VII. 
 
 26. To find the length of the common chord of the circles 
 
 {x-aY+iy-hf^r', (x-hf + {y-ay = r'. 
 
 27. Find the area between the two circles 
 
 a^+2x + y'+iy = 0, a^ + 2x+y' + iy=l. 
 
 ' 28. To find the length of the chord of a circle a^ + y' = r% made 
 by the straight line - + ^ = 1. 
 
 29. If from a given point S, a perpendicular be drawn to the 
 tangent PY at any point P of a circle, of which the centre is G, 
 and, in the line MF at right angles to CS and produced if 
 necessary, a point Q be taken, such that QM = ST, to find the 
 locus of Q. 
 
 30. Given the equation to a circle, and the chord of a circle; 
 shew that a perpendicular let fall upon the chord from the centre 
 bisects the chord. 
 
 31. Find the diameter of the circle 
 
 «" + y^ + 2xy cos <o = ax + hy. 
 
 32. In the equation Ax + By + C = 0, if C is constant, and 
 A and B vary, subject to the condition A" + £^=& constant, the 
 equation represents a series of tangents to a given circle. 
 
 33. Find the equation to the circle which passes through the 
 points (0, 0), (— 8a, 0), (0, 6cs), the axes being rectangular. 
 
 34. To find the locus of middle points of chords which pass 
 through a given point. 
 
 35. If on any radius vector through a fixed point 0, OQ be 
 taken in a constant ratio to OP, find the locus of Q. 
 
 36. The circles represented by the equation 
 
 (w + 1) (as^ + y')='ax + nby, 
 where n is arbitrary, have a common chord. 
 
EXAMPLES VU. 135 
 
 37. Prove algebraically that the angles in the same segment 
 of a circle are equal, and that the angle in a semicircle is a right 
 angle. 
 
 38. Two sides of a triangle are 6 and e, and they include an 
 angle A ; if these sides be taken as axes, the equation to the 
 circumscribed circle is 
 
 as" + y* + 2x1/ cos A — 6a; — cy = 0. 
 
 39. Given the base and vertical angle, to shew that the locus 
 of the point of intersection of the pei'pendiculars from the angles 
 on the sides is a circle. 
 
 40. Given base and ratio of sides of a triangle ; shew that the 
 locus of the vertex is a circle. 
 
 41. "When will the locus of a point be a circle, if the square 
 of its distance from the base, of a triangle be in a constant ratio to 
 the product of its distances from the sides ? 
 
 42. When will the locus of a point be a circle, if the sum of 
 the squares of the three perpendiculars from it on the sides of a 
 triangle be constant ? 
 
 43. Knd the locus of a point, the square of whose distance 
 from a, given point is proportional to its distance from a given 
 right line. 
 
 44. Given the base of a triangle, and m times the square of 
 one of its sides ± n times the square of the other = a constant ; 
 find locus of the vertex, find centre and radius of resulting circle, 
 and where it cuts base. 
 
 45. Find the equations to the circles which touch the three 
 lines, referred to rectangular axes, 
 
 x = a, y=2b, y=2b'. 
 
 46. The locus of the centres of all circles inscribed in all 
 right-angled triangles on the same hypotenuse is the quadrant 
 described on the hypotenuse. 
 
136 EXAMPLES VII. 
 
 47. The equation to a circle is y V a" = a (y + a); what is the 
 equation to that diameter which passes through the origin of co- 
 ordinates ? 
 
 48. To find the equation to a circle referred to two tangents 
 at right angles, as axes. 
 
 49. If through any point of a quadrant whose radius is S, 
 two circles be drawn, touching the bounding radii of the quadrant, 
 and r, r be the radii of these circles, rr = Ji'. 
 
 50. To find the equations to the straight lines which touch 
 both the circles 
 
 51. To find the equation to the circle which touches the 
 three straight lines, referred to rectangular axes, 
 
 . = 0, y = 0, %|=1. 
 
 52. To find the equations to two circles, which touch rect- 
 angular axes of X and y, and pass through a given point {ab). 
 
 53. The straight lines joining the angles of a triangle with 
 the points in which the escribed circles touch the opposite sides, 
 meet in a point. 
 
 54. In any circle draw a chord AB ; from the middle point 
 E of the lesser segment draw any straight line cutting AB in C, 
 and meeting the circumference in Z) j join AD, and in AD take 
 AF=AG', find locus of P. 
 
 55. The axes Ox, Oy cut a cii-cle in points A, A', B, B re- 
 spectively; to compare the values of x, y, at the intersection of the 
 chords AB', A'B. 
 
 56. Determine the magnitude and position of the circle 
 
 p' - 2p (cos e + 73 sin e) = 5. 
 
 57. Find the locus of a point, such that, if straight lines be 
 drawn to it from the four corners of a given, square, the sum of 
 their squares may be invariable. 
 
EXAMPLES VII. 137 
 
 58. ACB is the segment of a circle, and any chord AC is 
 produced to a point P, so that AG : CP in & given ratio; re- 
 quii-ed to find the locus of P. 
 
 59. Find the equation to a straight line which cuts a given 
 circle, when the straight lines drawn from the points of intersection 
 to the centre contain a right angle, and one of them is inclined 
 to the axis of aj at a given angle. 
 
 60. Two straight lines revolve uniformly in one plane about 
 one extremity, the one moving twice as fast as the other. Find 
 the locus of their point of intersection, supposing them to begin 
 to move together in the same direction, from the straight line 
 joining their fixed extremities. 
 
 61. If any number of circles touch one another in one point, 
 all their polars, which correspond to a common pole, pass through 
 a single point. 
 
 62. To find the locus of the pole, when the polar of a given 
 circle always passes through a given point. 
 
 63. In the sides AB, AG of a given triangle ABC, take 
 
 two points M, W, such that j-j^ = -=^ , and shew that the circle 
 
 described about the triangle AMN will always pass through a 
 given point. 
 
CHAPTER VIII. 
 General Equation of the Second Degree. 
 
 139. The equation of the first degree has given us but 
 one species of line, viz. the straight line. We saw, in the 
 case of the circle, that an equation of the second degree may 
 represent a curve limited in every direction ; and the case 
 where it represents two straight lines, shews us that it may 
 represent loci extending to infinity. These are but particular 
 cases of the equation of this degree, of whiph the most 
 general form is 
 
 Ax^ + Bxy + (7/+ DxJrEy + F= 0. 
 
 Our object in the present chapter is to interpret this 
 equation, which for brevity we shall write (xy) = ; and 
 we shall shew how, by a proper selection of origin and axes, 
 we may in every case reduce it to a form, from which we 
 shall be able to trace the locus, and deduce its most re- 
 markable properties. It will be shewn in a future chapter, 
 that much of this can be accomplished without reduction of 
 the equation; and properties so established have the ad- 
 vantage of applying to every particular form that the locus 
 may assume ; but the proofs are of necessity more cumbrous 
 and difficult to a beginner, than those which we are about 
 to use, 
 
 140. If a point be so situated with regard to a locus, 
 that all chords of, the locus, drawn through the point, are 
 bisected in it, the point is called th.Q. centre of the locus. 
 
THE CENTEE. 
 
 13.9 
 
 y 
 
 ^ 
 
 ^ 
 
 C M J X/ 
 
 Suppose C fo be the centre of a locus and the chord 
 PGQ to meet the locus in P 
 and Q ; then GP= CQ, and, if 
 we take G as the origin of co- 
 ordinates, it is evident that the 
 co-ordinates of Q are the same 
 as those of P, but with opposite 
 signs ; i.e. if the co-ordinates of 
 P be a;', y', the co-ordinates of 
 Q are — x', — y ; and this is true, whatever be the position 
 of P. Hence, the centre being origin, for every point {cc'y'), 
 whose co-ordinates satisfy the equation, there will be a cor- 
 responding point (—oo, —y), whose co-ordinates also satisfy 
 the equation. From this it follows, that, when the centre is 
 the origin, the equation will not be altered by writing — x, 
 — yfor x and j. 
 
 Also, if for every point P{x'y') there is a correspondiug 
 point Q{—x', — y'), the origin is the centre; for, since the 
 triangles PClf, ^CiNT will be equal in every respect, CP= CQ; 
 and, since the angles PCM, Q GN are equal, GP and CQ are 
 in the same straight line ; therefore every chord passing 
 through C is bisected. Hence, conversely, if an equation 
 is not altered by writing — x and — y for x and y, the origin 
 is the centre of the locus. 
 
 Now the equation <f) (xy) = cannot remain unaltered, 
 as above, unless the terms Px and Ey vanish from it ; hence 
 if we select the centre of the locus for origin, we shall 
 simplify the equation by getting rid of these two terms. 
 
 141. In order, then, to find the centre of the locus, we 
 must transfer the origin to a point (x'y), and then observe 
 what values of x', y , make the new coefficients of x and y 
 vanish. These values will be the co-ordinates of the centre 
 with reference to the original axes. 
 
liO GENERAL EQUATION OF THE SECOND DEGEEE. 
 
 Writing (Art. 70) a; + a;' for x,y+y' for y in the general 
 equation, we have ' 
 
 A{x+x'y+B(x+x')(jf+y')+C(jj-Vy'f+D{x+x)+E{r/+y')+F=0, 
 
 or Ax^ + Bxy+Gy^-it-2Ax' x + 2Gy y + (ji {x'y) = 0, 
 
 + By' +Bx' 
 +D +E 
 
 where ^ {a/y) = Ax'' + Bxy + Gy"" + Bd-\- Ey' + F. 
 
 Hence the co-ordinates of the centre, with reference to 
 the original axes, will be determined by the equations 
 
 2Ax' + By' + D = Q (1), 
 
 2Gy' + Bx' + B=0 (2). 
 
 These two equations wiU, in general, give one and only one 
 value of x and y' ; hence, loci of the second- degree have in 
 general one, and only one, centre. 
 
 Its co-ordinates are found, by solving the above equations, 
 to be 
 
 BE-2CD ,_ BD-2AE 
 ^~ B'-4iAG ' ^ ~ B'-iAG ' 
 
 Equations (1) and (2) may be thus remembered^. For (1), 
 take those terms only of the general equation, which involve 
 
 1 The reader of the Differential Caloulua 'will see, that the equations of the 
 centre are obtaaned by differentiating the equation <f> [ocy) = with respect to 
 X SiUd y. The equation ^(xy) = 0, when the origin is transferred to a point 
 (x'y') is ^ (aj'+a;, y'+y)=0, or, by a familiar expansion, 
 
 + terms which iuvolve higher differential coefficients, and therefore vanish; 
 hence, in order that the terms of the first degree in x and y may vanish, we 
 
 must have 13=0, ^^. = 0. 
 da! mf 
 
CENTRAL LOCI. 141 
 
 X ; multiply eacli term by tlie index of x in it, and diminish: 
 that index by unity. Equation (2) may be obtained similarly; 
 by substituting y for x in the above rule. Thus the equa' 
 tions for the centre of the locus represented by 
 
 3a;= + 2xy + 3/ - IGy + 23 = 0, 
 
 are 6a5 + 2^ = 0, and 2a; + 6y-16 = 0. 
 
 Cor. Since the equations for the centre do not involve 
 the constant term F, it follows that all central loci, whose 
 equations can be written so as to differ in the constant term 
 only, are concentric. 
 
 *142. The calculation of the value oi ^{xy), when {x'y) 
 is the centre, may be thus facilitated ; 
 
 24> {x'y) = {2Ax'+By'+D) x'+ (2 Cy'+Bx'+E)y'+Dx'+Ey'-\^2F; 
 
 but these two first terms are each = 0, when {dy) is the 
 centre ; therefore 
 
 2<l>{x'y')=Dx' + Ey'-\-'2.F; 
 
 hence the equation ^ {xy) = 0, becomes, when the origin is 
 transferred to the centre, 
 
 Ax' + Bxy+Of + n^+E^^+F^O; 
 
 that is, we can transfer the origin to the centre, by sub- 
 stituting the halves of the co-ordinates of the centre, for x 
 and y respectively, in the terms Dx and Ey. 
 
 Ex. The equations for the centre of 
 
 3a;« + 2xy + Sy" - 16y + 23 =0, 
 are 6x+2y=0, and 2a!+6y-16=0; whence x=-l, y=3 are the co-ordi- 
 nates of the centre 
 reduced equation, 
 
 nates of the centre; and writing ^ f or j^ in the term -ISy, we have for the 
 
 3x''+2xy+Sf-l=0. 
 
142 GENERAL EQUATION OF THE SECOND DEGBEE. 
 
 143. We see then, that it will be always possible to find 
 one, and only one, pair of values for the co-ordinates of the 
 centre, except when .B" - 4^ (7 = 0. Hence the loci of the 
 second degree may be divided into two classes : (i) loci which 
 ha'ie a centre, where B^ — 44 C is not zero ; and (ii) loci which 
 in general have not a centre, or rather, whose centre is in- 
 finitely distant, where 5" -44 (7=0. It will be seen in 
 Art. 157, why we say 'in general.' 
 
 144. We shall first consider the case of Central Loci. 
 
 We see by Art. 141, that, in the case of central loci, the 
 general equation may, by taking the centre {x'y) of the 
 locus as origin, be reduced to 
 
 Ax' + Bxy +Cf + ^ {x'y') = 0. 
 
 We next proceed to inquire, whether, by any change in 
 the direction of the axes, we can get rid of the term involving 
 xy, as it will be seen hereafter, that this will greatly facilitate 
 our inquiries into the form and properties of the curve. 
 Now it is manifest, that, if we can so transform the axes as 
 to get rid of the term involving xy, the equation will be left 
 in the form 
 
 where, if any value be given to one of the variables, the 
 other will have two equal values with opposite signs ; hence, 
 ill this case, each axis will bisect all chords parallel to the 
 other. 
 
 145. We have not hitherto supposed the axes necessarily 
 rectangular ; but the generality of our reasoning would not 
 have been affected by such a supposition, since, if they had 
 been oblique, by transforming them to rectangular axes, we 
 should (Art. 75) have obtained an equation of the same 
 degree, and of the sanie form as the one we have assumed. 
 
AXES OF THE LOCI. 
 
 143 
 
 We shall now, for the sake of simplicity, suppose such a 
 change, if necessary, to have been made, and the axes to 
 be rectangular; then, if we turn them round through any 
 angle d, we must (Art. 71) substitute in the equation 
 
 A^^Bxy + Cy' + <p {x'y') = (1), 
 
 for X, xcQs9 — y sin 6 ; for y, xsmO + y cos d. 
 
 Substituting, and arranging^ the terms, we have 
 
 AcQsW si?-2Asixieoosexy+AaW6 y''+<j}{x'y')=0. 
 + 5 sin ^ cos 6 +5003"^ -isin^cos^ 
 
 + Csin''^ -Bsm'd +Gcos'd 4. .(2). 
 
 + 2Csin0oose 
 
 If now we put the new coefficient of xy = 0, we obtain 
 
 B 
 
 tan 20 = ■ 
 
 (3), 
 
 A-G 
 
 an equation from which we may determine the angle 0, 
 through which axes must be moved, in order that the term 
 involving xy may vanish. 
 
 As the tangent of an angle may have any magnitude, it 
 follows that this equation will always give real values for 20. 
 There will be an infinite number of solutions ; for, if 2a be 
 one value of 20 which satisfies the equation, then the equa- 
 tion is satisfied, if 2^ = wtt + 2a, where n is any integer. 
 
 WTT 
 
 Hence the values of are expressed by ^ + a, and we 
 obtain a series of angles 
 
 a, 
 
 + a, ir + a, 
 
 "2" + a, &c. 
 
 The only difference, that will be made by selecting dif- 
 ferent values of 0, will be, that the axis of x and y in one 
 
 ^ The student will find these transformations easy to write, if he will 
 arrange the framework as aboye, and theUj insert the 00-effioients of x\ xy, 
 &c., as they occvu: in the expansions. 
 
144 GENERAL EQUATION OF THE SECOND DEGREE. 
 
 case may occupy the position of the axis of y and x in 
 another, or the positive and negative directions of the axes 
 may change places. Hence, when the centre is origin, there 
 exists one system of rectangular axes, and one only, about 
 ■which the curve is so situated, that each axis bisects all 
 chords parallel to the other. Also there can be no such 
 system with any other origin ; for a straight line bisecting ,a 
 system of parallel chords must pass through the centre, 
 which is the bisection of one of them. These axes are 
 called the axes of the curve. We shall see hereafter that 
 their position is that of the figures. 
 
 Cor. 1. Since the direction of the axes depends upon 
 the quantities A, B, G only, it follows that loci, whose 
 equations can be written so as to have the first three terms 
 the same, have their axes parallel. 
 
 CoK. 2. If we have B = (), and A=C, the new coeffi- 
 cient of xy vanishes for every value of 0, and tan 20 becomes 
 indeterminate. Hence in this case we may take any rect- 
 angular axes whatsoever, without introducing the tenn xy 
 
 into the equation, 
 is then a circle. 
 
 This agrees with Art. 106, for the curve 
 
 *146. The axes of the locus, then, make angles and 
 90° -1- with the axis of x, where is determined from the 
 equation 
 
 tan 20 = - 
 
 B 
 
 A-C 
 
AXES OF THE LOCUS. 145 
 
 hence, the centre being origin, the equations to the axes are 
 
 y — x tan 0=0, j^ -f a; cot = 0, 
 or, as one locus, 
 
 2/" + (cot - tan 0) a;?/ - a;' = 0, 
 
 or, since cot Q — tan Q = ^tt; = 2 — =r — , 
 
 tan 26' B 
 
 The student -will observe that this is (Art. 68) the equa- 
 tion to the straight lines which bisect the angles between 
 the straight lines represented by 
 
 Ax' + Bx2/ + Cf=0, 
 
 a coincidence which will be hereafter (Art. 186) explained. 
 
 147. We have shewn (Art. 141) that the coeflficients of 
 the first three terms of the equation ^ (tcy) = 0, are aot 
 altered by a transfer of the origin ; we shall now shew, that, 
 when the axes are moved through an angle 0, and the new 
 coefficients denoted by A', B', C", we have the relations, 
 
 A'+G'=A+0, B"-4<A'0' = B'-iA0. 
 From equation (2) of Art. 145 we have 
 
 A' = A cos'0 + Bsme cos + Gsm^d, 
 C = A sin'(9 -B&mO cos 6 + Ocos'e; 
 .■.A'+C' = A + 0, 
 
 A' - 0' = Bsm20 + (A-C) cos20; 
 .: iA'C = {A' + Of - {A' - Cy 
 
 = {A+Gy-{Bsm29+{A-0)cos2eY. 
 
 Also B" = {B cos 20 -{A- G) sin 20}" ; 
 
 .-. F^-4iA'C'^B' + {A-Cf-{A+Gf = B'-4:AG. 
 
 10 
 
146 GENERAL EQUATION OF THE SECOND DEGREE, 
 
 This result is very important, because we shall shew that 
 the nature of the curve depends upon the sign of B^ — 4A G. 
 
 *148. The preceding article furnishes us with an easy- 
 method of arriving at the actual values of A' and C. For, 
 since the equation can be reduced to the form 
 
 A'x'+Gy + tl>(xy) = 0, 
 
 B' is =0, and we have 
 
 A'+C' = A+C, -iA'G'^B'-iAG; 
 
 hence, since A' and G' are the roots of the equation 
 
 z''-{A'+G')z + A'G' = 0, 
 
 they are the roots of 
 
 , , . , ^, B'-4>AG _ 
 
 an equation which need not be remembered, since it can 
 easily be written down from the reasoning above. 
 
 The above method does not enable us to say, which of 
 the roots is A' and which C". If we simply wish to find the 
 nature of the locus, without accurately fixing its position 
 with regard to the new axes, this uncertainty is not im- 
 portant ; for the two equations 
 
 A'x' + G'f + ^ («y) = 0, (7V + ^y + ^ (a;'^') = 0, 
 
 wiU represent the same locus, the axes of x and y respec- 
 tively in the one being called the axes of y and x in the 
 other, 
 
 *14!9. If we wish to select the appropriate roots for A 
 and C", we may proceed as follows. As in Art. 147, we 
 have 
 
 ^'-C' = (^-0)cos2^ + 5sin2^ (1). 
 
THE LOCUS KEFEEBED TO ITS AXES. 147 
 
 And, from the value of tan 29 in Art. 145, 
 
 O = {A-C)sm2e-Bcos20 (2). 
 
 Squaring and adding (1) and (2), we have 
 
 XA'-Gy={A-Cf+^, ov A'-G'=±J{A+Cy+B''-4!AG. 
 
 If we determine to take for 29 the smallest positive angle 
 that satisfies equation (3), Art. 145, the value of A'— G' must 
 be taken with the sign the same as that of B. For, if we 
 eliminate cos 20 between equations (1) and (2), we have 
 
 B {A! - C) = sin 29 {{A- Gf + BF]. 
 
 But 29 must be less than 180°, since for angles between 0° 
 and 180°, the tangent passes through every possible value ; 
 hence sin 29 is positive, and so also is {A — GY + E'. It 
 foUows then that A — G' is of the same sign as B. 
 
 Hence, according as B is positive or negative, 
 
 A'+G' = A + G, A'- G' = ±J{A+Gy + B'-4AG; 
 
 whence A' = ~{A+G± JiA+ Gf + B'-iAG}. 
 
 That is to say. A' is the root of the equation 
 
 W — 4AG 
 z'-{A + G)z-^-^^ = 0, 
 
 which is taken with the radical of the same sign as B. 
 
 150. We see then, that the equation (j) (xy) = 0, may, in 
 the case of central loci, always be reduced to 
 
 ^v+'oy+</.(ajy)=o. 
 
 We shall now divide these loci into two classes. 
 
 Glass I. W — 'iiAG negative. Since the sign of this 
 quantity remains the same after transformation, and the 
 coefficient of xy is now = 0, -^A'C' must be negative, i. e. 
 
 10—2 
 
148 GENEKAI, EQUATION OF THE SECOND DEGREE. 
 
 A' and C must have the same sign, and the equation may be 
 written in the form 
 
 where P and Q are positive quantities. We shall now have 
 three cases. 
 
 (i) If S is positive, we have a locus not yet investigated, 
 which is called an Ellipse. 
 
 (ii) If R = 0, the locus (Art. 65) is the two imaginary 
 straight lines 
 
 '/Px + \r^^ = 0, VPa;-V^y = 0, 
 
 which meet in the real point a; = 0, ^ = ; or, as will appear 
 hereafter, is the Ellipse indefinitely diminished. 
 
 (iii) If B is negative, the equation (Art. 66) can be satis- 
 fied by no real value of x and y, and the locus may be called 
 an imaginary Ellipse. 
 
 The circle belongs to this class ; for its most general equa- 
 tion may (Art. 110) be written, so that A=C=1, and 
 5 = 2 cos £0 ; hence B' — iAC = 4! cos^'to — 4, which is always 
 negative. It is in fact a particular case of locus (i), when 
 P = Q. 
 
 Class II. ^ - 4^ C positive. Here -4^'C'mustbe 
 positive, i.e. A' and C must have different signs, and the 
 equation may be written in the form 
 
 Px'-Qf = R, 
 
 where P and Q are positive. In this case the sign of E will 
 make no difference in the natwre of the locus, since the equa- 
 tion Qf — Px" = B will represent one of the same form, the 
 axis of X in the former equation having the same position 
 with reference to the locus, as the axis of y in the latter. 
 Hence we have two cases. 
 
SUMMARY OF THE METHOD OF EEDUCTION. 149 
 
 (i) When E is not = 0, we have a locus not yet investi- 
 gated, -which is called an Hyperbola. 
 
 (ii) When ^ = we have the two intersecting straight 
 lines 
 
 'JPx+'jQy = 0, 'JPx-^Qy = {i. 
 
 *151. To sum up hriefly. In order to reduce the equa- 
 tion to a Central Locus, 
 
 (i) Find the Class (Art. 150) to which the locus belongs ; 
 and then, if £*— 4J.C is not = 0, 
 
 (ii) Write the equations of the centre (Art. 141), and 
 obtain the values of its co-ordinates. 
 
 (iii) Substitute the halves of these co-ordinates (Art. 142) 
 in the terms Dx and Ey, and so reduce the equation to the 
 form 
 
 Ax^ +Bxy+0f+(J3 {x'y') = 0. 
 
 The origin is now transferred to the centre of the locus. 
 
 (iv) To reduce the equation to the form 
 
 A'x+G-y+<f>{x'y') = 0, 
 
 we know (Art. 148) A' and C to be the roots of 
 
 £'-{A'+C')z + A'C' = 0, 
 
 , ,., ^ B'-4,A0 . 
 or z^-(A+C)s T = 0. 
 
 where A' (Art. 149) is the root of the latter equation taken 
 with the radical of the same sign as B. The axes have 
 now been turned througb an angle determined by the 
 equation, 
 
 TO 
 
 tan 29 = —: r~ . 
 
 A— G 
 
150 GENERAL EQUATION OP THE SECOND DEGREE. 
 
 .(1). 
 
 •Ex. Let the proposed equation be 
 
 5x^+2!e!/ + 5y^-12x-12y=0.. 
 
 Here 5' -440= -96, or the locus belongs to Class I. The equations 
 of the centre are 
 
 5a!+y-6=0, 
 x+5y-6=0, 
 
 whence x-=y=l. If then we transfer the 
 origin to a point C, so that 0M=CM=1, 
 C mil be the centre, and the equation with 
 the new axes Ox', 0)f, parallel to the old 
 ones, is found by writing x=\,y = \va. the 
 last two terms of (1), to be 
 
 5a!'' + 2^^ + 53;' -12=0 (2). 
 
 
 y 
 
 r 
 
 
 11' 
 
 \ 
 
 > 
 
 \ . 
 
 \ 
 
 /* 
 
 \ \" 
 
 / 
 
 
 m \ 
 
 K 
 
 Next, if we take tan£ 
 
 'A 
 
 -7V=^-P = oo, we have 29=90», e=45«; 
 - O o — o 
 
 hence, if the axes are turned through 45", equation (2) becomes 
 
 4'a;« + Cy-12=0 (3), 
 
 where A' and G' are the roots of 
 
 z'-102+24 = 0. 
 
 If we take A' with the positive sign of the radical, since 5 is positive, (3) be- 
 comes 
 
 8a!3+22^i'-6=0. ; 
 
 This is now seen to be the locus which (Art. 150) we have called an 
 Ellipse; and its position with regard to the new axes will be seen hereafter to 
 be that of the figure. As the original equation contained no absolute term, 
 it is evident (Art. 109) that is a point on the locus. 
 
 *152.* We may also shew that B'^ — iA'C is of the same 
 sign as ff - 4: A C, after ani/ transformation. Suppose the 
 axes to be rectangular, and we wish to make them inclined 
 at an angle a>. Let the axis of x remain the same, and let 
 the axis of 1/ be moved through an atogle 3t)° — w, as in Art. 
 73 ; then we must write 
 
 for X, 
 
 x + y cos w ; for y, 
 
 y sm (o, 
 
 1 From Salmon's Conic Sections. 
 
PARABOLA, 151 
 
 in equation (1) of Art. 145. In this way we obtain the 
 equation 
 
 Ax' + 2 A cos a xy + A cos" eo y' + 4> {xy) = 0. 
 
 + -B sin <o + B sin m cos m 
 + (7 sin'' ft) 
 
 Hence £'" - 4A' C = {B" - 4^ C) sin" «, 
 
 and, since sin'w is positive, the proposition is proved for 
 this transformation. It follows that it is true, if we change 
 from oblique to rectangular, and then to any other system of 
 oblique axes. Hence it is true for any transformation. 
 
 153. Class III. We shall now consider the case where 
 B^ — 4iAG = 0. We saw (Art. 143) that in this case the co- 
 ordinates of the centre become generally infinite; hence we 
 cannot destroy the terms Dx and Ey, by removing the origin 
 to the centre ; but, if we proceed as in Art. 145, and move 
 the axes through an angle 6, the new coefficient of xy will 
 vanish, when 
 
 B 
 
 tan 20 = 
 
 A-C 
 
 for the introduction of the terms Bx and Ey will not affect 
 the proof But, since (Art. 147) 
 
 5'" - 4J.' O'=B^-4tAC=0, 
 
 if 5' = 0, either A' or 0' must = 0, and the equation will 
 assume the form 
 
 G'f-^B'x + E'y + F' = (1), 
 
 where we have supposed A' to vanish. If we supposed C to 
 vanish, the equation would not represent a different form of 
 locus, but one having a situation with regard to the axes of x 
 and y, similar to the situation of the supposed locus, with 
 regard to the axes of ^ and x respectively. 
 
152 GENEEAl EQUATION OF THE SECOND DEGREE. 
 
 If vre transfonn the origin to a point {x'y), equation (1) 
 becomes 
 
 C'f + D'x + (2 0'y' + E')y + G'y" + Ux + E'y' +F' = 0; 
 
 and, if we take for d and y the only pair of values that can 
 be obtained from the equations 
 
 2 C'y' + JS'=0, Cy" + D'x + E'y + J" = 0, 
 
 the equation assumes its simplest form, namely, 
 
 Oy + D'x = 0, or y" = Lx.' 
 
 Precisely as in Art. 145, it will be seen that there is one' 
 system of rectangular axes, and one only, which gives the 
 equation to the locus in this form. The axis of x is now 
 called the axis of the locus, and the origin of co-ordinates is 
 called the vertex. 
 
 If D' = in (1), the equation becomes 
 C'f + E'y + F'=0, 
 
 which represents (Art. 64) two straight lines parallel to the 
 new axis of x, which are real and different, real and coinci- 
 dent, or imaginary, according as 
 
 ^'^-4C"i?"> = <0. 
 
 Hence, when E' — ^AC^O, we have 
 
 (i) A locus not yet investigated, which is called a 
 Parabola. 
 
 (ii) Two parallel straight lines. 
 
 (iii) Two coincident straight lines. 
 
 (iv) A locus, of which no geometrical conception can be 
 formed, called two imaginary parallel straight lines. 
 
 *154. The preceding article is valuable, chiefly on ac- 
 count of the general conclusions to which it leads us. The 
 
PARABOLA. 153 
 
 values of A' and C may be obtained as in Arts. 148, 149, 
 and the student may prove that C or A' vanish, according as 
 B is positive or negative, and that F' = F. But the calcula- 
 tion of the values of D' and H' is 'tedious; and the student 
 will find the method of Art. 156 more convenient for the 
 working of numerical examples. 
 
 155. The reduction of Art. 153 may be made by a 
 method suggested by the form of the equation itseE For, 
 since £^ — 4iA G=0, the first three terms iji the general 
 equation form a perfect square, and it may be written 
 
 {ax + ci/y+I)x + Fy + F=0 (1), 
 
 where a and c are written for /J A and i\/G. Now we 
 see from equation (2) Art. 52, that the lengths of parallel 
 straight lines, drawn from a point (xy) to meet the line 
 {Ax + By + C = 0), vary as Ax'+ By + C. since Ac + Bs does 
 not vary in that case. Suppose then, that the lengths of 
 straight lines drawn from {soy) to meet 
 
 ax + cy = (2), Dx + Ey + F=Q (3), 
 
 respectively, and parallel to (3) and (2) respectively, are 
 
 P{ax + cy), Q{Dx + Ey + F), 
 
 where F and Q are constants. Now equation (1) may be 
 written 
 
 P" {ax + cyY = --^Q{Dx + Fy + F); 
 
 this equation asserts, that the square of a straight line drawn 
 from any point {xy) on the locus, parallel to (3) to meet (2), 
 varies as the straight line drawn from the same point parallel 
 to (2) to meet (3) ; hence, if we take (2) for the axis of x 
 and (3) for the axis of y, the equation will assume the form 
 
 y^ = Lx. 
 
154 GENERAL EQUATION OF THE SECOND DEGREE. 
 
 *156.' The new axes used in the last article are in 
 general not rectangular. We shall now shew how to trans- 
 form equation (1) to the same form, the new axes being 
 rectangular. We shall suppose the co-ordinates in equation 
 (1) to be rectangular. 
 
 If h be any constant, (1) may be written 
 
 {ax + cy^-'kY^{B-1ak) x^ (E -^ch) y -^-F -¥ =^ ...i^). 
 
 The condition {AA + BB-=CS, Art. 47, Cor.) that the 
 two lines 
 
 ax + cy + Ji; = (3), 
 
 {D-2ak)x+{E-2cJc)y + F- k'' = (4), 
 
 should be at right angles, is 
 
 a{D-2ak) + c{i;-2ck)=0 (5), 
 
 from which we may always obtain a value, = « suppose, for k. 
 Substitute this value for k in (2), and for brevity write it 
 
 (ax + cy + Ky + J)'x + E'y + F' = (6), 
 
 where the two lines 
 
 ax + cy + K = (7), J)'x + E'y + F'=0 (8), 
 
 are at right angles to one another. Now (6) may be written 
 
 (ax+cy+Ky _ 'JB^+F"' B'x+F'y+F' 
 
 a' + c^ - a' + c' ^ ^JWTW ^ '' 
 
 and we know from Art. 62, that 
 
 ax + cy + K , B'x + E'y+F' 
 V^N^ 's/B'^ + F" ' 
 
 are the lengths of the perpendiculars from a poiat (xy) on (7) 
 and (8) ; hence, if we construct the two lines (7) and (8), the 
 equation (9) asserts, that the square of the perpendicular 
 
 ' From Salmon's Conic Sectims. 
 
PARABOLA. 155 
 
 from any point [xy) of the locus on the first line, varies as the 
 perpendicular on the second line. 
 
 Hence, if we transform our axes, and make the line (7) 
 our new axis of x, and (8) our new axis of y, then our new 
 y will be the perpendicular on (7), and our new x will be the 
 perpendicular on (8), and equation (9) is reduced to the form 
 
 y' = Lx. 
 
 If the lines (aa; + cy = 0) and {Dx + Ey + F=Qi) are at 
 right angles to one another, equation (1) may be written in 
 the form of equation (9) at once. 
 
 In working numerical examples, if we wish to illustrate 
 the transformation by a figure, we must examine the per- 
 pendicular on the right-hand side of equation (9), by the 
 rules of Art. 54, to see on which side of the axis of y it is 
 drawn. We must then write a? or — a; for this perpendicular, 
 according as we choose that side for the positive or negative 
 direction of the axis of x. The left-hand member will not 
 require this examination, since in either case it becomes y^. 
 If we simply desire to know the nature of the locus, it is not 
 important to decide on which side of the axis of y it is 
 drawn. 
 
 COE. Since the axis of the parabola {ax + cy 4- « = 0) is 
 always parallel to [ax + cy = 0), it follows that, if the equa- 
 tions to two parabolas can be written so as to have the first 
 three terms the same, their axes will be parallel. Also, since 
 K and L do not depend upon the constant term F, if the 
 equations can be written so as to differ in the constant term 
 only, the parabolas will have their axes coincident, and will 
 be equal, 
 
 Ex. (i). Let the proposed equation be 
 
 x''-2xy+y^-Sx + l&=0 (1), 
 
 therefore (x-y+h)'-{S + 2h)x+2h/ + ie-ty'=0 ....(2), 
 
156 GENERAL EQUATION OF THE SECOND DEGREE. 
 
 Then, that the lines may he at right angles, 
 
 - (8 + 2i) -2i=0, whence Te=- 
 
 Equation (2) then becomes 
 
 (a!-2/-2)2-4(a!+y-3)=0. 
 
 2 
 
 = 2^2^ 
 
 ^2 
 
 .(3). 
 
 a; +-2^-3 = 0. 
 
 .(5), 
 
 If then CX and CT are the lines 
 x-y-2=0 (4), 
 
 we know by Art. 54, that 
 
 x-y-2 , x+y-S 
 ^ — and 2;^- 
 
 represent the lengths of the perpendiculars drawn from (xy) on the sides of 
 CX and CY remote from the origin; 
 but the former might,, without caus- 
 ing any change in equation (3), be 
 written with the opposite signs, and 
 so represent a perpendicular drawn 
 on the origin side of GX; hence 
 equation (8) asserts that 
 
 PM'=2 J2PN, 
 
 ■ where PM may be drawn on either 
 side of CX, but PN is always on 
 the Bide of CY remote from the 
 origin. If we take OX and CY 
 as the positive directions of the axes 
 Of x and y respectively, equation (3) 
 becomes 
 
 2/" =2 ^x. 
 The form of the locus wiU be seen hereafter to be that of the figure. 
 Ex. (ii). The equation 
 
 4a?-ixy + y^-Wx-'i0y=0, 
 may at once be written 
 
 since (2s:-y=0) and (x+'iy=(i) are at right angles. The perpendicular on 
 the left hand may be on either side of the line ('ix-y=0), but that on the 
 right must (Art. 56) be above the line (a;-|-2i^=0). The figure will resemble 
 that of Ex. (i), the point O coinciding with 0. 
 
 s/ 
 
 \ 
 
 p/ 
 
 
 
 / 
 
 yd^ 
 
 1-, 
 
 
 
 
 \ 
 
 \ . 
 
EXAMPLES VIII. 157 
 
 157. "We said, Art. 143, that, when B'-4<AO=0, the 
 co-ordinates of the centre become generally infinite. If 
 however we have BE—2GI) = at the same time that 
 B^-4tAC = 0, we also have 
 
 B' = 4<AG=4:A ~, or BD - 2AE= 0, 
 
 and both the co-ordinates of the centre in Art. 141 become 
 
 = - , and are therefore indeterminate. The two equations 
 
 2Ax + By + D = 0, 2Gy + Bx + E=0, 
 will now represent two straight lines that coincide, as may 
 be seen by multiplying the former by E and the latter by JD : 
 hence there are an indefinite number of centres, all situated 
 in that line. The proposed equation, with the above rela- 
 tion between its coeflficients, no longer represents a parabola; 
 for the solution of Art. 62 becomes 
 
 2Cy + Bx+E±JJE'-4<aF=0, 
 
 which represents two parallel straight lines equidistant 
 from the line 2Gt/ + Bx + E=0, which is therefore a line 
 of centres. 
 
 *EXAMPLES VIII. 
 
 Transform the following equations, illustrating each transfor- 
 mation by a figure, as in Arts. 151, 156. 
 
 1. l + 2a! + 32/^ = to s/''=-|a;. 
 
 2. 3af + 2y''-2x + y-l = to 72x' + 4:8y' = 35. 
 
 3. 3s<f + 2xy + 3y''-16y + 2S = to ix' + 2y'=l. 
 
 4. y^~lOxy+sje' + y + x+l = to 32a;' - 482^' = 9. 
 
158 GENERAL EQUATION OF THE SECOND DEGEEE. 
 
 5. f-2(ci/ + a^-6y-Gx+9=0 to y''=3>j2x. 
 
 6. y' + xt/ + x' + ij + x-5 = to 9ajV32/' = 32. 
 
 7. y'-2xy-x'' + 2 = to y'-x''+j2 = 0. 
 
 8. /-a!^-2/ = to 4a;'-4/+l = 0. 
 
 9. Shew by transformation that the equation 
 
 12x1/ + 8x-21y- 18=0, 
 represents two straight lines parallel to the axes. 
 
 10. Shew by transformation that the equatioa 
 
 f-2xy + 3si^-2y~10x+l9=0, 
 
 represents two imaginary straight lines passing through the Joint 
 (3, 4). 
 
 11. Shew by transformation that the equation 
 
 y" - ixy + 5s(^+2y-ix + 2 = 0, 
 represents an imaginary ellipse. 
 
 12. Shew that any point on the line (y = «+ 1) is a centre of 
 the locus. 
 
 y" - 2xy + 0? -2y-¥2x = 0. 
 
 13. Shew by transformation that the equation 
 
 '^■>r2xy + a?+\=0, 
 represents two imaginary parallel straight lines. 
 
 14. What is the equation to the axis in Ex. 5, and to the 
 axes in Art. 151 Ex. '? 
 
 1 5. Transform Ty' + 1 6a;y + 1 6a^ + 32y + 64a! + 28 = 0, the axes 
 being inclined at an angle of 60°, to 2/^ + 4a!^= 9, the axes being 
 rectangular, and the axis of x remaining the same. 
 
CHAPTER IX. 
 
 Central Conic Sections'-, referred to their axes. 
 
 158. In this chapter we shall consider the eqiiations, 
 whose loci (Art. 150) we have called the Ellipse and Hyper- 
 bola, namely, 
 
 Px' + Qf = B, Px' - Qf = E, 
 where P, Q, R are supposed positive. 
 
 The equation to the ellipse may be written 
 
 R^ R • 
 P Q 
 
 Put -^ = 0^, -7=. = 6^ then the equation becomes 
 P V 
 
 where a and b are evidently the intercepts of the curve on 
 the axes of x and y respectively. 
 
 The equation to the hyperbola, which differs from that 
 
 ^ The term ' conic section,' or ' conic,' must be iinderstood to mean 'locua 
 of the second degree,' and to embrace every variety of locus mentioned in 
 Chap. VIII. It can be proved (Art. 322) that the section, made by any plane 
 in a cone standing on a circular base, is a locus of the second degree. It was 
 as sections of the cone, that the properties of these curves were first exa- 
 mined. Hence the name. 
 
160 CENTRAL CONIC SECTIONS. 
 
 ,of the ellipse in the sign of the coefficient of y" only, may 
 be written in the corresponding form 
 
 2 2 
 
 but, in this case, when x = 0, y^= — V, so that the intercept 
 on the axis of y is an imaginary quantity, or the curve does 
 not meet that axis. 
 
 159. The figure of the curves may now be deduced from 
 the siraple form to which we have reduced their equations. ■ 
 We wiU begin with the ellipse, and, since we may choose 
 whichever axis we please for axis of x, we shall suppose that 
 we have so chosen the axes, that a may be greater than h. 
 
 160. To find the polar equation to the ellipse; the centre 
 being the pole. 
 
 Writing (Art. 13) p cos 6 for x, and p sin 9 for y in the 
 
 x^ y^ 
 equation -j + ^ = 1, we have 
 
 1 _ cos' e sin'g 
 
 "'' '' «■' sin" 6^ + 6" cos" 61' 
 
 which is the required equation. It may be written 
 
 ^ a' -{a'- V) cos' y+ (a' - V) sin'' 6 ' 
 
 and it will appear hereafter, that it is convenient to use the 
 
 «" - 6" 
 abbreviation — ^- = e". Hence, dividing numerator and 
 
 denominator by a^ we have 
 
 f^'l-e'co&^d' 
 the form most commonly used. 
 
FIGUBE OF THE CUEYESJ 161 
 
 161; This equation -will be found the most convenient for 
 tracing the ellipse'. The least value that 6' + (a' - 6") sin' 
 can have, is when ^= 0° or 180° ; therefore, since 
 
 the greatest values of p are the intercepts on the axis of x, 
 which are each = a. 
 
 Again, the greatest values of J" + (a" — J") sin' 9 are, when 
 sin'^ = l, i.e., when ^ = 90° or 270"; hence, the least values 
 of p are the intercepts on the axis of y, which are each, = h. 
 The greatest chord then that can be drawn through the 
 centre is the axis of x, and the least chord, the axis of y. 
 
 From this property, these lines, A' A (=2a) and £'5(=2S); 
 are called the axis major and the axis 
 minor of the curve. It is plain, that the 
 smaller sin'^ is, the greater p will be; 
 hence, the nearer any radius vector is to 
 the aads major; the greater it will be. By "°' 
 
 taking the two values of p, positive and negative, for each 
 value of 0, we shall, as varies from 0° to 90°, trace the 
 portions AB, A'B'; and, as varies from 90° to 180°, the 
 portions BA', BA. The form of the curve will therefore be 
 that of the figure. The points J. ',-4 are called the vertices. 
 
 162. We obtain the same value for p, whether we sup- 
 pose = 0., or = — a. Hence, Two radii vector es which 
 make equal amghs with the aaois will be equal. And it is easy 
 to shew that the converse of thi^ theorem is true. 
 
 163. The figure of the ellipse may also be seen from the 
 following construction. 
 
 1 The Articles on the figure of the cnrres are taken chiefly from Salmon's 
 Gonic Sections, 
 
 n 
 
162 
 
 CENTRAL CONIC SECTIONS. 
 
 Solving the equation to the ellipse for y, we have 
 
 y 
 
 ^±-Ja'-as'. 
 
 Now, if we describe a concentric circle with the radius a, its 
 equation will be 
 
 y = + Ja'-x'. 
 
 Hence, if a circle be described on the axis major, and on 
 each ordinate MQ a point P be taken, 
 such that MP may be to MQ in the con- 
 stant ratio h : a, then the locus of P will 
 be the required ellipse; hence the circle 
 described on the axis major lies wholly 
 without the curve. 
 
 We might, in like manner, construct the ellipse, by de- 
 scribing a circle on the axis minor, and increasing each ordi- 
 nate in the constant ratio a :h; hence the circle described 
 on the axis minor lies wholly within the curve. 
 
 We see also, that the equation to the circle is the par- 
 ticular form which the equation to the ellipse assumes, when 
 h = a. 
 
 *164. Let CQ be joined, and let the angle QGM=j>; 
 then, if as and y are the co-ordinates of P, 
 
 X = CQ cos ^ = a cos ^ ; y = - QM= - a sin ^ = 5 sin ^. 
 
 Thus the co-ordinates of any point may be expressed in 
 terms of a single variable <p. These values of x and y evi- 
 dently satisfy the equation to the ellipse ; for, when substi- 
 tuted in it, they produce the equation 
 
 cos* (j) + sin' (^ = 1, 
 
.PiatJEE OF THE CURVES. 163 
 
 which is always true. The angle <f> is called the eccentric 
 cmgle of the point P. 
 
 165. To find the eqvation to the ellipse, when one of the 
 vertices is origin, the direction of the axes being the same as 
 before. 
 
 The problem is, to transfer the origin of co-ordinates to 
 the point A' (- a, 0) ; hence, writing as - a for 05 in the equa- 
 tion to the ellipse, 
 
 h' 
 we have y'= -^ (2aa; - a;'), 
 
 for the equation to the ellipse when the origin is the ver- 
 tex A, 
 
 166. We shall now investigate the figure of the- hyper- 
 bola from its equation 
 
 2 la ~~ -^^ 
 
 a or 
 
 The intercept on the axis of x is evidently = + a, but 
 
 that on the axis of y, being found from the equation ^ = — 1, 
 
 is imaginary; the axis of y then does not meet the curve in 
 real points. 
 
 If, however, we take an hyperbola whose equation is 
 
 V a' ~ ' 
 
 the axis of y wiU meet this curve at a distance =±h from 
 the origin, and the axis of « will not ™eet it in real points. 
 
 11—2 
 
164. 
 
 CENTRAL CONIC SECTIONS, 
 
 This (for reasbns evident hereafter) is. 
 called the conjugate hyperbola, and pos- 
 sesses properties which will be of use to 
 us in considering those of the original 
 Qurye. We shall then call the distance 
 A' A (=2a), between the two vertices, 
 or points where the curve meets the 
 axis of X, the transverse axis, and we stall call the distance. 
 £'B between the two points where the conjugate hyperbola 
 meets the axis of y, the conjugate axis. For we have chosen 
 as axis of x that which meets the hyperbola in 'real points, 
 and aie therefore not entitled to assume a greater than b, 
 so that the terms axis major and axis minor would not here 
 be applicable. 
 
 The conjugate hyperbola will evidently be an hyperbola, 
 whose transverse and conjugate axes correspond to the con- 
 jugate and transverse axes of the original curve. 
 
 167. To find the polar equation to the hyperbola, centre 
 pole. • 
 
 Transforming the equation to the hyperbola to polar co- 
 ordinates as in the case of.the ellipse, we get 
 
 g'y ^ aV g'y 
 
 Since formulae concerning the ellipse are altered to the 
 corresponding formulae for the hyperbola by changing the 
 sign of &^ we must, in this case, use the abbreviation e* 
 
 2 j^ 7 2 
 
 for — J — . Dividing numerator and denominator by a", we 
 
 have for the polar equation to the hyperbola, the centre 
 being pole, 
 
 ^~e'cos'^-l' 
 the form most commonly used. 
 
.FIGURE OF THE CtJEVES; 165 
 
 168. The hyperbola may be conveniently traced from 
 this equation. The denominator b'' - (a' + b") sin^ 6 will 
 plainly be greatest when ^= 0° or 180°, therefore, for these 
 values, p will be least; or, the transverse axis is the least 
 chord which can he drawn through the centre to the curve, • 
 
 As increases from 0", p continually increases, until 
 sin 
 
 in^ = — = (or tah5 = -), 
 
 when the denominator of the value of p becomes =0, and 
 p becomes infinite. After this value of 0, p' becomes nega- 
 tive, and the radii vectores cease to meet the curve in real 
 points, until again 
 
 sin = for tan = ) , 
 
 when p again becomes infinite. It then decreases regularly as 
 increases, until becomes = 180°, when it again receives its 
 minimum value == a. The lower part of the curve evidently 
 corresponds exactly with the upper. 
 
 The form of the hyperbola, therefore, is that represented 
 by the dark line in the figure, where the branches liAR,, 
 L'A!B! extend to infinity. We have drawn the curves al- 
 ways concave to the axis of x. We shall hereafter (Art. 188) 
 prove the correctness of the figures in this respect. 
 
 169. It was shewn that the radii vectores answering to 
 
 tan ^ = + - meet the curve at infinity. These radii vectores, 
 ~ a 
 
 indefinitely produced, are, for reasons given in Article 171, 
 
 called the asymptotes of the curve. They are the lines R'R, 
 
 L'L of the figure, and evidently separate the lines which 
 
 meet the curve in real points, from those which meet it in 
 
 imaginary ; i. e. the whole of the (jurye is included in the 
 
166 CENTRAi CONIC SECTIONS. 
 
 angles RCL, KGU. Hence the equations to the asymp- 
 totes are 
 
 11 = -^- X, or, as one locus, -» — ^=0. 
 
 Similarly for the conjugate hyperbola, the equations to 
 the asymptotes are 
 
 x = ±Ty, or, as one locus, -5 — ^ = 0- 
 
 Hence the asymptotes of the conjugate hyperbola coin- 
 cide with those of the original curve, and it lies wholly 
 within the angles RCL', LCB', corresponding to the dotted 
 line in the figure. 
 
 170. If a = 5, the hyperbola and its conjugate become 
 equal in every respect, and, since (Art. 169) 
 
 - = l = i3isiRaA, 
 a 
 
 BOA will =45°, or the angle between the asymptotes is a 
 right angle. This is called the rectangular or equilateral 
 hyperbola, and is to the hyperbola what the circle is to the 
 ellipse. Its equation is 
 
 ^-y' = a' (1). 
 
 Hence, if the equation to a conic referred to its axes is 
 Aa?+Gy' = F (2), 
 
 the locus is a rectangular hyperbola, if J. + (7=0; but we 
 have shewn (Art. 147) that this quantity A+ G does not 
 change by transformation, so long as the axes remain rect- 
 angular ; hence, if the axes are rectangular, and A + 0=0 
 in the general equation of the second degree, it will also = 
 in the reduced equation (2). Now the locus must in this 
 case belong to the hyperbola class, (Axt. 150) since, A and G 
 being of different signs, B^ — iAO must be positive; the 
 
ASYMPTOTES. 
 
 167 
 
 equation may therefore always be reduced to the form (2), 
 and will represent either a rectangular hyperbola, or (Art. 67, 
 Cor. 2) two straight lines at right angles. The latter is the 
 case (Art. 150), when ¥= in (2), which therefore becomes 
 »* — ^ = 0, representing the asymptotes of the rectangular 
 hyperbola (1), 
 
 171. Def. An asymptote is a straight line, the distance of 
 which from a point of a curve diminishes without limit, as 
 the point in the curve moves to an infinite distance from the 
 origin. 
 
 It must not be assumed, if the value of any radius vector 
 becomes infinite, as in Art. 169, that it is therefore an 
 asymptote to the curve. In fact, it will be proved hereafter, 
 that any lines drawn parallel to EK and LL' have one point 
 of intersection with the curve at an infinite distance, just 
 as in Art. 42 we found the co-ordinates of the intersection 
 of parallel lines to be infinite : but in neither case do the 
 loci approach indefinitely near to one another. We shall 
 therefore shew that the lines BR and LL' correspond to the 
 above definition. 
 
 If the ordinate MF be 
 produced to meet BB' in 
 Q, the distance of the point 
 P from RR' = PQ sin PQG, 
 and therefore varies as FQ. 
 Now if GM=x', FM=y', 
 QM=y^, we have from the 
 equations to the curve and 
 asymptote. 
 
 _ 
 a" 
 
 J2 - -^ ^' 
 
 Vi-yl^V", ox y^-y' = 
 
 Vx + y 
 
16S centeAl conic si^tionS. 
 
 But this value of y^ — j/, which = PQ, becomes indbfiiiitely 
 small, when y^ and y' become indefinitely large, and there- 
 fore BE is an asymptote. Similarly LL' is an asymptote. 
 
 *VJ2. The co-ordinates of toy point in the hyperbola 
 may, as in the ellipse, be expressed by a single variable by 
 -means of the eccewfrican^fe.; for we may put 
 
 x=a sec ^, y = i tan ^, 
 since these values, when substituted in the equation . 
 
 a" ,6'~ • ^ '' 
 
 will give sec'^ - tan" ^ = 1 (2), 
 
 which is always true. If a tangent MQ be drawn from the 
 foot of the ordinate MF, to the \ ^ ^ 
 
 circle on the transverse axis^ the 
 angle QGM is the eccentric angle 
 of thie point P; for - 
 
 CM =CQ sec QOM, 
 or x = a sec ^, 
 
 and therefore, from (1) and (2), y = Jtan <p. 
 
 173. As in the case of the ellipse, we shall find the 
 equation to the hyperbola, when the vertex A is taken for 
 origin, by writing a; + a for a; in the equation 
 
 The result is y' = -j. (2aai + a^, 
 
tHE ECCENTEICftTT. 169 
 
 174!. The quantity e is called the ecoentrmty of the 
 curve ; 
 
 in the ellipse, 6= = ^-^ ; 
 
 in the hyperbola, «« = "^ \^\ 
 
 Hence in the ellipse c is less than unity ; and in the 
 hyperbola greater than unity. When a = J in the ellipse, or; 
 the curve becomes a circle, e = 0, and, if a remain the same, 
 e increases as h diminishes, or as the curve passes from a. cir- 
 cular to an oval form. If is the angle between the asymp- 
 totes of the hyperbola, we have 
 
 tan^ = -; .-. sec^^ = -^ = e». 
 
 175. : In the following investigations we shall in most 
 cases consider both the ellipse and hyperbola together, as 
 they have many properties in common, resulting from the 
 similarity of their equations ; and the properties of the one 
 may be deduced from those pf the other by changing the 
 sign of b\ The properties of the conjugate hyperbola may b^ 
 deduced from those of the ellipse by changing the sign of 
 o", or from those of the given hyperbola by changing the 
 signs of a" and h\ . We shall, whenever we speak of the 
 ellipse and hyperbola together, use the signs which apply 
 to the ellipse. 
 
 176. To find the length of a straight line drawn to the curvt 
 
 fr&m the point (x-if). 
 
 Let the equation to the line be 
 
 .f:z±=.yiil^l, (1). 
 
 „s + 2,2 ~" ^r 
 
170 CEirrRAL CONIC SECTIONS, 
 
 Then, as in Art. 114, for the distances (Z) between (aj'y') 
 and the points of section of the line and curve, we have 
 
 a' "*■ b" ~^' 
 
 or FP+Ql+Ii = 0. 
 
 Now this equation wUl always give two values for I; 
 hence every straight line meets the carve in two real, coin- 
 cident, or imaginary points, according as the roots of (2) are 
 real and unequal, real and equal, or imaginary. 
 
 We shall hereafter have occasion to consider the follow- 
 ing particular forms that this equation may assume : 
 
 If P= 0, one value of 7 (Appendix) becomes infinite. 
 
 If F= and Q = 0, both values of I become infinite. 
 
 If ^ = 0, the point {oo'y') is on the curve, and one value 
 of I becomes = 0. 
 
 If i? = and Q = 0, both values of I become = 0, and 
 the line passes through two coincident points of the curve, 
 and is a tangent. 
 
 If C = 0, the roots of the equation are equal and of 
 opposite signs, and (x'y') is therefore the middle point of the 
 chord. 
 
 177. If P = in equation (2), we have, according as the 
 curve is an ellipse or hyperbola. 
 
INTERSECTION OF STRAIGHT LINE AND CURVE. 171 
 
 The former equation gives imaginary values for -; the 
 
 latter gives 
 
 5 
 ■a' 
 
 s . 
 
 but - is the tangent of the angle that the line (1) makes with 
 c 
 
 the axis of x; hence (Art. 169) one value of I becomes infi- 
 nite, if the Kne (1) is parallel to either asymptote, i.e. a straight 
 Une drawn parallel to an asymptote meets the curve in one 
 point only at a finite distance from the origin, and in one 
 point at an infinite distance. 
 
 178. If P = and Q = 0, we have in the hjrperbola 
 s _ b. 3/ _ cb _ b 
 
 a X 
 
 sa 
 
 Hence by the first condition the line (1) Is parallel to an 
 asymptote, And by the second .the point (a;'y') is on the same 
 asymptote; i.e. the line (1) is one of the asymptotes, and 
 both the values of I are infinite. 
 
 179. We have said that a line parallel to 'an asymptote 
 meets the curve in one point at an infinite distance. This is 
 a short way of enunciating 
 the property, which may be 
 stated more clearly as follows. 
 Let K be the point {x'y'), and 
 PKD a straight line cutting 
 the hyperbola in F and D, 
 and let KL be a straight 
 line parallel to the- asymp- 
 tote CF; theni, if the point 
 D moves along the curve to 
 an infinite distance, and the 
 line PKD tUriis about K, it 
 
172 • CENTRAL CONIC SECTIONS. 
 
 will, tend to coincide with KL, as its limiting position. 
 Without this explanation, the statement would probably 
 perplex the student, inasmuch as he would see that KL does 
 not actually meet the curve at infinity, for the distance 
 between it and the curve can never become less than the 
 (iistance between it and the asymptote. In the same way we 
 have said (Art. 42) that parallel straight lines meet at in- 
 finity; and our statement here simply asserts, that the curve 
 tends to become a straight line parallel to KL, i.e. the 
 straight line OV. 
 
 180. To find the equation fe a straight Une touching 
 tiie curve 
 
 oi the point {x'y'). . 
 
 The method employed is precisely the same as that used 
 in the case of the tangent to the circle (Art.. 117), to which 
 the student may refer for the. figure. We shall simply indiy 
 cate the steps. 
 
 Xet the equation to a straight line cutting the curve in 
 the point Q (a;'/), be 
 
 x — x 
 
 = ^=Z (1), 
 
 C 8 
 
 then, for the distances (J) between {xy) and the points of 
 section of the line and curve, yrfi obtain equation (2) of Art. 
 175, which becomes 
 
 a;'' «" 
 Since -2 + ^ = 1; for (^x'y) is on the curve. 
 
THE TANGENT. ) 17S 
 
 Equation .(2) gives us 1 = 0, as it should, for (x'y) coin- 
 cides with one of the points of section, and also 
 
 {?+p)'+H?'-f)=°' 
 
 the value of I in which equation is the distance ( QR) of (x'y) 
 from the othetpoint of section B. If the line be a tangent at 
 Q [x'y), this distance vanishes, and, we have 
 .,..,, .............cj^ s£_ 
 
 «'■*■ 6»~ ' 
 
 Miminating c and », by means of this equation and equa- 
 tion (1), we have 
 
 • («'-«'')^+(y-y')^=o. 
 
 .. .. . ..XX yy 
 
 or —2-+ — 
 
 
 , • xx' yi/ , 
 
 whence — »" + '^ = 1> 
 
 which is the equation to the tangent at the point (x'y), and 
 is easily remembered from' its siipilarity to the equation to 
 the curve. The form of the equation, when the origin is 
 transferred, may be seen ftom Art. 118. 
 
 181. If m be the trigonometrical tangent of the angle, 
 which the tangent at {x'y') makes with the axis of x, we have 
 
 s Vx 
 
 »» = - = — 
 
 ,2-.' » 
 
 c ary 
 
 Since this value of m does not change, when a/, y' 
 are replaced by — x, — y , we see that tangents at the extre- 
 mities of chords through the centre are' parallel. Also at 
 the points B, B', for which a;' = 0, y' = ±h, m = Q, or the 
 tangents are parallel to the axis of x; and at A, A', for which 
 a; = ± ci'^y = 0, m = oo , or the tangents are perpendicular to 
 the axis of a, . 
 
174 CENTRAL CONIC SECTIONS. 
 
 182. To find the equation to the tangent in terms of its 
 inclination to the aoois ofx. 
 
 Proceeding as in (Art. 180), we have, for the equation to 
 the tangent, 
 
 x-x' ^y-i/ ,j^ 
 
 C 8 
 
 with the condition -» +-^ = (2); 
 
 a or 
 
 and we must now eliminate x' and y' instead of s and c : 
 
 from (2) ^ = -% 
 
 ^ ' be ay 
 
 , oV + JV JV + aV' a'6» ,„, 
 
 whence — t^ — = 8 ,, = -j-ia w> 
 
 oV a'^' <ry 
 
 for, i^o^y') being on the curve, & V + a'^'" = a'J'. 
 
 From equations (2) and (8), we have 
 
 Wc ,__ o*s 
 
 .', from (1) cy — sx = cy' — sx' 
 
 = ±VaV + 6V, 
 according as we take the upper or lower signs. 
 
 If - =»w, where «i is 'the tangent of the angle which the 
 line (1) makes with the axis of x, the equation becomes 
 
 y = mx ± Jm^a^ + V. 
 
 The double sign refers to the two tangents at the ex- 
 tremities ctf a chord through the centre, since, by the last 
 article, these tangents have the same ijiclination to the axis 
 of a;. . 
 
THE TANGENT. 175 
 
 The student will have no difficulty in obtaining this 
 equation without introducing the point (sc'y'), as in the circle. 
 Art. 120. 
 
 CoK. Hence if a straight line and a conic be represented 
 
 by 
 
 y = mx + ^, and-, + ^=l, 
 the condition of tangency is 
 
 183. If the tangent pass through the centre, we have 
 VmV + 6' = 0. (Art. 28, Cor. 3.) This gives an imaginary 
 value for m in the case of the ellipse, but for the hyperbola 
 we have 
 
 mV — &' = 0, or m = + - ; 
 a 
 
 hence a tangent to the hyperbola drawn from the centre 
 coincides (Art. 169) with the asymptote, and meets the curve 
 at an infinite distance only. The asymptotes may there- 
 fore be considered as straight hues touching the curve at an 
 infinite distance. 
 
 184. The result of the last article may be obtained from 
 the equation to the tangent at any point (pe'y'), as follows : 
 
 The equation to the tangent to the hyperbola at {x'y'), is 
 (Art. 180), 
 
 ^'_M'_i m or- ?^-^-l m- 
 
 a' b''~ .^'' a^'y' b''~y ^'' 
 
 hence, when y' is indefinitely large, or the point of contact is 
 indefinitely distant, the right-hand member of (2) vanishes ; 
 also from the equation to the hyperbola, since {ns'y') is a point 
 on the curve,' 
 
 ^-^ o" 
 
176i CENTEAI. CONIC SECTIONS. 
 
 and therefore, when x and y become infinitely great, lilti- 
 
 mately 
 
 ^^ _a^' x' _ a 
 
 f^'V""^ y'~-b' , 
 
 Hence equation (2) becomes, after reducing and transt 
 posing, , 
 
 X V 1 ^ y • n fa\ 
 
 - = ±r> or as one locus, -3— fs = •(2), 
 
 a o do 
 
 which represents the two asymptotes. 
 
 In the ellipse the same process would give ultimately- 
 
 y- V • 
 
 which gives for the asymptotes of the ellipse the two imagin- 
 ary straight lines . 
 
 - = + V— 1 r . or as one locus, -» + ts = 0. 
 a ~ b a b 
 
 It will perhaps be observed, that in the above proof we 
 began with the equation to OTie straight line, (1), and, on 
 endeavouring to trace its limiting position, consequent on 
 the indefinite increase of x' and y', we have obtained an 
 equation which represents two straight lines. Can each of 
 these be the limiting position of (1) ? The difficulty vanishes, 
 if we take into consideration that (1) represents, not one, but 
 & class of lines, indefinite in their number, and subject only 
 to the limitation that {x'y) should be on the curve. The 
 suppositions we have made with regard toa;' and y', are 
 eqiially applicable to the whole class. Hence the result- 
 ing equation (2) represents the limiting position, not of one 
 tangent but of all of them. 
 
 185. Since the asymptote touches the branch RAL, 
 (fig. Art. 166) at infinity, it must also, by the symmetry of 
 the figure, touch the brattch RA!Ii at infinity. But we saw 
 
TANGENT AND ASYMPTOTE. 177 
 
 (Art. 176) that, when we combine the equatiorls to a straight 
 line and the curve, the result is a quadratic, which wiU 
 determine two points of intersection, and only two. That is, 
 a straigl^t line can only intersect a curve of the second order 
 in two points, and, when these two points coincide, the line 
 is said to touch the curve ; hence the two points of contact 
 of the asymptote would seem to coincide. This difficulty 
 wiU be removed, when we consider that the asymptote BE' 
 does not really touch the hyperbola at infinity, and that this 
 expression is only a short way of saying, thaty as the distance 
 of the point of contact becomes very great, whether on the 
 branch AB ox A'R, the tangent in each case tends to coin- 
 cide with the line BR. 
 
 186. We see then that, if a central conic is referred to 
 its axes, the equations to the conic and its asymptotes are 
 Fa?+Qf=B, Fx'+Qy' = 0, 
 
 the asymptotes being real, only when P and Q are of oppo- 
 site signs. Now let any transformation of axes be made, 
 the centre remaining the origin, so that we wfite (Art. 72) 
 ax + iy for x, and a'x + b'y for y, where a, a', b, V represent 
 constant quantities ; then, if the equation to the conic becomes 
 
 Aii^ + Bxy^-G')f = R, 
 (for B will evidently not be altered), the equation to the 
 asymptotes will become 
 
 Ax^ + Bxy^-Gy^^i^. 
 
 Hence, with any axes whose origin is the centre, the 
 equation to the asymptotes is found by equating to zero 
 the terms of the second degree in the. equation to the 
 conic. Also, since (Art. 141) these terms are not altered, 
 when the origin is transformed from the centre to any point 
 whatsoever, the direction of the axes remaining the' same, we 
 shall, by equating them to zero in any equation, find a pair 
 of straight lines drawn through the origin parallel to the 
 
 12 
 
178 CENTRAL CONIC SECTIONS. 
 
 asymptotes. This article explains the coincidence mentioned 
 in Art; 146, since the axes bisect the angles between the 
 asymptotes. 
 
 Cor. Since the equation -4a;'' + Bxy + Cy' = determines 
 the direction of the asymptotes, it follows tha.t, ii A = O'in 
 the general equation of the second degree, one asymptote 
 will be parallel to the axis of x, for its direction is given by 
 the equation 2/ = 0. Similarly, if (7=0, one asymptote will 
 be parallel to the axis of y. The locus will in either case 
 (Art. 150) belong to the hyperbola class, since B' — iAG 
 becomes £', and must be positive. 
 
 Ex. if the equation to a conic is ^ (xy) = 0, the equation to the asymptotes 
 is 4> [niy) - 't> ip'y') =0, where [xfy') is the centre. 
 
 Since the equation to a conic and the equation to its asymptotes differ in 
 the constant term only, they may be represented respectively by 
 
 i>{xy)=0, 0(!ty) + i=O; 
 hut since the asymptotes pass through the centre {x'y'), we have ^{x'y') + Jc=0, 
 _ and their equation becomes 
 
 Otherwise, let 0(a!y)=O become :l/{xy)=0, when the centre (x'y') is 
 origin ; then (Art. 144) f (xy) - 4> (x'y") = is the equation to the asymptotes, 
 or, returning to the first origin, ^(xy) - ^ (kY) = 0. 
 
 187. To determine the equations to the tangents to an 
 ellipse or hyperbola, which pass through a given point (x'y'). 
 The equation to the tangent is (Art. 182) 
 
 y — mx = ± \/a^m^ + b^. 
 Since it passes through {x'y), we have 
 
 (j/-mx'Y = a^m^+V; 
 
 '^+^^'" + ^4^ = W' 
 
 which equation gives two values of m ; let them be /i and /*' ; 
 then the equations of the tangents required are 
 
 y-y' = li{x-3^), y-y' = f^{x-x') (2). 
 
TANGENTS DEAWK "raiROTJGH A GIVEN POINT. 179 
 
 188. The roots of equation (1) of Art. 187 are real and 
 different, real and equal, or imaginary, according as 
 
 ^Y>^<{^'-y")(fl'-^') (3), 
 
 or ^ ^ + ^-l> = <0 (4), 
 
 a 
 
 V 
 
 and a little consideration will shew, that inequality (4) gives 
 the conditions that {x'y) shall be without, on, or within the 
 ellipse. Hence no tangent can be drawn to the ellipse from 
 within the curve. 
 
 To prove the same property for the hyperbola, we must 
 write — 6" for S' in (3), and we shall then obtain, instead 
 of inequality (4), 
 
 J-|^-l< = >« (^)- 
 
 Condition (5) will evidently always give possible roots, 
 
 x^ 
 when x' does not exceed + a ; for then — ^ is less than unity, 
 
 and consequently the whole expression is negative; but, 
 
 x^ 
 when x' exceeds those values, —^ becomes greater than unity, 
 
 and, in order that the whole expression may be negative, the 
 value of y' must not be less than that of the ordinate of 
 the curve corresponding to x, since it. is that value which 
 makes 
 
 ^-l 1 = 0. 
 
 2 1,2 
 
 a 
 Hence inequality (5) gives the condition that (x'y') should 
 be without, on, or within the curve (where the foci are con- 
 sidered as within) ; and no tangent can be drawn to the hyper- 
 bola from within the curve. The above reasooing shews 
 that the curves are, as we have drawn them, always concave 
 to the axis of x ; for, if this were not the case, it would 
 evidently be possible to draw a straight line from within, 
 which should pass through two coincident points of the curve. 
 
 12—2 
 
180 CENTRAL CONIC SECTIONS. 
 
 The reader will observe, that the expressions 
 
 2 2 2 3 
 
 , ,. a^^V ' a' W ' 
 
 are negative when (xy) is on the origin side of the curves, 
 and change sign when (xy) crosses the curves. This result 
 should be compained with Arts. 55, 121. 
 
 189. To determine the locus of the intersection of two 
 tangents at right angles to one another. 
 
 If, in Article 187, the two tangents are at right angles, 
 fjifjb' = — 1. Hence, from equation (1), since fifj.' is the pro- 
 duct of the roots, 
 
 -2—^2 = w =-1; 
 
 therefore x'^ + y"" = a= + 6' ; 
 
 or the locus, of the point {^oy'), where the tangents intersect, 
 is a circle, whose centre is C and radius = Va" + 6". 
 
 The corresponding equation for the hyperbola is, of course, 
 
 which is a circle, unless J" be greater than a^ in which case 
 the locus is impossible; i.e. two tangents cannot be drawn at 
 right angles to one another, when V is greater than d^. 
 In the equilateral hyperbola h=a, and the circle, is reduced 
 to a point, namely the origin; hence only one pair of 
 tangents at right angles to one another can be drawn to 
 the equilateral hyperbola. These pass through the centre, 
 and are the asymptotes, which are tangents at an infinite 
 distance. 
 
 190, To find the ■perpendicular from the centre on the 
 tangent, in terms of the angle which it makes with the axis 
 o/x. 
 
TANGENT AND NOEMAL. 181 
 
 If the perpendicular p make an angle a with the axis of 
 X, the equation to the tangent is (Art. 27) 
 
 X cos a + y sin a=p, 
 and hence, by the condition given in Art. 182, Cor., 
 
 p^ = a' cos" a + J" sin'' a, 
 or since (Art, 174) 5' = a" (1 - e') , 
 
 y = a''(l-e''sin»a). 
 Hence the equation to the tangent in the form given in 
 Art. 27 is 
 
 a; cos a + y sin a — a Vl — e* sin" a = 0. 
 If we had used the signs suitable to the hyperbola, we 
 should have obtained the same result for that curve. 
 
 191. To find the equation to the normal at any point 
 
 By reasoning similar to that used in the case of the 
 circle (Art. 122), since the normal passes through the point 
 ifl'y') and is perpendicular to the tangent, whose equation is 
 
 • ■ . ■ • , aV , fv 
 
 its equation is y—y= jf-< (aJ — a ), 
 
 X y 
 
 192. To find the equation to the normal in terms of its 
 inclination to the major axis, we may write it 
 
 aV 
 y-y'=m{x-x') (1), where m,=-^, (2), 
 
 and is the tangent of the angle which the normal makes 
 with the axis of x. We have then to express x and y in 
 terms of m ; that is, we must eliminate x and y' between 
 the three equations, (1), (2), and the equation to the curve.. 
 
182 
 
 
 CENTBAL CONIC SECTIONS. 
 
 From 
 
 (2) 
 
 hm ay 
 a ~bx" 
 
 whence 
 
 
 bW + a? aY + Vx" a" 
 a' 6V^ x"' 
 
 since 
 
 
 ' ay= + 6V' = aTi 
 
 therefore 
 
 3!' = 
 
 r— ; aiTirl RiTnilarly y' = 
 
 b'm 
 
 Substituting these values in (1), we have 
 '{a'-b'')m 
 
 y = mx-^ 
 
 VSW+o"' 
 
 193. The intercepts of the tangent and normal on the 
 axis oi'x mky be found by putting^ = in their equations. 
 
 Let them meet the axis in Tand G; then, putting y = 
 in the equation to the tangent, we have 
 
 ^ = 1, or CT=-,; 
 
 and in the same manner from the equation to the normal, 
 when y=0, we have 
 
 X 
 
 =<-D. 
 
 or CG = e'x'. 
 
 The portion MT, intercepted on the axis between the 
 tangent and the ordinate of the point of contact, is called the 
 
TANGENT AND NOEMAL, i IBS'; 
 
 svibtangent, and MG is called the subnormal. The length of 
 the subtangent is, 
 
 in the ellipse, CT- GM= -,-x' = ^^~ '"" 
 
 X X 
 
 and in the hyperbola, CM- 0T= a--~, = ?-=^' . 
 
 X X 
 
 In the same manner the length of the subnormal is, 
 
 w 
 in the ellipse, MG = x — eV = -j a;', 
 
 Oi 
 
 in the hyperbola, MG = eV — a;' = -^ji/, 
 
 194. If in the equation to the ^ngent 
 
 we write a;', x for a;, a;', and j/', y for y, ^' respectively, the equa- 
 tion remains unchanged. Hence (Art. 131) all the theories 
 of poles and polars proved for the circle in Arts. 123 — 127, 
 129, 130 are equally true for the ellipse and hyperbola, and 
 the proofs will require no alteration, except that we must 
 write the equations to the ellipse or hyperbola and its tan- 
 gent, in the place of the equations to the circle and its 
 tangent. These properties are so important, that the student 
 is recommended to convince himself thoroughly of the truth 
 of the above assertion, by writing out the articles with the 
 requisite changes. 
 
 195. If, in the equation to the polar of the point (a?y) we 
 make y' = 0, the equation becomes —2=1, the equation to a 
 
 Co 
 
 straight line parallel to the axis of y ; hence the polar of any 
 point on the axis of x, is parallel to the axis of ^ ; and simi- 
 
184 
 
 CENTRAL CONIC SECTIONS. 
 
 larly the polar of any point on the axis oiy, is parallel to the 
 axis of X. 
 
 If we take two points, H, 8, on the axis of x, so that 
 
 in the ellipse C8=CE= Va' - b" = ae, 
 
 and in the hyperbola C8= CH = 4a^ + V = ae, 
 
 the equations to the polars of these points will be, writing 0, 
 ae and 0, ^ ae for y', x', 
 
 a; = - for the polar of If, aj = — - for the polar of 8. 
 
 These points are called the^^ci, and their polars are called 
 the directrices of the curve. 
 
 In the ellipse, since e is less than unity, the foci lie between 
 the centre and the vertex, and the directrices KX, K'X', the 
 polars, respectively, of H and 8, lie beyond the vertex. In 
 the hyperbola, since e is greater than unity, the reverse will 
 be the case. In the circle a = J, and the foci coincide with 
 the centre ; also (Art. 174) e = 0, so that the directrices are 
 infinitely distant. It will be seen hereafter, that the focus and 
 its polar, the directrix, possess many remarkable properties in 
 connexion with the curve. 
 
 Cob. From the results of this article we have in the 
 ellipse, 
 
 AH=AC- CH=a-ae = a{l-e), 
 
THE loci. 183 
 
 AX= ex- GA = --a = ''^^'^\ 
 e e 
 
 HX= CX - CH= --ae = "^^lzH, 
 e e 
 
 In the hyperbola, 
 
 AH=CH-CA = ae-a = a{e- 1), 
 
 AX=CA-CX=a'-~ = '!^^^^, 
 e e 
 
 EX= CH- OX=ae-"^=^"^l^ . 
 e e 
 
 196. To find the distance of any point in the ellipse from 
 the focus, in terms of the abscissa of the point. ' 
 
 Since the co-ordinates of the point jffare (x = ae,y = 0), 
 the square of the distance of any point P, {x'y) from it is 
 (Art. 7) 
 
 {x - aef + y'^ = x" - 2aea;' + aV + y" ; 
 
 and, if P be a point in the curve, 
 
 y- = l^ia''-x") = il-e^){a'-x'% 
 
 since V = c? — aV. Hence 
 
 SP» = a=_2aea;' + eV, 
 
 or HP = a — ex'. 
 
 We do not notice the value {ex' — a), obtained by giving 
 the negative sign to the square root. For e is less than 1, and 
 a; less than a, hence ex' — a is constantly negative, and need 
 not be taken into consideration, since we are now examining 
 the magnitvde, not. the direction, of the radius vector BP. 
 
186; CENTRAL CONIC SilCTIONS. 
 
 Writing —ae for ae in the preceding proof, we have for 
 the distance of P from the other focus 
 
 SP = a + ex'; 
 hence 8P + HP = 2a, 
 
 or the sum of the distances of cuny point in an ellipse from the 
 foci is constant, and equal to the aosis major, 
 
 197. In the case of the hyperbola, we obtain the same 
 value for HP", but, in extracting the root, we must take 
 the value 
 
 EP=ex'-a, 
 
 since in the hyperbola x is greater than a, and e is greater 
 than unity, and consequently a — ex' is constantly negative. 
 In like manner, we have 
 
 8P=e(io' + a; 
 
 hence 8P-HP = 2a, 
 
 or, in the hyperbola the difference of the focal radii is constant, 
 and equal to the , transverse aosis. 
 
 198. The property proved in the two last articles wiH 
 enable us to describe an ellipse or hyperbola mechanically ; 
 for, evidently, if a string 8PI£ be fastened to two points jSi 
 and H, a pencil P, moved so as to keep the string always 
 stretched, will describe an ellipse of which 8 and H are the 
 foci, since 8P + HP will be a constant quantity. 
 
 Also, toy portion of an hyperbola may beifflescribed by a 
 ruler and cord; for let a ruler ,SS revolve 
 round, /Sim the plane pfihe "paper, and let a 
 cord be fastened to 2", shorter than 8B by a 
 given "difference c ; then a pencil P, which 
 should always keep the string stretched 
 against 8B, would describe an hyperbola; for the difference 
 
FOCAL PROPERTIES. , 187- 
 
 SP—PH wonld always equal a constant quantity c, the differ- 
 ence in the lengths of the ruler and cord. 
 
 199. The distance of any point on the curve from the 
 focus, is in a constant ratio to its distance from the directrix. 
 
 The equation to the directrix KX is (Art. 195) 
 
 ex — a = (1), 
 
 and the length of the perpendicular (fig. Art. 195) from any 
 point P {xy") in the ellipse on (1) is, (Art. 54), since P zis on 
 the origin side of KX, 
 
 _ ex' ~a ^l ^p , 
 
 e e ' 
 
 In the hyperbola, since P is not on the origin side of 
 KX, the perpendicular is 
 
 e e 
 
 Hence in both curves the distance of any point P froni 
 the focus is to its distance from the directrix in the constant 
 ratio of e to 1. 
 
 ■In the ellipse e = — — ^- — , and is less than 1, or the dis- 
 a 
 
 tance from the focus is less than the distance from the directrix. 
 
 a/ 2 I 2.8 
 
 In the hyperbola e = , and is greater than 1, or the 
 
 distance from the focus is greater than the distance from the 
 directrix. It wiU afterwards be seen, that the parabola has 
 a focus, and that the distance of any point from the focus is 
 equal to the distance from the directrix. 
 
 Hence it is often given as a definition of a conic section^ 
 that it is the locus of a point, whose distance from a given point 
 has ajixed ratio to Us. distaneejrom^ ^vea s^aight line. 
 
188 CENTRAL COKIC SECTIONS. 
 
 200. The double ordinate through the focus is called the 
 Latus Rectum. Putting x = aein the equation to the curve, 
 we have 
 
 or y = + - ; hence the latus rectum = — = 2a (1 — e") in the 
 ellipse, and = 2a (e* — 1) in the hyperbola. 
 
 *201. In the ellipse the normal bisects the interior angle 
 between the focal distances, and in the hyperbola the exterior 
 angle; and the focal radii make equal angles with the tangent. 
 
 The equation to HP, since it passes through {x'y), 
 (ae, 0) is 
 
 -^=^y- (1), 
 
 x — ae X —ae ^ ' 
 
 and similarly, the equation to 
 8P\b 
 
 ^JL^ = ^Jl— (2). 
 
 X ■{■ ae X ■{■ ae 
 
 Hence, forming the equation to the bisector PG of the 
 angle between (1) and (2), we have, by the rules of Art. 59; 
 
 — {fC — ae) y + y'x — aey _ (a;' + ae) y — y'x — aey' .„. 
 
 {y'-+{x--aey]k ~ {y^+(a,' + ae)'}i --W. 
 
 which we might shew by reduction to be the equation to the 
 normal at {x'y"). This may however be proved briefly as fol- 
 lows. The denominators of (3) are evidently HP and SP, 
 and therefore reduce, for the ellipse, to a — ea;' and a + ea;', as 
 in Art. 196 ; hence, to find where (3) meets the axis of x, we 
 have, making y = Q, 
 
 x — ae_—{x + ae)_ 
 a— ex' a + ex' ' 
 
FOCAL PEOPERTIES. 
 
 189 
 
 hence x = GG = eV, and therefore (Art. 193) the bisector PQ- 
 is the normal, 
 
 The reasoning for the hyperbola is precisely the same ; 
 but the denominators of (3) will now become ea;' — a and 
 ea;' + a, and we shall have, to determine the point T where 
 the bisector PT meets the axis, 
 
 x — ae _— (x + ae) ^ 
 ex' — a ex' + a ' 
 
 hence x= CT=-,, 
 
 X 
 
 and therefore (Art. 193) the bi- 
 sector PT is the tangent, and 
 consequently the normal PQ bisects the exterior angle EPS. 
 
 \ 
 
 ^^ 
 
 y 
 
 ^-Jk^ 
 
 X 
 
 o 
 
 yT 
 
 Kr 
 
 202. The following is an easy geometrical proof of the 
 preceding results. In the ellipse and hyperbola GG = eV ; 
 hence in the ellipse, 
 
 8G = ae + eV, HG = ae- eV ; 
 
 also 8P= a + ex', HP = a —ex' ; 
 
 therefore SG : EG=SP : MP, 
 
 and therefore, by Euc. VI. 3, PG bisects the angle SPH. In 
 the hyperbola, 
 
 SG = e^x' + ae, HG = ^x'-ae; 
 
 also 8P= ex' + a, HP = ex' —a; 
 
 therefore 8G : HG = SP: HP, 
 
 and therefore, by Euc, Vl. ^,PG bisects the exterior angle 
 HPE. 
 
190 CENTEAL CONIC SECTIONS. 
 
 203. To find the locus of the extremities of perpendiculars 
 dropped from the foci upon the ta/ngent. 
 
 Let 8Y, HZ be these perpendiculars ; then the equation 
 to PTis, taking the fig. of the ellipse, 
 
 y = mx + >/mV + b", 
 
 and, since HZ -passes through H(\/a' — b', 0), and is perpen- 
 dicular to FT, its equation is 
 
 y = --fx-'/a'-b\ 
 ^ m^ ' 
 
 between which equations if we eliminate m, we shall obtain 
 the equation to the locus of their intersection. These equa- 
 tions may be written 
 
 y-'mx = \lm^d' +6', my + x = Va" — V ; 
 
 adding their squares, 
 
 or a? + y' = a', 
 
 the equation to the locus of Z, which represents a circle on 
 the axis major as diameter. 
 
 204. If HZ is the perpendicular from H {ae, 0) on the 
 tangent, whose equation is 
 
 -^ + f=l (1)- 
 
 we have for the ellipse (Art. 54), since H is on the origin 
 side of (1), 
 
 ea;' _ ^ 
 
 HZ=— "" °^° ('^ ~ ^'"') 
 
 V ^ + F 
 
rOCAL PEOPEETIES. 191 
 
 But ay + h'x" = a'V (p y'' + 1 a;'=) 
 
 and therefore HZ=h ^ /a- ex' ,^. 
 
 V a+ex' ^ ' 
 
 Similarly 8Y=h J"^^, (3). 
 
 SJa — ex 
 
 Hence SY.HZ=W (4). 
 
 In working this problem for the hyperbola, we must 
 remember that, in the case of HZ {?ig. Art. 201), H is hot on 
 the origin side of (1). We mention this, because the student 
 would otherwise probably be puzzled, by getting — b^ as the 
 right-hand member of (4). 
 
 If 8Y=p, 8P = p, equation (3) may be written 
 
 p« = p P 
 
 2a + p' 
 
 the upper or lower sign being taken, according as the curve 
 is an ellipse or hyperbola. 
 
 205. Any focal chord is perpendicular to the line joining 
 its pole with the focus. 
 
 By definition, the directrix is the polar of the focus, and 
 conversely, by Art. 129, the polar of 
 any point in the directrix passes 
 through the focus. Hence assume 
 the pole of any focal chord PHQ to *^ ^ ^ 
 
 be a point -K"!-, y'j on the directrix, 
 
 then the equation to the chord is (Art. 123) 
 
 ae "^ b' ~ ' 
 
 X 
 
 E 
 
192 
 
 CENTRAL CONIC SECTIONS. 
 
 and the equation to a perpendicular HK to this chord, drawn 
 through the focus, is 
 
 aey' , , 
 
 and, when x = 0X=- in this equation, y=y' = KX, or the 
 
 6 
 
 line which it represents passes through K, the pole of the 
 chord. 
 
 206. We have deduced all the above properties of the 
 ellipse and hyperbola from their equations alone ; we shall 
 now shew how, conversely, their equations may be deduced 
 from a knowledge of their properties ; for example, let it be 
 required to find the locus of a point (P), the distance of which 
 from a given point [S) has a constant 
 ratio to its distance from a given straight 
 line (KX), 
 
 Suppose 8P:PK = e:l; draw 8X 
 perpendicular to KX, and divide 8X in 
 0, so that SO : OX=e : 1; then 
 will be a point in the locus; take OSx, Oy as axes, and 
 let OM{=x), MP {=y) be the co-ordinates of P; let 08 = d, 
 
 and, therefore, 0X=- , Then 
 
 S M X. 
 
 or 
 
 + PM'=8P\ 
 {v-dy + y'=e'.PK'; 
 d 
 
 whence we obtain 
 
 {I- e')x' + y^-2 {14- e) dx=tO 
 
 an equation of the second degree between x and y. 
 
 •(1), 
 
FOCAL PROPERTIES, ,193 
 
 207. We proceed to interpret this equation by the 
 methods of Chapter viii. The locus belongs to the Ellipse, 
 Hyperbola, or Parabola class, according as B'—4!AGis nega- 
 tive, positive, or zero, where A, B, C are the coefficients of 
 ^"i ^y, 2/°/respectively ; that is, according as 
 
 4 (e° — 1) <> = 0, or as e <> = 1. 
 
 I. Suppose 6 to be less than unity, or the locus to belong 
 to Class I. The equation may be written 
 
 Transferring the origin to a point x = ;j , «/ = 0, we 
 
 obtain 
 
 . , f _ d' 
 
 which represents an ellipse with the centre for origin, unless 
 d=0, or the given point is on the given line, in which case 
 the equation (Art. 107- ll.) is satisfied only by the values 
 
 x = 0, y = 0. If we replace the known quantity = by a, 
 
 and put a^ (1 — e") = h", we obtain the equation 
 
 which has been already discussed. 
 
 II. Suppose e to be greater than unityj or the locus to 
 belong to Class ii. In this case 1 — e and 1 — e" are negative ; 
 we shall therefore write the equation, 
 
 2dx y 
 e-1 ,e*-l ' 
 
 e— I , (T— I 
 
 13 
 
194 CENTRAL CONIC SECTIONS. 
 
 or 
 
 / d V f d' 
 
 Transferring to the point x = ^ , y = 0, we obtain 
 
 „. / 
 
 X —• 
 
 e'-l (e-iy 
 whicli represents an hyperbola- with the centre for origin, 
 unless <Z=0, or the given point is on the givea line, in which 
 case the locus is the two straight lines 
 
 If we replace the known quantity ^ by a, and put 
 
 we obtain the equation 
 
 which has been already discussed. 
 
 III. Suppose e to be equal to unity, or the locus to 
 belong to Class ill. The equation in this case becomes 
 
 j" = idx. 
 
 This represents the curve called the Parabola, which we 
 have not yet discussed, unless d=0, in which case it repre- 
 sents two straight lines coinciding with the axis of x. 
 
 Cor. Hence we see that the simplest form, by which 
 the three conic sections may be represented, is 
 
 where the curve is an ellipse, hyperbola, or parabola, accord- 
 ing as n is negative, positive, or zero. 
 
 The point will plainly be one of the vertices of the 
 curve, S the focus, and KX the directrix. 
 
FOCAX PROPERTIES. 195 
 
 208. We may find the equation to the above locus more 
 generally as follows. 
 
 Let the given point be {xy), and the given line 
 
 Ax + By+G=0, 
 
 then, if {xy) be any point on the locus, its distance from 
 
 =^[{x-x'Y + (y-yy]\ 
 
 and its distance from the given line is 
 _ Ax + By+G 
 
 Hence, by the conditions of the problem, we have 
 (X - xy + (r/-yy = -j^—^ (^^ + ^2/+ Of, 
 
 an equation of the second degree, which may be interpreted 
 by Chapter viii, 
 
 209. To find the polar equation to the ellipse or hyper- 
 bola, the focus being pole. 
 
 Here we have to find the polar equation to the locus of 
 Art. 206, the given point being a focus, and the given line 
 the corresponding directrix. In the figure of Art. 206 let 
 
 8P = p, aagleP8x = d, 8X=c; 
 then SP= e . PK= e (8X+ 8M). 
 or p=e(c + pcos^) (1), 
 
 Let = 90° ; then p = ec, or the whole chord through 8, 
 i. e. the latus rectum, is equal to 2ec ; hence, if we denote 
 the latus rectum by 21, (1) may be written 
 
 p = ^ — - — a' or- = l-ecos5 (2), 
 
 •^ 1 — ecos0 p 
 
 the equation required. 
 
 13— a 
 
196 CENTBAl CONIC SECTIONS. 
 
 By the figure used it will be seen, that this is the equa- 
 tion, when the left-hand focus is taken in the ellipse, and 
 the right-hand focus in the hyperbola, with their cori-espond- 
 ing directrices, and the point P has laeen. taken in that 
 branch of the hyperbola, in which the pole lies. If we take 
 the other foci and directrices in each, and the point P in 
 the left-hand branch of the hyperbola, the equation will be 
 
 - = l-l-ecos0 (3). 
 
 P 
 
 210, The student is recommended to trace the curves 
 from equations (2) and (3) of Art. 209, substituting for I its 
 value, a (1 — e^) in the ellipse, and a (e* — 1) in the hyperbola. 
 He will find, in the latter, that each equation represents both 
 branches, the positive values of p tracing the branch in which 
 the pole lies, and the negative values of p (Art. 12) tracing 
 the other branch. For example, if he traces from equation 
 
 (2), he will observe, that, when 6 = cos"* - , the value of p 
 
 e '^ 
 
 becomes infinite, and the radius vector is (Art. 174) parallel 
 to the asymptote CR (fig. Art. 166). Let this angle = aj 
 then as 6 varies from 
 
 to a, a to IT, TT to 27r — a, 217 — a to 2ir, 
 
 p varies from 
 
 — (ae + a) to — 00 , oo to ae — o, ae — a to oo , — oo. to — (ae+ a), 
 
 thus tracing the branches, A'E, RA, AL, L'A'. Thus it 
 appears that B, R, and L, L' are consecutive points on the 
 hyperbola, a result which has been noticed in Art. 185. 
 
 ' 211. To find the locus of a point P, the sum of whose 
 distances from two given points S and H is a constant quantity. 
 
FOCAL PROPERTIES. 197 
 
 Let 8P = p, HP = p', 8P + HP = 2a, 8ff = 2o, angle 
 P8H=0; then 
 
 p + p'=2a, 
 whence p^=p^ — 4tap + ^c?, 
 
 but p'" = p^+ 4c^ — 4cp cos 6 1 
 
 therefore d^ — ap = <? ~cp cos, 6, 
 
 ^ - a" 
 
 or 
 
 c cos d — a' 
 
 now, if we write ae for c, where e is less than unity, since 
 8P + HP > 8S, and therefore a> c, the equation becomes 
 
 ^ aCl-e") 
 '' 1-ecos^' 
 
 which is, evidently, (Art. 209), the equation to' an ellipse, 
 whose foci are 8 and ff, and whose eccentricity = e, '' 
 
 ' 212. To find the locus of a point P, ths difference of whose 
 distances from two given points S and H is a constant quantiti/. 
 
 Let EP = p, 8P= p, IT -PES = 6, then 
 
 p' — p = 2a, 
 
 whence p" = p' + 4tap + 4a'', 
 
 also p'^ = p' + 40" + 4cp cos 6, 
 
 hence, as before, we have 
 
 '' ~ 1 - e cos 6» ' 
 
 where e is greater than unity, since SE is greater than 
 8P- EP, 0x0 a. 
 
 This equation (Art. 209) represents an hyperbola, whose 
 foci are 8 and E, and whose eccentricity is e. 
 
198 CENTRAL CONIC SECTIONS. 
 
 EXAMPLES IX. 
 
 The following problems are enunciated, some for the ellipse, 
 and some for the hyperbola, though many of them are equally 
 applicable to both curves. 
 
 1. If the tangent to an hyperbola, at a point ■whose abscissa 
 CM is positive, meet the transverse axis in T ; A'M, A'A, A'T 
 will be in harmouical progression. 
 
 2. The distance of the centre of an ellipse from a tangent 
 inclined to the major axis at an angle ^, is = a (1 — e" cos'' ^)*, 
 
 3. The distance of the focus of an ellipse from a tangent 
 inclined to the major axis at an angle ^, is 
 
 a {e sin ^ + (1 - e^ cos' 0)4}, 
 
 4. rind the angle (5) at which the focal distance SP is in- 
 clined to the major axis, when SP is a mean proportional between 
 the semi-axes of an ellipse, when a = 50, 6 = 30. 
 
 5. If in any hyperbola, three abscissae be taken in arithmetical 
 progression, the focal distances of the extremities of the ordinates 
 of these points will also be in arithmetical progression, 
 
 6. Shew that the equations to the tangents to an ellipse 
 (3a:' + y = 3), inclined at an angle of 45° to the axis of x, are 
 y=a;-(-2, y = x — i. 
 
 7. If the semi-axes of an ellipse are 5 and 4, find the angle 
 at which CP is inclined to the major axis, when an arithmetic 
 mean between CA and CB, 
 
 *8. Find the eccentric angle (i) at the extremity of the latus 
 rectum of an ellipse, and (ii) at the point where x=y. 
 
EXAMPLES IX. 199 
 
 9. If any number of hyperbolas be described, Having the same 
 transverse axis, the tangents drawn at the extremities of their 
 latera recta will all pass through one point. 
 
 10. If the tangent at any point P in an hyperbola intersect 
 the axis in T, and CP meet the tangent at A in E, ET is parallel 
 iaAP. 
 
 1 1. Shew tan — -z^ — tan — -^ — = = , where P is any point in 
 
 2 2 1 + e' -^ 
 
 an ellipse. 
 
 12. Find the points of intersection of the ellipse and hyper- 
 bola whose equations are 
 
 a:' + 22/'=l, 3a^-6y' = l, 
 
 and shew that at each of these points the tangent to the ellipse is 
 the normal to the hyperbola. 
 
 13. If GA, CB be the semi-axes of an ellipse, shew that, when 
 SBH is a right angle, GA" : GB'=-2 : \. 
 
 14. Find the condition that the line ( — + - = 1 ) should touch 
 
 \m n / 
 
 the hyperbola ( ^ - jJ = 1 J ■ 
 
 15. The tangent to an ellipse is inclined to the major axis at 
 an angle ^; shew that the area included by this tangent and the 
 axes is = ^ («' tan tf> + h' cot <^). 
 
 16. The circle described on any radius vector iSfP of an ellipse 
 as diameter, will touch the circle on the axis major. 
 
 17. Find where the tangents from the foot of the directrix 
 will meet the hyperbola, and what angle they will make with the 
 transverse axis. 
 
 18. Find the equation to the tangent at the extremity of the 
 
 . a? if 
 latus rectum of an ellipse whose equation is g— , + — , = 1. 
 
200 EXAMPLES IX. 
 
 19. A. tangent at the extremity of the latus rectum of an 
 hyperbola meets any ordinate PM produced in B ; shew that 
 SP = MR, where S is the focus through which the latus rectum 
 passes. 
 
 *20. Shew that the equation to the normal, at the point 
 whose eccentric angle is <f), is ax sec <l>-by cosec <j> = a'' — ¥. 
 
 21. Find the radius of a circle inscribed in a semi-ellipse, 
 touching the axis minor. 
 
 22. From the point where the circle on the major axis is 
 intersected by the minor axis produced, a tangent is drawn to the 
 ellipse J find the point of contact, 
 
 23. If from the extremities of the minor axis two straight 
 lines be drawn through any point in the ellipse, and intersect the 
 axis major in Q and R, then GQ . OR=CA'. 
 
 24. If a rod slide between a vertical wall and a horizontal 
 plane, any point on it traces out an ellipse. 
 
 25. If a tangent be drawn to the interior of two concentric 
 ellipses, the axes of which are in the same straight line, meeting 
 the exterior one in P, Q, and at P, Q tangents be drawn to the 
 latter, intersecting in R, prove that the locus of R is an ellipse. 
 
 26. Shew that the locus of one end of a given straight 
 line, whose other end and a given point in it move in straight 
 lines at right angles to one another, is an ellipse. 
 
 27. If with the co-ordinates of any point in an elliptic quad- 
 rant as semi-axes, a concentric ellipse be described, the chord of 
 the quadrant of the one will be a tangent to the other. 
 
 28. The locus of the centre of a circle touching two circles 
 externally is an hyperbola. 
 
 29. The locus of the centre of a circle touched by one circle 
 externally and one internally is an hyperbola. 
 
 30. Find the locus of the extremity of the perpendicular from 
 the centre on the tangent to the hyperbola. 
 
EXAMPLES IX. 201 
 
 *31. If 6, <j) be the eccentric angles of two points on an 
 ellipse, the equation to the chord joining the two points is 
 
 X 6 + dt V . 6 + d) 6 — A 
 
 - cos — ^ + ^ sin —;— = cos -s— . 
 a 2 2 2 
 
 Hence deduce the equation to the tangent at the point whose 
 eccentric angle is tfi. 
 
 *32. Prove Arts. 203, 204 by means of the equation to the 
 tangent obtained in Ex. 31. 
 
 33. If 3AG= 2CS in an hyperbola, find the inclination of the 
 asymptotes to the transverse axis. 
 
 34. If from a point P in an hyperbola, FK be drawn parallel 
 to the transverse axis, cutting the asymptotes in J and -2", then 
 PK. PI=a\ or, if parallel to the conjugate, PK.PI=h\ 
 
 35. Is the point (2, 3) wUhout or within the hyperbola 
 2x^ -Zy'' = 1 % Shew that the straight line, joining this point 
 with the point (6, 4) cuts the curve, 
 
 36. If A, A' be the extremities of the major axis of an 
 ellipse, T the point where the tangent at P meets AA', QTR a 
 line perpendicular to A A', and meeting -4P, A'P in Q and R re- 
 spectively, then QT= TR. 
 
 *37. Find the eccentricity and latus rectum of the conic 
 
 22^ + a;'' + 4y-2a!-6 = 0, 
 
 the axes being rectangular. 
 
 38. Find the equations to the asymptotes of the curve 
 
 3ar'-10a!y + 32^-8 = 0; 
 and find the angle between the asymptotes of 
 
 y^-lOay + as^ + y + iB + l = 0, 
 the axes in the latter case being rectangular. 
 
 39. How many normals can be drawn to an ellipse fi^om 
 a point on the major axis, and how many from a point on the 
 minor axis ? 
 
202 , EXAMPLES IX. 
 
 40. In the equilateral hyperbola, the eccentricity is the ratio 
 of the diagonal of a square to its side. 
 
 41. A tangent at any point P of an ellipse meets the axis 
 major produced in T, and the axis minor produced in i ; to find 
 the locus of a point Q in Tt, such that, QT : Qt = m :n. 
 
 42. To find the locus of the intersection of the ordinate of 
 any point in an ellipse produced, with the perpendicular from 
 the centre upon the tangent at that point. 
 
 43. If the normal at P meet the axis major of an ellipse in 
 G, and GK be drawn perpendicular to SP, GK=e.PM, -where 
 PM is the ordinate of P. 
 
 44. If SQ be drawn, always bisecting the angle PSG, in an 
 ellipse, and equal to a mean proportional between SG and SP, find 
 the eccentricity of the curve which is the locus of Q. 
 
 45. Two straight lines, such that the product of the tangents 
 of their inclinations to the axis of x is constant, touch an ellipse ; 
 shew that the locus of their intersection is an ellipse, or hyperbola, 
 according as the product is negative or positive. 
 
 46. Shew that the locus of the summit of a moveable right 
 angle, one side of which touches one, and the other side the other 
 
 ' of two confocal ellipses, is a concentric circle. 
 
 47. An ellipse and hyperbola have the same foci and coin- 
 cident axes ; they cut each other at right angles. 
 
 48. If P be any point in the hyperbola, S and H the foci, 
 find the locus of the centre of the circle which is inscribed in 
 SPH. 
 
 49. If a tangent at any point of an hyperbola be intersected 
 by the tangents at the vertices in E and K, the circle on HK as 
 diameter passes through the foci. 
 
CHAPTER X. 
 
 Central Conic Sections. Conjugate Diameters. 
 The Hyperbola re/erred to its Asymptotes. 
 
 213. We saw (Art. 145), that there is anly one system 
 of rectangular axes, about which the ellipse and hypetbola 
 are so situated, that each axis should bisect all chords parallel 
 to the other. "We shall now shew, that there are an infinite 
 number of oblique axes, which possess the above property 
 with regard to the two curves. 
 
 214. To find the locus of the middle points of any system 
 of parallel chords. 
 
 Let QQ' be one of the chords, M its middle point {x'y'), 
 
 J« a 
 
 and let the equation to Q^ he 
 
 x — x'_y — y 
 
 s 
 
 = t. 
 
 (!)• 
 
 Now, if we substitute for x and y from this equation in 
 the equation 
 
 ^+^ = 1 
 
 _2 ~ 12 '- 
 
 (2), 
 
204! CENTRAL CONIC SECTIONS. 
 
 we obtain equation (2) of Art. 176, to determine the dis- 
 tances (I) of the point (x'y) from the intersection of (1) and 
 (2). But, since these distances {MQ, MQ') are equal in 
 magnitude; the two values of I are equal and of opposite 
 signs (Art. 34, Cor.) ; hence the coefficient of Z = 0, or 
 
 a^ ^ F ' 
 which is a relation between the co-ordinates of the middle 
 point of the chord QQ'. But, since the chords are parallel, 
 s and c are the same for them all, and the same relation 
 holds for the middle points of all ; hence the equation to the 
 locus required is 
 
 which represents a straight line {CM) through the centre. 
 
 If m be the tangent of the angle which the chords make 
 with the axis of a;, m = -, and the equation is 
 
 " a'm 
 The equation for the hyperbola is, of course, 
 
 " am. 
 
 Cor. If (a;y) be the point P, where CM meets the 
 curve, we have 
 
 m = — —,, 
 
 ay 
 
 but the right-hand member (Art, 181) represents the tan- 
 gent of the angle, which the tangent at {xy) makes with ' 
 the axis of x. Hence the tangent at the extremity of CP is ; 
 parallel to the chords bisected by that line, as it evidently- ; 
 should be, since it may be considered as the limiting position 
 of any chord QQ', as it moves parallel to itself up to P. 
 
CONJUGATE DIAMETERS. 
 
 205 
 
 215, The straight line, which bisects any system of 
 parallel chords, is called a diameter of the curve, and the 
 chords are called the ordinates of that diameter. We have 
 seen above (Art. 214), that, if the equation to one of the 
 chords be 
 
 y = mx + c, 
 
 the equation to the diameter, of which those chords are ordi- 
 nates, is 
 
 V 
 " am 
 
 We see then, that all diameters pass through the centre, 
 and, conversely, since m may have any value, all lines passing 
 through the centre are diameters. We shall see hereafter, 
 that the same is true in the case of the parabola; but, the 
 centre of the parabola being infinitely distant, all its dia- 
 meters will consequently be parallel, 
 
 216. If two diameters be such, that one of them bisects 
 all chords parallel to the other, then the second will bisect all 
 chords parallel to the first. 
 
 Let the diameter CP{y = mx) bisect all chords parallel 
 
 to the diameter CD {y = m'x). Then, by Art. 215, 
 
 m = — 
 
 arm 
 
 or mm = — »; 
 
206 
 
 CENTRAL CONIC SECTIONS. 
 
 and this is the only condition that must hold, in order that 
 {y = m'x) should bisect all chords parallel to {y = mx). 
 
 217. Diameters so related, that each bisects every chord 
 parallel to the other, or, more commonly, such portions of 
 these diameters, as are intercepted by the curve, are called 
 Conjugate Diameters; and the condition that the diameter 
 (y = mx) should be conjugate to the diameter {y = m'x) is, 
 by the last article, 
 
 mm = — :.. 
 a' 
 
 We shall see hereafter that only central curves can have 
 conjugate diameters. 
 
 If 0, be the angles which the conjugate diameters make 
 with the major axis, we have 
 
 y 
 
 m the ellipse, tan 6 tan ff = — 5 , 
 
 in the hyperbola. 
 
 tan 6 tan 6' = 
 
 hence; in the ellipse the conjugate diameters fall on different 
 sides of the axis minor; for, if one of the angles 6, & be 
 acute, and its tangent positive, the other must be obtuse, 
 and its tangent negative; and we see by similar reasoning, 
 that in the hyperbola the conjugate diameters lie on the same 
 side of the conjugate axis. 
 
 218. Also, of any two conjugate diameters, only one can 
 meet the hyperbola; for if 
 one of the tangents (tan ff) 
 
 be less than - , the other 
 a 
 
 (tan &) must be greater 
 
 than -, and it is evident 
 a 
 
CONJUGATE DIAMETEES. 207 
 
 from Art. 169 that the diameter, which makes the angle 
 •with the axis, will fall within the angle BOL, made by the 
 asymptotes, and meet the curve, while the diameter, which 
 makes the angle ff with the axis, will faU within the angle 
 RCL', and will not meet the curve, for it has been shewn 
 
 that taia.ROA=-. 
 a 
 
 If, in the hyperbola, tan 6 = tan 6' = ±- , the two dia- 
 meters coincide with the asymptote BB' or with LL', accord- 
 ing as we take the upper or lower sign; hence each asymptote 
 is its own conjugate diameter. 
 
 If tan d = — tan 6' = — , in the ellipse, the angles d and ff 
 
 are supplementary, and, by the symmetry of the figure, the 
 conjugate diameters are equal; hence the equal conjugate 
 diameters in an elUpse are parallel to chords joining the 
 extremities of the major and minor axes ; and if an ellipse 
 and hyperbola have the same centre and axes, the equal 
 conjugate diameters of the ellipse coincide with the asymp- 
 totes of the hyperbola. 
 
 219. When the ellipse becomes a circle, V = a', and 
 
 tan 6 tan 6' = — \, 
 or all conjugate diameters of the circle are at right angles to 
 one another, and when the hyperbola is rectangular, 
 tan 6 tan & = 1, 
 
 therefore - d + ff = -aOt=-^. 
 
 220. We saw (Art. 195), that, when the curve is referred 
 to its axes, as axes of co-ordinates, the 
 polar of any point in the axis of x is 
 parallel to the axis of y, and vice versd. 
 This may be now seen to be a property 
 of aU coiljugate diameters, including 
 
208 CENTRAL CONIC SECTIONS. 
 
 the axes as a particular case. For suppose the equations to 
 OF and CD to be 
 
 y=^<^ W' y^--^"" (^)- 
 
 Now, if («'/) be any point on CB, the equation to the 
 polar of {xy) is 
 
 which is the equation to a line making with the axis of x 
 
 ¥x' 
 an angle whose tangent is —, ; but since {x'y') is on (2), 
 
 V = 5 — X, or m = 5-7, 
 
 " am ' ay 
 
 and therefore (3) is parallel to (1) ; hence the polar of any 
 point on CD is parallel to CP, and vice versd. 
 
 CoE. If ips'y) be the point D, where the diameter meets 
 the curve, the equation to the polar 
 
 will represent the tangent at D; hence, the tangent at the 
 extremity of any diameter is parallel to its conjugate, as we 
 saw in Art. 214, Cor. 
 
 221. The co-ordinates of ike extremity of any diameter 
 heing given, to find those of the extremity of the diameter con- 
 jugate to it. 
 
 Let CP, CD (fig. Art. 220) be a pair of conjugate diame- 
 ters in the ellipse, and let x, y be the co-ordinates of P; then 
 the equations to CPand CD (Art. 216) are 
 
 y4^ W' y—'^^ (2). 
 
CONJUGATE DIAMETERS. 209 
 
 To find the co-ordinates of Z) {xy), we have from (2) the 
 relation 
 
 ay _ hx 
 bx ay 
 
 (3), 
 
 also ay + &V = aV =aY + V'x' (4), 
 
 since {xy), (x'y) are points on the curve ; 
 
 hence from (3) 
 
 6V dY 
 
 or from (4) 
 
 I'a? =aY, 
 
 
 ^=±y; 
 
 hence 
 
 --V- 
 
 CN=-iy\ or- = -f . 
 b'^ a b ■ 
 
 .and from (2) J)N=-x', or f = - 
 
 b a 
 
 The other pair of values f a; = t y', y = — x'j have refer- 
 ence to the extremity D'. 
 
 222. In the hyperbola, if the diameter OP meet the 
 curve, CD will not (Art. 218) meet it ; and, indeed, it is evi- 
 dent that the method of the last article will give us imagi- 
 nary values of the co-ordinates of D; for, as in equation (3) of 
 the last article, we have for the co-ordinates of J) (xy), 
 
 ay^i_^ /jv 
 
 bx ay ^ ^' 
 
 ay-6V -(ay-5V°) 
 hence 6V = ^/^ ^^^' 
 
 which would ^ve JV = — ay. Ifj however, we take D {xy) 
 
 14 
 
210 
 
 CENTEAL CONIC SECTIONS. 
 
 as the point where CD meets the con- 
 jugate hyperbola, whose equation is 
 
 we have 
 
 ay - &V = a^W = - [aY - V-x>"), 
 whence, from (2) and (1) 
 
 x=CN= \^, y = DN=-x'. 
 
 The other pair of values have reference to D'. 
 
 We shall therefore define the extremity of the diameter 
 conjugate to GP, as ' the point where it meets the conjugate 
 hyperbola.' 
 
 It is evident, by exactly the same reasoning as we have 
 used in the case of the hyperbola itself, that, if we consider 
 CD as a diameter of the conjugate hyperbola, GF wiU be the 
 diameter conjugate to GD ; and whatever is proved of the 
 point P, as a point in the hyperbola itself, is true of the 
 point D, as a point in the conjugate hyperbola ; for instance, 
 (Art. 220), the tangent at D is parallel to GP, and so forth. 
 
 *223. If ^ is the eccentric angle of the point P, we shall 
 have (Arts. 164, 221) in the ellipse, 
 
 for P,x=a cos ^, ?/= S sin ^ ; for -D, x = — a sin^, y= h cos ^; 
 
 and (Arts. 172, 222) in the hyperbola, 
 
 for P, £e = a sec ^, y = l tan ^; for Z), x =a tan ^, y=h sec ^. 
 
 These results will be found very useful in the solution of 
 problems which relate to conjugate diameters. 
 
 *224. If 4>, (j) be the eccentric angles of P and Z> in the 
 ellipse, we have for JD 
 
 X = a cos (ji = — a sin <j), y = i sin ^' = J cos ^ ; 
 
CONJUGATE DIAMETERS. 211 
 
 therefore cos j>' — — sin ^, sin <^' = cos <f>, 
 
 whence <^' = 90° + (j). 
 
 If the eccentric angle 4>' of ^iny point in the conjugate 
 hyperbola, be constructed as in the given hyperbola, the co- 
 ordinates of the point may be written, 
 
 x = a tan <}>', y = h sec ^', 
 since these values satisfy the equation ^ — 2 = 1; hence, if <^, 
 
 ^ be the eccentric angles of P and D in the given and con- 
 jugate hyperbola respectively, we have for D 
 
 x=ata.n<f)' =atan^, y = i sec<f>' = &sec0; 
 
 therefore tan tji' = tan </>, sec ^' = sec <}>, 
 
 whence <f> = <}>'. 
 
 CoE. The above properties afford a simple method of 
 constructing geometrically the diameter CD, conjugate to a 
 given one GP. For, if we construct the eccentric abgle of P 
 by Arts. 164, 172, we may construct the eccentric angle of D 
 as above, and thus find the point D. 
 
 225. In the ellipse, the sum of the squares of any two 
 semi-conjngate diameters is equal to the sum of the squares^ 
 of the semi-axes; and, in the hyperbola, the same is true of 
 their difference. 
 
 Let P {<x^y') and D {x"y") be the extremities of any two 
 semi-conjugate diameters CP (a'), and CD (6') ; then 
 
 a"=GP'=oi>''+y'\ 
 
 and b''==CD' = x"'+y" 
 
 =^+-5- (^^^-^^i-) 
 
 14—2 
 
212 ' CENTEAL CONIC SECTIONS. 
 
 Therefore a" + h" = (a" + 5^ K + |J) 
 
 = a" + i', in the ellipse ; 
 
 /2 '2 
 
 , se y -, 
 
 since m that case — a + tt = 1- 
 
 a 
 
 Also a" - h'' = (a' - V) (|J - •^') 
 
 = c^ — h^ in the hyperbola ; 
 
 '2 '2 
 
 since in that case —3—^ = 1. 
 
 Cob. If a' = J', we have in the hyperbola, 
 
 or every diameter in the equilateral hyperbola equals its con- 
 jugate. 
 
 226. The rectangle contained by the focal distances of 
 any point, is equal to the square of the corresponding semi- 
 conjugate diameter. 
 
 In the ellipse Giy = a^ + ¥- CP', (Art. 225) 
 
 but ■ CF^ = a;' + y" = x^ + w{l-'^X 
 
 = 5= + eV''; 
 
 therefore GD^ =a^- eV = (a - ex') (a + ex) 
 
 = RP.8F. (Art. 196.) 
 
 A similar investigation for the hyperbola would lead us to 
 the same result. 
 
 227. To find the length of the perpendicular from the 
 centre on the tangent at any point P (x'y'), in ternfis of the 
 semi-diameter, conjugate to CP, 
 
CONJUGATE DIAMETEES. 213 
 
 If p be the length of a perpendicular from the origin on 
 the line ^ + ^ = 1, we have (Art. 52, Cor. 2) 
 
 1 aV 
 
 P 
 
 W h*) V a= ^ V J 
 
 therefore p = ^; for 6 ^ = —^ + —^^ , as m Art. 225. 
 
 Qi 
 
 228. -4?Z parallelograms, whose sides are formed hy straight 
 lines parsing through the extremity of one diameter and parallel 
 to its conjugate, are equal in area. 
 
 By Art. 220, these lines are tangents to the curve, for 
 it is there proved that the tangent at the extremity of any 
 diameter is parallel to its conjugate ; and, by Art. 181, the 
 tangents at the extremities of any diameter are parallel to 
 one another. 
 
 Let PP, DU be two conjugate diameters, and let the 
 
 sides of the parallelogram be tangents at P, P', D, If. Then 
 the area of the whole parallelogram 
 
 = 4 times the parallelogram CPFD 
 
 = 4i.CQ. CD, 
 
214 CENTRAL CONIC SECTIONS. 
 
 . where CQ is the perpendicular from the centre on the tan- 
 gent at P, 
 
 = 4 ^ . &' (by Art. 227) 
 = 4a&. 
 
 229. From the last article may be found the angle be- 
 tween any two conjugate diameters which are given ; for, if 
 this angle be = 7, and the given semi-conjugate diameters 
 be a', b', we have 
 
 a'b' sin y = parallelogram CPFD 
 
 = db; 
 
 ah 
 hence sm 7 = —n-, • 
 
 ao 
 
 This equation together with the relations 
 a!" + h'" = a" + b\ in the elhpse, 
 a'" — b'^ = a^ — V, in the hyperbola, 
 
 determine also the magnitude of two conjugate diameters 
 that contain an angle 7. Their position is known from the 
 equation 
 
 tan 6 tan (^ + 7) = 5 , 
 
 Oj 
 
 where 6 is the angle that GP makes with the axis of x. 
 
 230. The angle 7, POD, in the ellipse, is always, except 
 in the case of the axes, greater than a right angle, P being 
 supposed in the first and D in the second quadrant; for 
 if TO and m' be the tangents of the angles, that CP and CD 
 make with the major axis. 
 
 m —lib 
 
 tanv^^j— ; (1), 
 
 1 + mm ^ " 
 
CONJUGATE DIAMETERS. 215 
 
 and since m = 5— . 
 
 am' 
 
 '-^-^^i (^>'. 
 
 and therefore, since m is always positive, and a greater than 
 S, 7 is > 90°, unless m = 0, in which case 7 = 90°. 
 
 When a — h, tan 7 = 00 , or the conjugate diameters of 
 the circle are all at right angles to each other, as we found in 
 Arts. 132, 219. 
 
 Solving equation (2) as a quadratic in m, we have 
 
 - (a' - V) tan 7 + \/(a''-6')Han''7^4a'6' ,„, 
 
 Hence the smallest negative value that tan 7 can have, 
 (since it is always negative,) is that which makes the quan- 
 tity under the root in (3) vanish, which is 
 
 This gives m = - .. from (3), which is therefore the value 
 
 of m which makes 7 the largest ; hence (Art. 218) the equal 
 conjugate diameters include the greatest angle. 
 
 231. To find the equation to the ellipse or hyperbola, when 
 referred to any two conjugate diameters (2a', 2J') as axes. 
 
 We saw (Art. 145), that there is only one system of rect- 
 angular axes, to which, when a central curve is referred, its 
 equation is of the form 
 
 an equation which asserts that all chords parallel to one axis 
 are bisected by the other. But every diameter (Art. 216) 
 bisects the chords parallel to its conjugate; hence there are 
 an infinite number of oblique axes, which will give the equa- 
 
216 CENTRAL CONIC SECTIONS. 
 
 tion in the above form, the only limitation being, that they 
 should pass through the centre, and that the angles {6, ff), 
 which they make with the major or transverse axis, should be; 
 subject to the condition 
 
 tan^ tan 6' = — ^ , in the ellipse, 
 
 V 
 tan d tan ff = --^,m the hyperbola. 
 
 Hence, the reasoning of Arts. 150, 158 applies to conju- 
 gate diameters, in precisely the same manner as to the axes 
 of the curve ; and the equations to the ellipse and hyperbola, 
 referred to any pair of conjugate diameters, the parts of 
 which intercepted by the curve are 2a' and 25', are 
 
 _, + |,= l, and^-|,= l. 
 
 232. Since this equation is of precisely the same form 
 as the equation to the curve referred to its axes, it follows 
 that every property, that has been deduced from the latter, 
 may be deduced from the former, so long as those properties 
 do not depend upon the inclination of the axes ; and, with 
 this limitation, everything which has been proved of the 
 axes, is true of any pair of conjugate diameters. 
 
 For example, the equation to the tangent at any point 
 {xy), when the. curve is referred to any conjugate diameters 
 (2a', 26'), is 
 
 a"= "^ b"" ~ ' 
 
 and the intercept on the axis of x, and the subtangent, are, as 
 
 '2 '2 '2 
 
 before, — r and ; — . Also, if we wish to draw a tangent 
 
SUPPLEMENTAL CHORDS. 217 
 
 to the curve from an external point {x'y), we have, as in 
 Article 123, 
 
 £'-j.2^-i ^^' ,yy' -^ 
 
 as equations to determine the points of contact. 
 
 233. The equation to the asymptotes, when the hyper- 
 bola is referred to any pair of conjugate diameters, is (Art. 
 186) 
 
 also, by Arts. 222, 186, the equations to the conjugate 
 hyperbola and its asymptotes, when referred to the same 
 axes, are 
 
 234. We may obtain a simple geometrical method for 
 constructing a pair of conjugate diameters, containing a given 
 angle, as follows. 
 
 The reasoning of Art. 216 is equally applicable, when the 
 curve is referred to any pair of conjugate diameters^ ; hence 
 we see that, when the curve is referred to a pair whose semi- 
 lengths are a', b', the condition that the two lines 
 
 y = mx, y = mx 
 
 should represent two conjugate diameters is 
 
 mm = — r,- . 
 a 
 
 ^ In the proof of Art. 214, if u is the angle between the conjugate dia- 
 meters, we have (Art. 34), 
 
 Rin sin (u-S) , sin 9 , sin 9' 
 
 1 = —. — , c= — 5 -, and.*, m— - "•'— 
 
 sin M sm w sin {la-B) sin (w - 9^ 
 
218 CENTRAL CONIC SECTIONS. 
 
 Now, if the curve be referred to these two diameters, the 
 equation may be written 
 
 which equation may be split into the two 
 h'^ 1 
 
 where i is a perfectly arbitrary quantity; now these two 
 straight lines (i) pass through the extremities of the diameter 
 which is the axis of x; (ii) intersect in the curve, since by 
 eliminating h between them we have the equation to the 
 curve ; (iii) fulfil the same conditions, with regard to their 
 inclinations to the axis of a; as a pair of conjugate diameters, 
 for 
 
 GJr^a'J-"^' 
 
 Straight lines drawn in this manner from the extremities 
 of any diameter to a point in the curve, are called Supple- 
 mental Chords ; hence diameters parallel to any pair of sup- 
 plemental chords are conjugate. 
 
 Hence, to draw a pair of conjugate diameters containing 
 any given angle, describe on any diameter a segment of a 
 circle containing that angle, and join the point where it 
 meets the curve with the extremities of the assumed dia- 
 meter. We thus obtain a pair of supplemental chords in- 
 clined at the given angle. The straight lines drawn through 
 the centre parallel to these, will be the required conjugate 
 diameters. 
 
 The property of supplemental chords was demonstrated 
 for the circle (Art. Ill), when it was shewn that all supple- 
 mental chords in the circle are at right angles, as are all 
 conjugate diameters. 
 
SUPPLEMENTAL CHORDS. 219 
 
 235. The above property is evident geometrically thus. 
 Let PP' be any diameter, and Q any q 
 
 point in the curve ; draw GR, OS to y^ /~7^ 
 the middle points oi PQ, P'Q respec- / si/ 
 tively; then CB and CR' are (Euc. vT /y^^ 
 VI. 2) parallel to P'Q, and PQ respec- ^^ 
 tively ; but CR bisects all chords pa- 
 rallel to PQ, or GR! ; and GE bisects all chords parallel to 
 F Q, or GR. Hence GR, OR' are conjugate, and PQ, P'Q 
 are parallel to conjugate diameters. 
 
 236. Tangents at the extremities of any chords intersect 
 in the diameter of which the chord is an ordinate. 
 
 If we take that diameter and its conjugate as the axes 
 of X and y, the equation to the tan- « 
 
 gent will be 
 
 
 a' - V^ ~ ' 
 according as we take Q {x'y) or 
 Q'{x', — y'\ as the point of contact. 
 
 '2 
 
 In each case, when y=0, x has the same value = — ;- , or 
 
 X 
 
 the tangents meet CP produced in the same point. 
 
 This may also be proved as follows. Let the equation 
 to the chord be 
 
 Ax+By+G=0 (1); 
 
 then, by Art. 215, the equation to the diameter that bisects 
 it may be -vvritten 
 
 Bh'x-Aa'y = (i (2). 
 
 If {xy') be the point of intersection of tangents at the 
 extremities of (1), then, by Art. 123, (1) is equivalent to 
 
 — -h^-1 Ci-S 
 
 a" ^^~^ ^'*^' 
 
220 CENTRAL CONIC SECTIONS. 
 
 Aa" Bb' O ,,. 
 
 whence — - = — r = — ^ W > 
 
 X y 1 
 
 and the values of x and y' in (4) will be seen to satisfy (2). 
 
 Hence the point of intersection is on the diameter. 
 
 237. The Asymptotes. 
 
 The following articles relate exclusively to the hyperbola, 
 since the asymptotes of the ellipse have been shewn to be 
 imaginary. 
 
 The diagonals of all parallelograms, described as in 
 Art. 228, are parallel to the asymptotes. 
 
 Take CP(=a') and CD{=b') as the axes of x andy; 
 then F (fig. Art. 228) is the point («'&'), and the equations 
 to CF, PD are 
 
 a 6' "' al^b' • 
 
 They are therefore parallel to the asjrmptotes, whose equa- 
 tions are (Art. 233) 
 
 a' b' ' d^y "■ 
 
 238. Hence, if any two conjugate diameters OP, CD be 
 given in position, we can find the asymptotes, by completing 
 the parallelogram CPFD, the diagonals of which will shew 
 the direction of the asymptotes; or, if the asymptotes be 
 given, we can find the position of the diameter conjugate to 
 any diameter CP, whose position is given ; for, if we draw 
 PO parallel to one asymptote to meet the other in 0, and 
 produce PO to D, making OD = PO, CD will be the dia- 
 meter conjugate to CP. 
 
 239. If any line {BR') cut an hyperbola and its asymp- 
 totes, the portions {SQ), {H'Q') intercepted between the curve 
 and the asymptotes a/re equal. 
 
THE ASYMPTOTES. 
 
 221 
 
 The eqiiation to the two asymptotes, considered as one 
 locus, is (Art. 169) 
 
 S 2 
 
 .(1). 
 
 Now let the equation to 
 RR, passing through a point 
 P(a!V),be 
 
 X 
 
 s 
 
 •(2); 
 
 then, for the distances of P 
 from B and R, we have from 
 {1} and (3) 
 
 M±fT_M±y)!_o 
 
 •(3); 
 
 and the condition that P {x'y') should be the middle point 
 of RR' is, exactly as in Art. 214, 
 
 
 (4); 
 
 but this is the condition (Art. 214) that (xy) should be on 
 the diameter bisecting QQ'. Hence, if P be the middle 
 point of RR, it is also the middle point of QQ, and we 
 have 
 
 PQ = Pq, PR = PR', and .: RQ = R' Q. 
 
 Cor. Equation (4), which is the condition that (a;y) 
 should be the middle point of the chord RR, is also 
 (Art. 180) the condition that the line should be a tangent 
 to the curve at the point {xy) ; hence, if the middle point 
 [^y) of RR be on the curve, RR is a tangent, or the portion 
 of the tangent intercepted by the asymptotes is bisected at the 
 point of contact. This follows directly from the considera- 
 tion, that the tangent SPS' is the limiting position of RR, 
 when it is moved parallel to itself up to P'. 
 
222 
 
 CENTRAL CONIC SECTIONS. 
 
 240. Frmn the equation to the hyperbola referred to its 
 axes, to derive the equation when referred to its asymptotes as 
 axes. 
 
 Let the lower asymptote be taken for axis of x, and let 
 CM, PM be the original co-ordinates, 
 CN, FN the new ones ; draw NQ per- 
 pendicular to CM, and ^F parallel to 
 it, meeting P-M" produced in V, and let 
 the angles BGM, LGM each = a; then 
 
 tan a = - , and, if d ■\-V = rr^, we have 
 a 
 
 a . h 
 
 cosa = — , sma = — . 
 m m 
 
 Now GM= NV+ GQ = PiVcos a + OZVcos a, 
 
 PM= PV- QN= PiV sin a - CiVsin a ; 
 
 hence we must write in the equation to the hyperbola. 
 
 for 
 
 ioty. 
 
 {y + x) cos a, or 
 
 a iy + x) 
 
 which gives 
 
 or 
 
 I \ ■ h(y — x) 
 
 [y — x) sin a, or — ' 
 
 m 
 
 {y + xy-{y-xY=.m\ 
 
 a^ + b' 
 
 which is therefore the required equation, 
 the conjugate hyperbola is (Art. 175) 
 
 a' + S^ 
 
 ^y = — 2— • 
 
 The equation to 
 
 241. The result of the last article may be obtained in- 
 dependently as follows. Let the equation to the hyperbola 
 referred to its asymptotes be 
 
 Ax' + Bxy+ Gy'' + Dx + Ey + F=-0 (1). 
 
HYPERBOLA EEFEREED TO ITS ASYMPTOTES. 223 
 
 Putting y=0, we have for the points where the curve 
 meets the axis of x, 
 
 Ax' + I)x + F=:0 (2); 
 
 but, since the two roots of this equation are infinite, we 
 have (Appendix) ^ = 0, Z> = 0. Similarly C=0, U=0, and 
 equation (1) becomes 
 
 Bxi/ + F=0, or xj/ = c' (3). 
 
 If now h and k be the co-ordinates of the vertex, and oi 
 the angle between the asymptotes, we have 
 
 hk = c'' (4), GA'' = h'' + Je' + 2hk cos CO (5). 
 
 But since OA bisects the angle between the axes, h = k, 
 and therefore from (4), = c ; hence from (5) 
 
 2c=(l + cosw)=a'' (6); 
 
 1 J. , (O b _ . n 2" ^a^ 
 
 but tan — =-, .'. l+cosa) = 2 cos Tj = ^ — r?; 
 •2 a 2 a^ + b" 
 
 therefore c' = — 3 — . 
 
 242. The equation then to an hyperbola referred to its 
 asymptotes is xi/ = k, where A; is a constant, and is positive 
 or negative, according as the curve lies in the angles where 
 X and y are both positive or both negative, or in those where 
 X and y are of different signs ; also the equations to the 
 asymptotes are a; = 0, y = 0. Hence, if any transformation 
 whatsoever of axes be made, so that (Art. 74) we have to 
 write 
 
 for a;, ax +hi/ + c; for y, a'x + b'y -^ c', 
 the equation to the hyperbola will become 
 
 (aa3 + % + c) (a'a; 4- % + c') = ^ (1), 
 
 where ax-^by + c=Q (2), a'a; + % + c'=0 (3), 
 
 are the asymptotes, since a; = 0, ^ = are transformed to 
 these equations. Hence the equation to any hyperbola, what- 
 ever be the origin or axes, may be written in the form (1). 
 
224 
 
 CENTRAL CONIC SECTIONS. 
 
 Conversely, if any equation of the second degree can be 
 written as (1), it represents an hyperbola with (2) and (3) for 
 asymptotes ; for suppose the lengths of straight lines drawn 
 from {xy), parallel to (3) to meet (2), and parallel to (2) to 
 meet (3), to be (Arts. 52, 155) respectively 
 
 p {ax + 1^+ c), q (a'x + Vy + c') ; 
 then, since equation (1) may be written 
 
 ■p [ax + hy -V c) x c[ [ax + Vy + c) = hpq, 
 it asserts that the product of two such lines, drawn from any 
 point [pay) on the locus, is constant ; hence, if (2) and (3) are 
 taken as axes, we shall have xy = SL constant, which equation 
 (Art. 150) must represent an hyperbola; for it cannot be 
 broken into two linear equations, unless the constant = 0, in 
 which case it represents the axes. 
 
 Since the equation xy = k represents the conjugate hyper- 
 bola if the sign of k is changed, so also does equation (1), 
 which is obtained from it ; and all the curves obtained from 
 (1) by varying k only, have the same centre and asymptotes. 
 Also, if we draw the lines (2) and (3), and fnark their positive 
 and negative sides (Arts. 55, 66), it is easy to see in which 
 angles the hyperbolas lie. If the positive sign is used with k, 
 
 the curves must lie in the ++ and compartments formed 
 
 by the lines ; for then the expressions 
 
 ax + ly + c, a'x + b'y + c 
 are both positive or both negative. If the negative sign is 
 used, the curves lie in the -\ — and — \- compartment. 
 Ex. The equation 2' 
 
 (y-a;)((j> + 2y-3)=7, 
 represents an hyperbola, of which the 
 asymptotes are the lines 
 
 y-x=0, x+2y-Z=0. 
 The curre lies in the + + and in 
 the — compartments formed by the 
 lines, as in the figure. 
 
HYPERBOLA EEFEBEED TO ITS- ASYMPTOTES. 225 
 
 243. To find the equation to the twngent at any point (x'y'), 
 when the asymptotes are the co-ordinate axes. 
 
 Let 0) be the angle between the asymptotes, and let the 
 equation to a line cutting the hyperbola in («;'/) be (Art. 34) 
 
 ^^=l:z^^i ; (1), 
 
 c s ^ ' 
 
 , sin a sinfcff— a) 
 
 where s= , c= — ? ; 
 
 sin 0) sm o) 
 
 then, substituting from this equation in the equation to the 
 hyperbola 
 
 ^-^ m. 
 
 we have {si + y') (cl + x') = — j — > 
 
 or scP+(sx' + cy')l = (3), 
 
 since (x'y) is a point on the cui-ve, and therefore 
 , , a^ + b" 
 
 Equation (3) will give us, by reasoning exactly similar to 
 that used in Art. 117, as a condition that equation (1) should 
 represent a tangent at {x'y), 
 
 sx' + cy' = (4), 
 
 and from (1) l^' = Z' 
 
 = -Jfrom(4); 
 
 hence xy' + yx' = 2x'y', or xy'+yx' = -—^ — , 
 
 which is the required equation, 
 
 15 
 
226 gj:nteal coniC' seotj^ns^ 
 
 244. Since the; equation 
 
 ,^ , «' + &"' 
 xy +yx = ^ 
 
 is not altered, when a; changes places with x', and y with y',, 
 it follows (Art. 131) that this is the equation to the polar of 
 (oiy), when the hyperbola is referred to its asymptotes, and 
 that the properties of Arts. 123 — 127, 129, 130, may be 
 proved by it. 
 
 245. We may make use of th,e equation to the chord of 
 contact in the above form, to shew on which branck of the 
 hyperbola the tangents fall, thajt are drawn from any external 
 point {x'y'). The points of contact are determiued (Art. 123, 
 Cor.) by the equations 
 
 ^y +y«'=-^— (1), ^y = —:^ (2)- 
 
 Eliminating y between (1) and (2), we have 
 
 ^y'x''-=L{c?^-W)x+{a^ + }?)ai=^ (3). 
 
 If ajj, ojg are the roots of (3), we. have 
 
 '»i + «'.,= ^ W> a'A = («'+^°),^ (*)• 
 
 Now, if X and y are both positive or both negative, the 
 abscissae of the, points of contact are so also; for from (5) 
 they must be of the same sign, and from (4) that sign must 
 be the sign of -if \ hence, if (»y ) lies in the same angle of the 
 asymptotes as the curve, the points of contact lie in that 
 angle. If the point (^y') lies in either of the two angles 
 which do not contain the curve, so that x and y have differ- 
 ent signs, the product of the roots of (3) is negative ; hence, 
 in this case; the abscissse of the points of contact have oppo- 
 site signs, and the tangents fall upon both branchesi 
 
EXAMPLES X. 227 
 
 EXAMPLES X. 
 
 *1. Pbove Arts. 221, 225—227 by means of tlie eceeatric 
 angle. 
 
 2. If GP, GQ be semi-diameters, at right angles to eacli 
 other, 
 
 1 J__2. 1 
 
 GP"'^ GQ'~ a''^ V 
 
 3. If, from the focus S of an ellipse, perpendiculars, be drawn 
 on GP, GD, conjugate diameters, these perpendiculars produced 
 backwards wUl intersect GD and GP ia the directrix. 
 
 4. If p, r and p, r' be respectively the focal distances of two 
 points, P, D, the extremities of a pair of conjugate diameters of 
 
 an ellipse, then 
 
 pr + p'r —a!' + ¥. 
 
 5. If a tangent to an hyperbola at P cut off GT, Gt from the 
 axes, then; PT . Pt = GD'', GD being the semi-conjugate diameter. 
 
 6. In the equilateral hyperbola the conjugate diameters make 
 equal angles with the asymptotes. 
 
 7. Erom the extremities P, D oi two conjugate diameters, 
 normals are drawn to the major axis of an ellipse; the sum of the 
 
 squares of these two = -5 (a? + ¥). 
 
 8. If the tangent at the vertex A cut any two conjugate 
 diameters of an ellipse produced in T and t, then, AT .At = V. 
 
 9. The lengths of the equal conjugate diameters of an ellipse 
 are ^2 {a^ + V), and the eccentric angles of their extremities are 
 45° and 135°. 
 
 15—2 
 
228 Examples x. 
 
 10. The locus of the middle points of chords of an ellipse, 
 •which pass through a fixed point, is an ellipse with the same 
 eccentricity, and if the fixed point be the focus, the major axis of 
 the ellipse is SG. 
 
 11. The tangent at any point of an hyperbola is produced to 
 meet the asymptotes j shew that the triangle cut off is of constant 
 area. 
 
 12. If the asymptotes of the hyperbola are axes, shew that 
 the equation to one directrix is a; + y — o = 0. 
 
 13. If any two tangents be drawn to an hyperbola, and their 
 intersections with the asymptotes be joined, the joining lines will 
 be parallel. 
 
 14. Shew that the locus of the points of quadrisection of all 
 parallel chords in a circle is a concentric ellipse. 
 
 15. If the angle between the equal conjugate diameters of an 
 ellipse is 60", find the eccentricity. 
 
 16. If a be the angle between two conjugate diameters which 
 make angles 6, ff with the axis major, 
 
 cos a = e° cos 6 cos 6'. 
 
 *17. GP, GD are semi-conjugate diameters of an ellipse, amd 
 PF is a perpendicular let fall from P on GD or GD produced ; 
 determine the locus of F. 
 
 *18. The chords joining the extremities of the conjugate 
 diameters of an ellipse will all touch "in their middle points a 
 concentric ellipse with axes aj'2, b Ji, coincident with those 
 of the original curve. 
 
 19. If a circle be described from the focus of an hyperbola, 
 with radius equal to half the conjugate axis, it will touch the 
 asymptotes in the points where they are cat by the directrix. 
 
EXAMPLES X. 229 
 
 20. Trace the curve, referred to rectangular axes, 
 
 4 + 9 =25. 
 
 21. The radius of a circle, ■which touches an hyperbola and 
 its asymptotes, ig equal to that part of the latus rectum produced, 
 which is intercepted between the curve and the asymptote. 
 
 22. The equation to the diameter conjugate to 
 
 aj « . . a; w . 
 
 -=Ois- + - = 0, 
 
 c « c s 
 
 the hyperbola being referred to its asymptotes. 
 
 23. An ellipse being traced upon a plane, draw the axes and 
 the directrix, and find the focus. 
 
 24. Mnd the angle between the asymptotes of the hyperbola 
 xy = bx' + c, the axes being rectangular j and write the equation 
 to the conjugate hyperbola. 
 
 .25. Tangents are drawn to an hyperbola, and the portions 
 intercepted by the asymptotes are divided in a given ratio ; shew 
 that the locus of the point of division is an hyperbola. 
 
 26. Draw the asymptotes of the hyperbolas 
 
 xj/-2x-3i/-2 = 0, a;y + 2a!' + 3 = 0, 
 and place the curves in the proper angles. 
 
 *27. Find the locus of the intersection of tangents to an 
 ellipse, which are parallel to conjugate diameters. 
 
 28. Find the equation to the locus of the middle points of 
 all chords of a given length, in an ellipse. 
 
 29. If two concentric equilateral hyperbolas be described, the 
 axes of the one being the asymptotes of the other, they will 
 intersect at right angles. 
 
 30. If -P be the middle point of a straight line AB, which is 
 so drawn as to cut off a constant area from the comer of a square, 
 its locus is an equilateral hyperbola. 
 
230 EXAMPLES X. 
 
 31. If S and ff be th« foci of an equilateral hyperbola, and 
 a circle be described upon SM, then the quadrantal chord of this 
 circle shall be a tangent to that described upon the transverse 
 axis. 
 
 32. If a be the acute angle between the axes of co-ordinates 
 of the ellipse (a;Vy = c^), find the lengths of the axes and the 
 eccentricity. 
 
 33. If AA' be any diameter of a circle, FQ any ordinate to 
 it, then the locus of the intersections of AF, A'Q is an equilateral 
 hyperbola. 
 
 34. In an equilateral hyperbola, focal chords parallel to con- 
 jugate diameters are equal. 
 
 3.T. If a series of straight lines have their extremities in two 
 straight lines at right angles to one another, and all pass through 
 a given point, the locus of their middle points is an equilateral 
 hyperbola. 
 
 36. FQ is an ordinate to the axis major AA' of an ellipse, 
 meeting the curve in F and Q ; draw AF, A'Q intersecting in F ; 
 the locus of F is an hyperbola with the same centre and axes. 
 
 37. If tangents be drawn, making a .given angle with the 
 axes of all ellipses having the same foci, the locus of the point of 
 contact is an equilateral hyperbola. 
 
 38. If normals be drawn to an ellipse from a given point 
 within it, the points where they meet the curve will all lie in an 
 equilateral hyperbola which passes through the given point, and 
 has its asymptotes parallel to the axes of the ellipse. 
 
 39. Find the locus of the middle points of chords in a circle, 
 which touch a concentric ellipse. 
 
 *40. If normals be drawn from the extremities of conjugate 
 diameters to an hyperbola, and the point of their intersection be 
 
EXAMPLES X. 231 
 
 joined to the centre, this line produced shall be perpendicular to 
 the straight line passing through the extremities of the conjugate 
 diameters. 
 
 41. Given in posititfh, a strtiight line AB and a point P out- 
 side it ; a straight line FM is drawn, intersecting AJB in G, from 
 the extremity M of which a perpendicular MD on A£ intercepts 
 CD of a given magnitude ; find the locus of M. 
 
 42. The locus of the centres of all circles, which cut off from 
 the directions of t^o sides of a triangle chorda equal to two given 
 straight -lin'eS, is an equilateral ij^perbola, having two conjugate 
 diaineters in the directions of these 'sidfes. 
 
 43. A. straight line passes through a given point and is ter- 
 minated in the sides of a given angle j find the locus of the point 
 which divides it in a given ratio. 
 
 44. From a point P perpendiculars are drc^p^d upon the 
 sides of a given angle, bo as to contain a quadrilateral of given 
 area ; shew that the locus of P is an hyperbola, whose centre is 
 the vertex of the given angle. 
 
 45. Given the base of a triangle and the difference of the 
 tangents of the base angles ; shew that the locus of the vertex is 
 an hyperbola, of which the perpendicular through the centre of 
 the base is an asymptote. 
 
 46. If about the exterior focus of an hyperbola, a circle be 
 described with radius equal to the semi-conjugate axis, and tan- 
 gents be drawn to it from any point in the hyperbola, the straight 
 line joining the. points of contact will touch the circle described on 
 the transverse axis as diameter. 
 
 47. If, from the centre of an equilateral hyperbola, a straight 
 line be drawn through any point P, and if <^ and 0' be the angles 
 which this line and the polar of P respectively make with the 
 transverse axis, then 
 
 tan <j> tan ^' = 1. 
 
232 EXAMPLES X. 
 
 48. Prove that the circle which passes through any three of 
 the four points, in which the equilateral hyperbola 
 
 x" + 2hxy -y' + 2gx + Ify + c = 
 
 cuts the rectangular co-ordinate axes, is equal to the circle 
 
 te' + 2/' + 2gx + 2/2/ = 0. 
 
 49. Find the locus of the middle points of a system of parallel 
 chords, drawn between an hyperbola and the conjugate hyperbola. 
 
 50. If, in two concentric hyperbolas, whose axes are coin- 
 cident, two points be taken, whose abscissae are as the transvei-se 
 axes of the hyperbolas, the locus of the middle point of the straight 
 liae joining them is an hyperbola, whose axes are arithmetic 
 means between those of the given hyperbolas. 
 
 51. If tangents be drawn from diflferent points of an ellipse, 
 of lengths equal to n times the semi-conjugate diameter at the 
 point, the locus of their extremities will be a concentric ellipse 
 with semi-axes equal to a Jn^ h- 1, 6 Jn' + 1. 
 
 52. If a length PQ = CD be taken in the normal to an ellipse, 
 the locus of the point ^ is a circle, whose radius = a — 6 or a + h, 
 according as § is taken within or without the ellipse. 
 
CHAPTER XL 
 
 The Parabola, 
 
 246. We saw (Art. 153) that there is one pair of rect- 
 angular axes, to which when the parabola is referred, its 
 equation may be written in the simple form 
 
 y" = Lx. 
 
 "We shall now proceed to determine its form and principal 
 properties from this equation, and shall suppose L to repre- 
 sent a positive quantity. 
 
 Since y = ± 'J Lx, when a; = 0, y = ± 0, or the origin is a 
 point in the curve, and the line (a; = 0) 
 meets the curve in two coincident points ; 
 that is to say, the axis of y is a tangent 
 
 to the curve. No part of the curve can ^ 
 
 lie on the left side of the origin, for 
 negative values of x would render y 
 imaginary. It must be symmetrical 
 with regard to the axis of x, since every value of a; gives 
 two equal values of y with opposite signs ; also, as x increases 
 indefinitely in a positive direction, y increases indefinitely in 
 both positive and negative directions; hence the form of 
 the curve is that of the figure : the point A is the vertex, 
 and Ax the axis of the parabola. 
 
 If the equation be 
 
 y^=-Lx, 
 
 ay 
 
234 THE PARABOLA. 
 
 no positive values of x will give real values for y, but x may- 
 have any negative value ; hence, in this case, no part of the 
 curve lies on the right of the origin, and, by exactly the same 
 reasoning as before, the curve may be seen to correspond to 
 the dotted line in the figure. It is proved to be concave 
 to the axis of x in Art. 258, aind this is also evident from 
 Art. 249. 
 
 247. The parabola, like the hyperbola, has infinite ■ 
 branches, with this important difference in their nature. 
 The tangent to the hyperbola, and consequently the direction 
 of the branch, tends ultimately to coincide with k straight 
 line making a finite angle with the axis of ic, viz. the asymp- 
 tote; while the tangent to the parabola, as will be shewn 
 hereafter (Art. 254, Cor.), tends ultimately to become parallel 
 to that axis and infiiiitely distant fto'm it. 
 
 248. We saw (Art. 153), that the parabola might be 
 considered as a central curve, with its centre removed to an 
 infinite distance. We may therefore regard the parabola as 
 an elongated ellipse ; and, as this analcfgy ife "^ery useful ih 
 enabling us to foresee the properties of the c'urve, We &hall 
 prove the following proposition. 
 
 249. If we suppose the distance between one veriex and 
 focus of an ellipse to be given, while the axis major increases 
 without limit, the curve will ultimately become a parabola. 
 
 The equation to the ellipse, wli'e& the vertex A' is origin, 
 is (Art. 165) 
 
 and, in order to find out wliat this 
 equation becomes under the proposed 
 circumstances, it will be necessary to 
 express b in terms of a and the dis- 
 
CONNECTION "WITH CENTRAL CONIC , SECTIONS. 235 
 
 taUce (d£) A'S iDetween the vertex and focUs, which is sup- 
 posed to remain finite. Now, 
 
 
 
 
 A'8+8C^a, 
 
 01- 
 
 d + '^a'-b' = a, 
 
 whence 
 
 
 
 b'=2ad-d'; 
 
 and the 
 
 equation 
 
 becomes 
 
 
 
 f 
 
 , 2ad-d\„ ,, 
 ^2 (2«a; x'). 
 
 or, when a = oo , y^ = idx, 
 
 which is the equation to a parabola. 
 
 Since -^ = = — , and therefore vanishes, when a is in- 
 
 a a 
 
 finite^ we have e^ = 1 a = 1> when a = ob ; hence the para- 
 bola may be considered as an ellipse whose eccentricity = 1. 
 
 The sam« property may be proved in the same manner 
 for the hyperbola: in that case, V = 2ad + d'^, which value 
 must be substituted in the equation 
 
 / = -^(2aa3-l-a:^). 
 
 We shall, for the future, use the equation to the para- 
 bola in the form 
 
 ^° = 4 Ja;, 
 
 derived from its analogy with the ellips6 '; tod 'wfe shall c&,ll 
 that point on the axis of x, at a distance = d from the vertex, 
 which was the focus of the ellipse, the focus of the para- 
 bola. 
 
236 THE PARABOLA. 
 
 *250. The co-ordinates of any point in the parabola, as 
 in the ellipse and hyperbola, may be expressed in terms of 
 a single variable. For draw PA from any point P in the 
 curve to the vertex, and let the angle PAx — (f> ; then 
 
 i/ = x tan 1^ ; also y' = 4>dx ; 
 
 id id 
 
 whence x = - — ^j- , y = - — 7 . 
 
 tan 9 tan^ 
 
 251. To find the length of a straight line drawn from a 
 point (x'j) to meet the parabola. 
 
 As in Art. 114, let the equation to the line be 
 
 gjl^^y-y'^; (1). 
 
 then, for the distances of (xy) from the points of section of 
 the line and the parabola (y^ = idx), we have 
 
 {sl + yy = 4,d{cl + x'); . 
 
 therefore s'/' + 2 {sy' -2dc)l + y"'-idx' = (2), 
 
 or pr+qi+R = o. 
 
 Now this equation will always give two values for I ; 
 hence every straight line meets the parabola in two real, 
 coincident, or imaginary points, according as the roots of (2) 
 are real and unequal, real and equal, or imaginary. 
 
 We shall hereafter have occasion to consider the following 
 particular forms that this equation may assume. 
 
 If P= 0, one value of I (Appendix) becomes infinite. 
 
 If P= and ^ = 0, both values of I become infinite. 
 
INTERSECTION OF STRAIGHT LINE AND CTJRTE. 237 
 
 If jR = 0, the point (^y) is on the curve, and one value 
 of I becomes = 0. 
 
 If ^ = and Q = 0, both values of I become = 0, and the 
 line passes through two coincident points of the curve, and 
 is a tangent. 
 
 If Q = 0, the roots of the equation are equal and of op- 
 posite signs, and {xy) is therefore the middle point of the 
 chord. 
 
 252. If P = 0, that is, if s = in equation (2), the line 
 (1) is parallel to the axis of the parabola, since s and care 
 the sine and cosine of the angle which (1) makes with the 
 axis. In this case then, one value of I becomes infinite; 
 hence a straight line drawn parallel to the axis of a parabola 
 meets the curve in one point only at a finite distance from the 
 origin, and in one point at an infinite distance. 
 
 If P=0, Q=0, both values of Z become infinite; but 
 this gives 
 
 „ , 2dG 2dc 
 5 = 0, 2,=— = — = 00; 
 
 so that the line is parallel to the axis of the parabola by the 
 first condition, and infinitely distant from it by the second. 
 Hence no straight line at a finite distance from the origin 
 meets the parabola, as the asymptotes meet the hyperbola, 
 altogether at an infinite distance. In other words, the para- 
 bola has no asymptotes. 
 
 253. We have said that a straight line, di-awn parallel 
 to the axis of a parabola, meets the curve in one finite point, 
 and in one point at infinity. We adopt this language, be- 
 cause it is in accordance with the algebraic result; it is a 
 short way of enunciating what may be stated more clearly 
 
238 
 
 THE PABABOLA. 
 
 as follows. Let K be the point («>'),. md KPD a straight 
 line cutting the parabola in P and 
 D, and let KL be drawn parallel 
 to the axis ; then, if the point D 
 moves along the curve to an in- 
 finite distance, and the line BK 
 tvirns about K, KD will tend to 
 coincide with KL as its limiting 
 position. Without this explana- 
 tion, the statement is not intel- 
 ligible, as the line. KL does npt actually meet the curve at 
 infinity ; indeed, as shewn in, Art. 246, the euj?ve becomes 
 indefinitely distant, from it. . As in the case of the hyperbola, 
 our statement must be held to assert, that the curve tends 
 to become a straight line parallel to KL. We have ex- 
 plained in Art. 42 the meaning of the. assertion, that paj'allel 
 straight lines meet at infinity. 
 
 254. To find the equation to a straight line touching the 
 parabola in the point (x'y'). 
 
 We shall proceed exactly as in the case of the circle and 
 ellipse, and shall simply point out the steps in the proof. 
 
 Let the eq^uation to a line cutting the curve in {x'y') be 
 
 ^^=^^ = Z (1); 
 
 then, for the distances of {xy') from the points of section of 
 the line and the parabola (^ = ^sdx), we haye, as in Art. 251, 
 
 sH"" -t- 2 (sy! - 2dc) l + y"- 4>dx' = 0, 
 
 or s'l^+2{sy'-2dc)l = (2), 
 
 since y'^ = Adx'. Equation (2) gives us 1 = 0, as it should, 
 and 
 
 s'l + 2(sy'^2dc)=0. 
 
THE TAliTGENT. 239 
 
 If the line (1) be ai tangen1< at {-vy'), I, vanishes, ap4 we 
 have 
 
 sy ^2(^=0.,...,.,.. ......(3). 
 
 E'litaifiating s and c by equations (1) and (3), we have 
 
 {3j-y')y' = 2d[x-d); 
 
 or yy' = Mx - Mx +^y'\ 
 
 or, since y'^ = Adx', we have 
 
 .yy'=U(x + x% 
 
 for the equation to the tangent at (ky ),. 
 
 CoE. If a be the angle which the tangent makes with 
 the axis of x, we have 
 
 2d 
 
 tana.= — 7. 
 
 y 
 
 When y = 0, tan a = oo , or the tangent at the vertex is per- 
 pendicular to the axis, as we saw in Art. 246. Also as y' 
 increases from to oo , t^ a decreases from oo to 0, or the 
 tangent, and therefore the direction of the curve, tends con- 
 tinually to become parallel to the axis. This agrees with 
 Art. 253. 
 
 255. If we make «/ = in the equation to the tangent, 
 we have x = — x or AT=AM. 
 Henee the subtangent MT= 2x', 
 and is bisected at the vertex. 
 
 Also, writing x = in the 
 equation to the tangent, we have 
 for the intercept A T, 
 
 _2dm' _y' 
 y~ 2,' -2' 
 
240 THE PARABOLA. 
 
 hence, when x and y become infinite, the intercepts of the 
 tangent at {ody') on the axes become infinite, or the tangent 
 has no limiting position at a finite distance from the origin, 
 as it has in the case of the hyperbola, where it becomes the 
 asymptote. 
 
 256. To jind the eguation to the tangent in terms of its 
 inclination to the axis. 
 
 Proceeding as in Art. 254, we have for the equation to 
 the tangent 
 
 x-x y-y . . 
 
 ~~r ~ s ^^^' 
 
 with the condition 
 
 sy'-2dc = (2), 
 
 where we must now eliminate a/ and y instead of s and c. 
 
 From (2) and the equation y' = idx' 
 
 , , 2c?c , da' 
 
 we have y = — , x = -^ , 
 
 S a 
 
 and from (1) cy — sx = cy' — sx' ; 
 
 dc^ 
 therefore cy — sx = — , 
 
 the equation required. 
 
 If m be the tangent of the angle made by (1) with the 
 axis of a;, iri =-, and the equation becomes 
 
 y = mx+ — . 
 m 
 
 This equation may be obtained directly, without intro- 
 ducing the point {x'y), as in the case of the circle (Art, 120). 
 
THE TANGENT. 241 
 
 Cos. Hence if a straight line and a parabola be repre- 
 sented by the equations. 
 
 y — mx + h, y^ = 4dx, 
 
 the condition of tangency is J = — . 
 "^ m 
 
 257. To determinethe tangents to a parabola which pass 
 through a given point (x'y'). 
 
 The equation to the tangent is 
 
 y = rnx+- (1), 
 
 and, since it passes through {x'y'), we have 
 
 y = mx H — ; 
 or m'-^m +—,= »(2); 
 
 which equation gives two values of m, and therefore shews 
 that, in general, two tangents may be drawn to the curve 
 through a given point. 
 
 If fi, fi be the two roots of the equation, the equations 
 to the two tangents will be 
 
 d , d 
 
 y = lix + -, y = fj.x+-,. 
 fi /* 
 
 258. The rOots of equation (2) (Art. 257) are real and 
 different, real and equal, or imaginary, according as 
 
 y — 4idx' > = < ; 
 
 and it is easy to see, that this inequality gives the condition 
 that [x'y) should be without, on, or within the curve. Hence 
 no tangent can be drawn to the parabola from within the 
 curve, &c., as in Art. 188, 
 
 16 
 
242 THE PARABOLA, 
 
 By reference to Arts. 55, 121 Cor., 188, it will be seen 
 in every case, if ^ [xy) = is the equation to the locus, and 
 {x'y') any point in the plane, that ^ (%') changes sign when 
 {x'y) crosses the lociis. 
 
 259. To determine the locus of the intersection of two 
 pingents at right angles to one another, • 
 
 If in (Art. 257) the two tangents are at right angles, 
 
 /*/*' = -!. 
 
 Hence, from (2), since /j-fi is the product of the roots, 
 
 d , ^ 
 
 -, = /i^=-l, 
 
 or x = —d is the equation required, which represents a 
 straight line perpendicular to the axis of x, at a distance = d 
 on the negative side of the origin. It will be seen hereafter 
 (Art. 263) that this is the directrix of the parabola, 
 
 260. It will be remembered that the equation to the 
 same locus in the case of the ellipse (Art, 189), was found to 
 be a circle whose equation is 
 
 x' + y' = a' + h' (1), 
 
 and we shall now shew that, when the ellipse passes into a 
 parabola, this circle becomes the directrix. For, transfening 
 the origin in (1) to A' by writing a; — a for w, we have 
 
 x'-2ax + y''-P = (2); 
 
 but iiA'S = d, V='2.ad-d^ (Art. 249), and (2) becomes 
 
 a:'-2to + /-2acZ + <Z'=0 (3). . 
 
 Dividing (3) by a and then making a infinite, we obtain 
 the equation 
 
 x= — d, 
 
 the equation (Art 263) to the directrix of the parabola. 
 
EQUATIONS TO NOEMAL AND POLAR. 243 
 
 261. To find the equation to the normal to a parabola at 
 any point (x'y'). 
 
 Since the normal passes througli {x'y'), its equation is 
 y~y' = m,{x-x'\ 
 and, since it is perpendicular to the line 
 
 yy'=2d{x-{-x'} (1), 
 
 we have m = — ^, and the equation is therefore 
 
 y-t/' — I^C^'-^') • (2)- 
 
 Putting y — in this equation, we have (fig. Art 255), 
 for the intercept of the normal on the axis of x, 
 
 x = 2d + x', OT AG=2d + AM. 
 
 Also MG = A0 — AM, or the subnormal = 2d. 
 
 Cob. If m is the tangent of the angle that the normal 
 makes with the axis of x, we have 
 
 m = —^; also y" = idx' ; 
 
 therefore v' — ~ 2dm, x' = dni? ; 
 
 hence, substituting these values in equation (2), we obtain 
 y = mx — 2dm — dm\ 
 
 262. If in the equation to the tangent 
 
 yy^=2d{x + x'), 
 
 we write x', x for x, x', aiid y\ y for y, y', respectively, the 
 equation remains unchanged. Hence (Art, 131) all the 
 theories of poles and polars proved for the circle in Arts. 
 123 — 127, 129, 130, are equally true for the parabola; and 
 
 16— a 
 
244 THE PAKABOLA. 
 
 the proofs will require no alteration, except that we must 
 write the equations to the parabola and its. tangent, in the 
 place of the equations to the circle and its tangent. The 
 student should convince himself of the truth of the above 
 assertion by writing out the articles with the requisite 
 changes. 
 
 263. If y' = 0, the equation to the polar becomes 
 
 x + oe' = 0, 
 
 which shews us that the polar of any point on the axis of x 
 is parallel to the axis of y. 
 
 The polar of the focus, whose co-ordinates are (Art. 249) 
 X =d, y=0, will have for its equation 
 
 x + d=0, 
 
 a straight line perpendicular to the axis of the parabola, 
 lying to the left of the vertex, and at a distance from it = d. 
 The polar of the focus is called the directrix of the parabola. 
 
 264. To find the dista/nce of arty point in the parabola 
 from the focus. 
 
 Since the co-ordinates of the focus (^S') are x = d, y = 0, 
 the square of the distance of any point P{x'y') from it 
 (Art. 7) 
 
 . ={x'-dy + y-; 
 
 but, if {x'y") be a point on the curve, 
 
 y"'=4!dx'; 
 
 hence SF" = {x - dy + idx' = (d + x'Y, 
 
 or SP = d + x'. 
 
FOCAL PEOPERTIliS. 
 
 245 
 
 265. The distance from the directrix of amy point in the 
 parabola, is equal to its distance from 
 the focus. 
 
 The equation to the directrix is 
 (Art. 263) 
 
 a; + d = , (1); 
 
 and, if a perpendicular be dropped 
 from any point P^x'if) in the curve 
 upon (1), we have (Art. 54) 
 
 PN=x' + d = 8R 
 
 This property is analogous to that proved for central curves 
 (Art. 199), since we shewed (Art. 249) that the parabola 
 might be regarded as an ellipse whose eccentricity (e) = 1. 
 
 j^ jB. 
 
 266. The property proved in the last article will enable 
 us to describe a parabola mechanically, by 
 means of a ruler and cord. For let a ruler 
 MNK, right-angled at N, slide along a line 
 LX, and let a cord whose length is = NB be 
 fastened at R and at a point 8; and while 
 NK slides along LX, let a pencil P be moved, 
 so as to keep a portion of the string stretched 
 against RN. Then P ynVL trace out a par- 
 abola ; for the distance PN will be always ~ 
 equal to SP. LX will be the directrix, and ;S' the fociis. 
 
 ■ar 
 
 267. The double ordinate through the focus may be 
 found, by putting x = d'va. the equation y = Mx; then 
 
 y'=4(f, or y = ±M; 
 
 hence the double ordinate =M. This quantity, as in the 
 case of the ellipse, is called the Latus Rectum of the curve. - 
 
246 THE PARABOLA. : 
 
 The latus rectum of the ellipse or hyperbola 
 
 = f (Art. 200) = ?^?^ (Art. 249). 
 
 = 4cZ, 
 when a becomes infinite, or the curve passes into a parabola. 
 
 *268. The tangent and the normal at any point make 
 equal angles with the focal distance of the point and the line 
 drawn through it parallel to the axis. 
 
 Draw PX parallel to Ax; then the equation to 8P, since 
 it passes through {x'y'), (d, 0) is 
 
 -^- = ^- m 
 
 • x-d x-d ^'' 
 
 and toPZ, y-y' = (2); 
 
 hence the equation to PO, the bisec- 
 tor of the angle between (1) and (2), is 
 
 -{d-d;)y+y'x-d^ 
 
 {y'+{x'-dY]^ u^ ^; w, 
 
 making y = in (3), we have, since (Art. 264) 
 
 y'' + (x'-df = {ai + df, 
 
 AQ=x = 2d + x'; 
 
 hence (Art. 261) PG is the normal at P. 
 
 It will be seen that this is a modification of the property 
 proved for the ellipse (Art. 201); for we may suppose the 
 line PX to be in the direction of another focus H, at an infi- 
 nite distance, and the angles tPX and 8PT to correspond to 
 the angles which the tangent makes with the two focal dis- 
 tances in the ellipse. 
 
FOClL PROPERTIES. 247 
 
 269. The above may be shewn geametrlcally thus : 
 
 8T=A8 + AT=d+a! (Ai-t. 255)i 
 80=SM+MG = x'-d + 2d = d^x'y 
 
 therefore (Art. 264) ST= 8G = 8P; 
 
 henee the angle S'PG'^the angle /S'G'P=the angle QPX, 
 and the angle tPX= the angle FT 8 = the angle 8PT: " . 
 
 Ex. The exterior angle Vetween two tangents to a parabola is half the 
 angle between the focal distances of the points of contact. 
 
 Let FT and P'T' be the two tangents, T being the nearer to A ; and let 
 them intersect in Q, and let TQ, be produced to F; then 
 
 PSx^iPTS, P'Sx=2P'T'x; 
 
 therefore ' PSP'=PSx-P'Sx, 
 
 =2{PTS-P'T'x), 
 
 =2VQP'. 
 
 270. To firid the locus of the extremities of p^pendiculars 
 dropped from the focus on the tangent. 
 
 The equation to the tangent in terms of its inclination to 
 the axis is (Art. 256) 
 
 y = mx ■{ — ; 
 ^ m' 
 
 hence the equation to a straight line SY, 
 3rawn through the focus [d, 0), and per- 
 pendicular to this tangent, is 
 
 y = (x — d). 
 
 If we combine these equations in any way, the resulting 
 equation will be satisfied by the co-ordinates of Y. Our 
 object is to eliminate m, which quantity particularizes the 
 tangent; hence, subtracting, we have 
 
 (m + 1) 
 
 x = 0. 
 
24!§ THE PARABOLA. 
 
 Therefore a! = is the equation to the required locus, 
 ■which evidently represents the axis of y, or the tangent at 
 the vertex. This equation may be obtained from the corre- 
 sponding equation for central conies, as in Article 249. 
 
 271. To find the length of the perpendicular from the 
 focus on the tangent. 
 
 Let SY=p, 8P = p (fig. Art. 270); then for the perpen- 
 dicular from S{d, 0) on the line 
 
 yy'-2d{x-\-x') = 0, 
 
 we have f= ^.,^^^, ^ ___j- = ^(c^ + ^); 
 
 therefore p^ = dp. 
 
 272. Any focal chord is perpendicular to the straight line 
 joining its pole with the focus. 
 
 By definition, the directrix is the polar of the focus, and 
 therefore (Art. 129) the polar of any point in the directrix 
 will pass through the focus. Hence we may assume the pole 
 of any focal chord to be a point (— d, y) in the directrix; then 
 the equation to the chord is (Art. 123) 
 
 yy'=2d{x-d) (1), 
 
 and, the equation to a perpendicular to this Une, tjirough the 
 focus {d, 0), is 
 
 y=-4d^^-^ (2)' 
 
 and, when 00=— d in (2), y='!/, or the line represented by (2) 
 passes through the point (— d, y'), which is the pole of the 
 chord; hence the truth of the proposition. 
 
 273. To shew that the locus of a point, whose distance 
 from a given point is equal to its distance from a given 
 straight line, is a parabola. 
 
DIAMETEES. 249 
 
 This is the converse of what was pi"Oved in Art. 265, where 
 it was shewn, that the distance of any point in the parabola 
 from the focus is equal to its distance from the directrix. 
 It has already been proved as a particular case of Art. 206. 
 
 274. To find the polar equation to the parabola, the focus 
 being the pole. 
 
 Here we have to find the polar equation to the locus of 
 Art. 273, the given point being the focus, and the given line 
 the directrix. Exactly as in Art. 209, if we make e = 1, we 
 obtain as the equation 
 
 - = 1 — cos y, 
 
 p 
 
 where I is put for half the latus rectum, and is equal to 2d. 
 
 275. Diameters. 
 
 We saw (Art. 153) that there is only one system of rect- 
 angular axes, which will give the equation to the parabola 
 under the form y' = Lx. Here the axis of a; is a diameter, 
 for it bisects aU chords parallel to the axis of y ; and the 
 axis of y is a tangent (Art. 246) at the extremity of that 
 diameter. The same wiU be the case if the general equa- 
 tion be reduced to the form y" = Lx with oblique axes, as in 
 Art. 155 ; so that, if the general equation of the second 
 degree represent a parabola, the equations 
 
 VAx + '/Cy = 0, Dx+Ey + F^'O, 
 
 will represent the diameter through the origin and the 
 tangent at its vertex. We shall now consider these other 
 diameters, and shew that the form of the equation, when the 
 axes are a diameter and the tangent at its vertex, is always 
 y^ = Lx. 
 
 276. To find the locus of the middle points of any system 
 of parallel chords. 
 
250 
 
 TH:E PAEABOM. 
 
 We shall proceed as iii the case of central curves. Let 
 QQf be one of the chords, M {x'y') 
 its middle point, and let the 
 equation to Q Q be 
 
 c s 
 
 Then for tbe distance Q) from 
 {x'y) of the points of section of 
 the chord and parabola {^=4idx), 
 we have equation (2) of Art". 251 to determine MQ and MQ'; 
 but, since these distances are equal in magnitude, the two 
 values of I are equal and of opposite signs; hence the co- 
 efficient of Z = 0, or sy — 2dc = ; and, since s and c are the 
 same for all the chords, this relation holds for the ordinateS 
 of all the middle points ; hence the equation required is 
 
 sy — 2clc = 0, 
 which represents a straight line (PX) parallel to Ax. 
 
 If m be the tangent of the angle that the chords make 
 with the axis, the equation becomes 
 
 ^2d 
 
 smce TO = - . 
 c 
 
 Cor. It is evident, as in central curves, that the tangent 
 
 at P, the extremity of PX, is parallel to the chords ; for the 
 
 2d . 
 
 ordinate y oi P= — , and the equation to the tangent is 
 
 yy =2d{x+ x), or y = m{x->r x), 
 
 which is the equation to a straight line parallel to the 
 cbords. 
 
 : 277, As, in. central curves, the straight line which bisects 
 any system of parallel chords is called a diam&ter of the 
 
1)IAMETEES. ' 251 
 
 parabola, and the chords are called the ordinates of the 
 diameter. We see then that the equations to a chord and 
 the diameter of which it is an ordinate are, respectively, 
 
 y = mx + c and v = — • 
 
 ■^ m ■ 
 
 Hence all diameters of a parabola are parallel to the axis; 
 
 and, conversely, all straight lines parallel to the axis may be 
 
 considered as diameters ; for by giving a suitable value to m 
 
 .2d 
 in the equation y= — , y may receive any value we please. 
 
 278. The polar of amy point in a diamieter is parallel to 
 the ordinates of that diameter. 
 
 We saw (Art. 263), that this is true when the diameter 
 
 2d 
 is the axis of the parabola. Let y = — be the equation to 
 
 any diameter ; then the equation to the polar of any point 
 
 {x'y') in it is 
 
 yy' = 2d{x + x% 
 
 , 2d . 
 and, since y = — , this equation becomes 
 
 y = m{x + x'), 
 
 which is the equation to a straight line parallel to the chords 
 which the diameter in question bisects. If the point {x'y) 
 be the extremity A' of the diameter (fig. Art. 280), the 
 equation 
 
 yy -2d{x + x') 
 
 represents the tangent at A'; hence, the tangent at the ex- 
 tremity of any diameter is parallel to the ordinates of that 
 diameter, as we saw in Art. 276. 
 
 279. Since all diameters of a parabola are parallel, it 
 cannot have conjugate diameters : it has however properties 
 
252 THE PARABOLA. 
 
 which correspond to the properties of conjugate diameters 
 in central curves, and which may be foreseen by regarding 
 the parabola as deduced from the ellipse by the method of 
 Art. 249. . 
 
 If we refer the ellipse to any diameter and the tangent 
 at its vertex, as axes, the equation will be 
 
 2^ = ^(2a'x-a;') (1), 
 
 which results from writing a; — a' for x in the equation to the 
 ellipse referred to any two conjugate diameters 2a and 26'. 
 Now this equation is of the same form as the equation to 
 the ellipse, when the vertex is origin, the major axis the 
 axis of X, and a tangent at the vertex the axis of y. Hence,' 
 regarding the parabola as an elongated ellipse, we may con- 
 ceive (fig. Art. 280) that any diameter A'X has a conjugate 
 at an infinite distance, parallel to the tangent A'Y, and we 
 foresee that the equation to the parabola, when referred to 
 A'X, A'Y as axes, will be in the sapie form {y^ = Lx), as 
 when it is referred to the axis of the curve and the tangent 
 at its vertex as axes. 
 
 For suppose fig. Art. 280 to represent part of an ellipse, 
 A' the extremity of a diameter and the origin in equation (1) ; 
 and suppose that A'X meets Ax in the centre G, and there is 
 a diameter CD{=h') conjugate to OA' (=a'); let 8A' = d'; 
 then (Art. 226) we have 
 
 8A'. HA' = CB\ or d' (2a - d') = b'^; 
 
 hence equation (1) may be written 
 
 ^=(«-?)(--3f 
 
 Now, when the centre is removed to an infinite distance, 
 both a and a become infinite ; also 
 
 a : a' = sin OA'A : sin CAA', 
 
DIAMETERS, 
 
 253 
 
 which is a ratio of equality when ^'C becomes parallel to 
 Ax, as it does when their point of intersection {G) becomes 
 jinfinitely distant ; hence in this case equation (1) becomes 
 
 2/° = ^d'x. 
 
 We shall, in the next article, prove this property inde- 
 pendently. 
 
 280. If we transfer the origin to the extremity A! (xy) 
 of one of these diameters, we 
 have, writing x-\-x for x and 
 y-'ry for y in the equation 
 y^ = ^dx, 
 {y+y'y = ^d{x + x'). 
 
 or y^+1y'y = Mx (1), 
 
 since y'^ = 4<dx'. 
 
 If we now preserve the axis of 
 X, and take a new axis of y, 
 (A'Y) inclined at an angle 
 to the axis of x ; then 
 
 the old y=PN = PM' sin 9, 
 
 the old X = A'N= A'M' + PM' cos 9, 
 
 A'M' and PM being the new co-ordinates of the point P, 
 when A'X, A' Y are axes. Hence, writing y sin 9 for y and 
 a; + ycos 9 for x in (1), the equation becomes 
 
 y^ sW9 + 2y'y sin 9 = 4>d{x + y cos 9), 
 
 or ^'sin'^ + 2/ (22/'sin^-4cZcos0) = 4^a; (2), 
 
 Now, in order that (2) may be reduced to the form y^=Lx, 
 
 we must have 
 
 2<? 
 2y' sin ^- 4c? cos = 0, or tan0 = — r; 
 
 y 
 
254 THE PARABOLA. 
 
 but this is the tangent of the angle which the tangent at 
 A {x'y') makes with the axis of x ; hence, when we use a 
 diameter ani the tangent at its vertex as co-ordinate axes, 
 the equation to the parabola will be 
 
 where 6 is the inclination of the axes. 
 
 But -^^a = dcomc^e = d(l +cot^6) 
 
 sm a ^ 
 
 = d^x'=8A'; 
 
 and the equation to the parabola, referred to any diameter 
 and the tangent at its vertex, is 
 
 2/" = M'x, 
 
 where d' is the distance of the origin from the focus. This 
 includes the case of the axis of the parabola and a tangent 
 at its vertex, 
 
 281. The quantity ^8A' is called the parcvmeter of the 
 diameter which passes through A' ; when the diameter is 
 the axis of the parabola, it is sometimes called the principal 
 parameter, as well as the latiis rectv/m. In all cases it equals 
 the double ordinate through the focus ; for draw QQ' through 
 the focus parallel to the tangent at A'; then (Art. 269) 
 
 A'B^ST=SA', 
 
 and QE' = 4^8 A' . A'R = iSA'^, 
 
 or QR = 28A', and QQ' = 4i8A\ 
 
 282. Since the equation to a parabola, referred to any 
 diameter and the tangent at its vertex, is of the same form 
 
DIAMETERS. ' 255 
 
 as when the diameter is the axis, it follows that every pro- 
 perty, which has been proved for the latter system, is true 
 for all the others, so long as the property does not depend 
 upon the inclination of the axes. For example, if we have 
 for the equation to the parabola referred to such axes, 
 
 we shall find, exactly as in Art. 254, that the equation to the 
 tangent at any point (a;'y') is 
 
 yy'='2.d' {x + x); 
 
 hence, as before, the subtangent is double the abscissa; also 
 the equation to a straight line joining the points of contact 
 of two tangents drawn from any point {x'y) is 
 
 yy' = M' (as + 03'), 
 and so on, 
 
 283. Tangents at the extremities of any chord will inter-^ 
 sect in the diameter of which the chord is an ordinate. 
 
 Let the parameter of the diameter be 4d' ; then, if we 
 take the diameter and the tangent at its extremity as axes, 
 the equation to the tangent will be (fig. Art. 280), 
 
 + yy = 2d' [x + x'), 
 
 according as we take Q {x'y), or Q {x, — y'), as the point of 
 contact. In each case, when y = 0, x has the same value 
 = — x', or the tangent meets the diameter produced in the 
 same point. 
 
 The student will find no difficulty in adapting the latter 
 part of Art. 236 to the case of the parabola 
 
256 EXAMPLES XI. 
 
 EXAMPLES XI. 
 
 1. Find the length of the side of an equilateral triangle, one 
 of whose angles is at the focus and the other two on the parabola. 
 
 2. Draw the curves af + Sy — O and y = 2x — iicf, and find their 
 points of intersection, the axes being rectangular. 
 
 3. -The rectangle contained between two ordinates y, , y^ of a 
 parabola (i/' = 4:dx) is equal to d'; find the magnitudes of y^ and y^, 
 the distance between them being = d. 
 
 4. Shew that a straight line, drawn from the point of the 
 parabola of which the abscissa is 8d, and cutting the axis at 
 the point x = id, will, if produced, meet the curve again at the 
 point x= 2d, and be a normal at that point, 4c^ being the latus 
 rectum. 
 
 5. From any point there cannot be drawn more than three 
 normals to a parabola. If the point be on the axis, and the 
 abscissa less than 2d, one only can be drawn. 
 
 6. The tangent at any point of a parabola will meet the 
 directrix and latus rectum produced in two points equidistant 
 from the focus. 
 
 7. Two equal parabolas have a common axis ; a straight line, 
 touching the interior and bounded by the exterior, will be 
 bisected in the point of contact. 
 
 8. Find a parabola which shall touch a given circle at a 
 given point, its axis being coincident with a given diameter. 
 
 9. To prove that the area of a triangle inscribed in a para- 
 bola is equal to 
 
 ^ {y' ~ y") (r ~ y ")(:>/'" -yl 
 
 where y, y", y'" are the ordinates of the vertices of the triangle, 
 y' = idx being the equation to the curve. 
 
EXAMPLES XI. 257 
 
 10. If a parabola intersect a circle in four points, prove 
 that the ordinates of the points of intersection which lie on one 
 side of the axis of the parabola are together equal to the sum of 
 the ordinates of the points of intersection which lie oh the other 
 side of the axis. 
 
 11. Two tangents are drawn to a parabola at points whose 
 co'-ordinates are a, h, a', h'. To find the point in which they 
 intersect. 
 
 12. Find the equation to the parabola, whose focus is origin, 
 and whose directrix is the line Ax + By + C = 0. 
 
 13. From points in the exterior of two equal parabolas having 
 the same axis, tangents, are drawn to the interior one ; they will 
 touch it at the extremities of diameters whose distance from one 
 another is constant. 
 
 1'4. Three parabolas, having their axes parallel, intersect ; 
 shew that the three chords passing through their points of contact 
 pass through one point. 
 
 15. Two tangents to a parabola make angles whose tangents 
 are a', a" with the axis ; find the equation to the tangent at the 
 extremity of the diameter of which the chord of contact is an 
 ordinate. 
 
 16. Find the locus of the middle points of chords passing 
 through any fixed point, and adapt the proof to (i) the focus, 
 (ii) the vertex, and (iii) the foot of the directrix of a parabola. 
 
 17. HA be the vertex, S the focus, and PSp the focal chord 
 of a parabola, prove that the rectilinear triangle PAp varies as 
 the square root of the distance Pp. 
 
 18. If a straight line be drawn from the foot of the directrix 
 of a parabola, making an angle 45° with the axis, it will touch all 
 parabolas having the same axis and directrix. 
 
 19. Find the equation to the normal at the extremity of the 
 latus rectum of the parabola whose equation is y^ — iidix — d), and 
 find its distance from the origin of co-ordinates. 
 
 17 
 
238 THE PAEABOLA. 
 
 20. Zp is a. normal to the parabola at L, the extremity flf 
 th* latus reetum, meeting the parabola again in p. Shew that the 
 diameter in which the tangents at Z and p intersect, passes 
 through the other extremity of the latus rectum. 
 
 21. Two ordinates to a parabola meet the axis in points 
 equidistant from the focus. If the vertex be joined with the 
 point where one of the ' ordinates meets the parabola; find the 
 equation to the locus of the point where this line intersects the 
 other ordinate. 
 
 22. Two tangents are drawn to a parabola, making angles 
 6, 6' with the axis. Prove that (i) if sin $ . sin 6' be constant, the 
 locus of the intersection of ,the tangents is a circle, whose centre 
 is in the focus ; (ii) if tan 6 . tan ff be constant, the locus is a 
 straight line perpendicular to the axis ; (iii) if cot + cot 6' be 
 constant, the locus is a straight line parallel to the axis ; (iv) if 
 dot 6 — cot 0' be constant, the locus is a pai-abola equal to the 
 original parabola. 
 
 23. A series of triangles are constructed on a given base, 
 their vertices being in a straight line parallel to the base ; shew 
 that the perpendiculars through the extremities of the base to 
 the sides of these triangles, will intersect in a parabola, whose latus 
 rectum is the distance between the lines. 
 
 24. To find the equation to a parabola, referred to the two 
 tangents at the extremities of the latus rectum as axes. 
 
 25. To find the area of a triangle included between the 
 tangents to parabolas y" = idx, if = 4Sflj, at points, the cominon 
 abscissa of which is as, and the portion of the ordinate intercepted 
 between the two curves. 
 
 26. To find the magnitude of the ordina:te of such a point 
 in a parabola {y^=A,dx) that the; intercepts on the axes of co- 
 ordinates of a tangent, drawn to the curve at this point may be 
 equal to each other. 
 
 27. The three altitudes of any triangle described about a, 
 parabola all pass through a single point in the directrix. 
 
EXAMPLES XI. 259 
 
 , 28.' To find the distance of the vertex- and focus from the 
 tfmgent in terms of the inclination of the tangent to the axis 
 of X. 
 
 ■ 29. Find the locus of the centre of the circle which ^hall 
 always, touch a given circle and a given straight line. 
 
 , . 30. Prom the vertex of a. parabola a straight line is driiwn 
 inclined at an angle. 45°, with the tangent at any point'; find the 
 equation to the curve .which is the locus of their intersection. ' 
 
 ' 31. In the focal distance SF take Sp equal to the ordinate 
 PM. Find the polar equation to the locus traced out by the 
 point p. ■ 1 ' • 
 
 32. From two points in a diameter of a parabola two pairs 
 of tangents are drawn to the curve ; the trigonometrical tangents 
 of. the inclination of one pair ^o the axis are /a.,, /m^j and of the 
 o;fcher [x^, /«4; to prove that 
 
 11-11 
 
 33. The vertex of a parabola is taken for the centre of a 
 given circle ; to find the equation to a straight line touching both 
 circle and parabola. 
 
 34. If from any point Q of the line BQ, which is perpen- 
 dicular to the axis GAB of a parabola whose vertex is A, QP be 
 drawn parallel to the axis to meet the parabola in P; shew that, 
 if CA be taken =^^, the locus of the intersection of AQ and GP 
 is .the original curve. 
 
 35. A parabola being traced upon a plane, draw the axis and 
 directrix, and. find the focus. 
 
 36. Two equal tangents cannot be drawn to a parabola, 
 except from a point on the axis. 
 
 37. Transform the equation to the tangent to the ellipse in 
 the form y =.mx + JaW,+.V, into the corresponding equation for 
 the parabola. .. ■, .. ••'■'' '■ > •' ' 
 
 17—2 
 
260 THE PARABOLA. 
 
 38. Given the radius vector, drawn from the focus to any 
 point of a parabola, and the angle it makes -with the curve ; find 
 the latus rectum and the position of the vertex. 
 
 39. The locus of the centre of a circle which passes through 
 a given point and touches a given straight line is a parabola. 
 
 40. If from the focus of a parabola straight lines be drawn 
 to meet the tangents at a given angle, prove that the locus 
 of their points of intersection will be that tangent to the 
 parabola, the inclination of which to the axis is equal to the 
 given angle. 
 
 41. The abscissa and double ordinate of a parabola are h 
 and h, and the diameters of the circumscx'ibed and inscribed cu'cles 
 are D and d. Prove that D + d—h + k. 
 
 42. If PQ be a chord of a parabola which is a normal at F, 
 and the tangents at F and Q intersect in a point T, shew that 
 FT is bisected in the directrix. 
 
 43. To find the equation to all parabolas which are touched 
 by the straight lines y = =•= ^ . 
 
 44. Find the equation to the parabola, referred to the tangent 
 and normal at the positive extremity of the latus rectum, as axes. 
 
 45. Two normals to a parabola (y^ = 4:dx) are always at right 
 angles to each other ; to find the locus of their intersection. 
 
 46. To find the equations to all the common chords of the 
 two curves y' = 2cx — x", if = 4:dx. 
 
 47. To prove that a series of circles, of which the centres 
 are in a parabola, and which pass through the focus, all touch 
 the directrix. 
 
 48. The distance of a point from one given straight line 
 varies as the square of its distance from another given straight 
 line ; shew that its locus is a parabola, having the second Irae as a 
 diameter, and the first as a tangent at its vfertex. 
 
EXAMPLES XI. 261 
 
 49. If the focus is origin, the equation to the tangent to the 
 parabola, in the form of Art. 27, is 
 
 d . 
 
 a; cos a + w sin a -J ■= 0. 
 
 cos a 
 
 50. From a point P, the concourse of two tangents {PQ, PQ') 
 to a parabola, PABG is drawn meeting the curve in A, C, and 
 QQ' in P. PA, PB, PC are in harmonical progression. 
 
 51. To find the locus of the intersection of perpendiculars 
 from the focus on the normal. 
 
 52. If two tangents be drawn to a parabola, prove that a 
 third tangent, parallel to. the chord joining the points of contact, 
 will bisect the parts of the other tangents, which are included 
 between their point of intersection and their points of contact. 
 
 53. The abscissae of two points in a parabola, reckoned along 
 the axis, are x, 3x, and the correspondiag focal distances r, 2r ; to 
 find the position of the former of these points. 
 
 54. Any number of parabolas are described having the same 
 vertex and axis, and any straight line is drawn at right angles 
 to the common axis. If any points whatever in this line be 
 taken as poles, to prove that all the polars belonging to all the 
 parabolas will intersect in a single point. 
 
 55. The centre of an ellipse coincides with the vertex of 
 a parabola, and the axis major of the ellipse is perpendicular to 
 the axis of the parabola ; required the proportion of the axes of 
 the ellipse that it may cut the parabola at right angles. 
 
 56. Given a point where a parabola intersects a given 
 diameter, and also the parameter of that diameter ; shew that the 
 locus of the vertex is an ellipse. 
 
 57. A parabola slides between rectangular axes; find the 
 locus of (i) the focus, and (ii) the vertex. 
 
CHAPITER XII 
 
 General properties of Conic Sections. 
 
 In this chapter -we shall shew how the' chief proper- 
 ties of the loci of the second degree may be deduced from 
 the general equation of the second' order without t ireduction. 
 "We. shall begin by tracing the loci. If the axes are rect- 
 angular, the method of Chap. VIII. Inay be used for this 
 purpose; and is tp be preferred, if we wish to determine the 
 elements of the locus, such as the axes, position pf foci,, &c. ; 
 but the following method may be used with great .adyantag^, 
 when the axes are pblique, and i;n all pases. where we wish 
 simply to trace the form of the locus, without determining 
 its elernents. 
 
 284. Solving the 'equation 
 
 Ax^ + Bxy+Cy^ + Px + Ey + F.^Q..,: (1), , 
 
 (which we shall call {xy) = 0,) so as, to obtain y in terms 
 of X, we have, as in Art. 62, 
 
 Bx+E . 1 
 
 1/ 
 
 ^_'^x+M^ ^ i_j^^2_^^ P)x'+2(Bi:-.20I)) x+E'-4!CF. 
 
 Let us draw the right line {DT in figs, of Arts.' 286, 
 287, 292) whose equation is 
 
 y=.-^^U^ or 2Oy+Bx + E = 0........,{f); 
 
 then, in order to obtain the ordmates corresponding to any 
 abscissa x' of the locus represented by (1), we have only to 
 
THE EOCI. TRACED. ;•., 263 
 
 increase and diminish the ordinate of (2) corresponding to a;' 
 fcy the quantity • , 
 
 i^JiB'-4,AC) x"'-+2{BE-2CI))x' + M' - 4>CF...{S). 
 
 This line (2) then bisects eyery chord of the locus parallel to 
 tlie axis of y, and is therefore a diameter. 
 
 Cor. Siiriila;rly, from the solution - as a quadratic in x, 
 we find that the line 
 
 2Ax + Bi/ + I> = (4) 
 
 is the diameter bisecting all chords parallel to the axis of a;. 
 Equations (2) and (4) are those obtained in Art, 141, to 
 determine the centre. 
 
 285. In order then to trace the locus, we must examine 
 expression (3), which we shall call Y; for, as long as the 
 values given to x render the quantity under the root positive, 
 we can find two points of the locus corresponding to evpry 
 abscissa. ; if they make it = 0, the two points coincide ; and, 
 if they make it negative, the value of T is imaginary, or no 
 point of the locus corresponds to the abscissa in question. 
 For the sake of brevity we shall write ' 
 
 Y= ^J{B' - 44 0) {x' + 2 Qx + B), 
 
 ^ BE-2CD „ IP-iCF 
 where ^= Bf-4iAQ' ^-WZIAU' 
 
 and may be either positive or negative quantities. The 
 student will observe, that -f Q is the abscissa of the centre 
 obtained in Art. 141/, -and this will be seen to agree with 
 the results hereafter obtained. 
 
 Now the expression x^ ■jr2Qx + B ma.y he^ written 
 
2G4 GENERAL PROPERTIES OF CONIC SECTIONS. 
 
 We have then three cases : 
 
 (i) If Q' — B is positive, that is, if the roots of the 
 equation 
 
 x^+2Qx + B = 
 
 are real and unequal, the expression can be broken up into 
 real factors, 
 
 {x+Q + ^(f-B) {x+Q-'J^^:^}, 
 
 or (x — a) (x — h), 
 
 where a + b = — 2Q, ab = B. This expression is positive for 
 aU real values of x, except for those which lie between a 
 and h; and it attains its greatest negative value, —[(^ — B), 
 when 
 
 Also if we make x = — Q + h ov =— Q — h, the values of the 
 expression are the same. 
 
 (ii) If ^^ — ^ = 0, or the roots of the equation are real 
 and equal, the expression becomes {x + Q)^. 
 
 (iii) If Q^ — B is negative, that is, if the roots of the 
 equation are imaginary, the expression cannot be broken up 
 into real factors. We shall write it 
 
 {x+QY + D\ 
 
 where the symbol D' is used to denote the positive quantity 
 B — Q'. This expression can never be negative for any real 
 value of X, and has its smallest positive value when a; = — Q ; 
 also, if we make x= — Q + h or = — Q - A, the values of the 
 expression are the same. 
 
 286. We shall now apply these results to the three cases, 
 when B'^ — 4iA G is negative, positive, or zero. 
 
THE LOCI TRACED. 
 
 265 
 
 I. When B^ — iAC is negative. In this case we know, 
 by Art. 150, that the locus is an ellipse or circle, a point, or 
 imaginary. 
 
 (i) Let the roots he real and unequal; then Y, which in 
 this case may be written 
 
 5^= 2^r J{B^-4>AC){x-a){x-h), 
 
 is = 0, when x = a and when x=h, and is real for those 
 values of x only that lie between a and 5, since these are 
 the only values that make {x — a)(x — b) negative ; hence 
 the locus lies wholly between the two lines {D'L and BB), 
 parallel to the axis of ?/, whose equations are 
 « — a = 0, a; — 6 = 0. 
 
 These lines are tangents to the curve at the points where 
 the diameter (2) BT cuts it; for, if we put a; = a in the 
 equation to the curve and diameter, the two ordinates to 
 the curve will become equal, and will equal the ordinate of 
 the diameter. The value of each is 
 
 Ba + JE 
 ^~ .20 ' 
 
'266 GENERAL PROPERTIES OF CONIC SECTIONS. 
 
 Hence the liae x — a='0 passes through two coincident 
 points of the curve, and is a tangent. Similarly a; — 6 = 
 is a tangent at the other extremity of the diameter,- It will 
 be observed that they are both parallel fo the ordinates of 
 the diameter. 
 
 The value of F will be greatest, ■when x= —x— {i.e. =— Q), 
 
 the .abscissa of G, the point midway between, i) and D'; for 
 we have shewn that this value of x gives the expression. 
 
 x^ + 2Qx + R, or (a; — a] (« — 6), 
 
 its largest negative value. This point is the ^centre of the 
 curve; for, if we write 
 
 a + 6 , T a + b , 
 x = — n hh, or a; = — ^ h, 
 
 we have shewn that the values of {x — a) {x — b), and there- 
 fore of Y, are the same. From this it follows, that if we take 
 QM = QN=h, we shall have (7F= OH, and 
 
 hence, as in Art. 140, 0G8' is a straight line, and is bisected 
 in C, which is therefore the centre. The curve then, which 
 is evidently an ellipse, has the form given in the figure. OP 
 is the diameter conjugate to GD, since it is parallel to the 
 ordinates of CD. • 
 
 Example. Let the equation be 
 
 ■whence 2y-x-2J^ J-{x-l) (a!-5) = 0. 
 
 The curve has then for diameter the line 2y~x-2 = 0; it lies wholly 
 between the lines a!!-l = 0, a!-5=0; and the largest value of T, ooirespond- 
 
 K 
 
 ing to a =3, is T=l. The co-ordinates of the centre are x=3, y==. 
 
 2 
 
tHE LOCI O^ACED. 26-7 
 
 (ii) Let the roots he real and equal. Then 
 
 and the equation to the locus is 
 
 heilce, since '^B'—AtAG is imaginary, the equation can only 
 
 be satisfied by the values of x and y which make ' ' 
 
 M - .\ -' A -.- r t;- 
 
 2Cy + ^a; + ^=0, and a; + Q = 0, 
 
 and the locus may' be considered as an ellipse reduced to'a 
 point, or as two imaginary straight lines which intersect in G. 
 
 Example. Let the equation be 
 
 y + iaiy'+ 5x^ -2y- 10a; + 10 =0, 
 ■whence y+2siy-liiJ~^{xTS)=Q, 
 
 an equation which is satisfied by one pair of (real values only, namely, 
 x=3, j=-5. 
 
 (iii) Let the roots he imaginary. Then 
 
 y=^ s/{B' - iAGl {(^ + QT,+^1> . 
 
 and, sin,ce {x+ QY + B^ can never be negative, Y is never 
 ,real; and the locus therefore may be called an imaginary 
 ellipse. 
 
 Example. • Let the equation be 
 
 solving for y, Yfe have for' the quantity under the root 
 
 -{x?fx + l), or; -jfp + g) +4l> 
 a quantity which can never be pbktive.' The locus is therefore imagiliary. 
 
268 GENERAL PROPERTIES OF CONIC SECTIONS. 
 
 287. 11. When jB' — 4 ^(7 is positive. In this case we 
 know by Art. 150, that the locus is either an hyperbola or 
 two intersecting straight lines. 
 
 (i) Let the roots he real and wnequal. Then 
 
 y=jQ'J[B'-'i>AG) {x-a) {x-h). 
 
 As in the case of the ellipse, the diameter DT, whose 
 equation is 
 
 2Gy + Bx^-E=0, 
 
 meets the curve in the points B', D, and the lines D'L, BR, 
 whose equations are x—a = 0, a;— 6=0, are tangents at those 
 
 points. Also, since {x—a) {x—b) is negative for values of 
 X between a and b, Y is imaginary for those values ; hence 
 no part of the curve lies between the parallels B'L, BS. 
 Beyond those limits any number of points of the locus may 
 be found, equidistant from the diameter BT; and, since 
 T is real when x has any real value, positive or negative, 
 except those which lie between a and b, we shall obtain four 
 infinite branches, and the curve, which is evidently an hyper- 
 bola, has the form given in the figure. As in the ellipse, C, 
 the middle point of BB', may be shewn to be the centre, and 
 CP the diameter conjugate to OB. 
 
THE LOCI TRACED. 269 
 
 Example, Let the equation be 
 
 'wlience 2y-2x + li= J3c^-3x-i=0, 
 
 or 2y~2x+lJ.^{x + l){x-i) = 0. 
 
 The curve then has 22^-2x+l=0 for a. diameter and a; + l = 0, a;-4=0 
 are the tangents at the points where it meets the curve. It has four infinite 
 branches, but no part of it lies between x= -1 and x=i. 
 
 (ii) Let the roots be real and equal. Then the equation 
 to the locus is' 
 
 which represents two straight lines which intersect in C, for 
 which point 
 
 2Cy + JBx + i:=Q, and x+Q = 0. 
 
 The line DT is still a diameter of the locus, which may- 
 be called a Eectilinear Hyperbola. 
 
 (iii) Let the roots be imaginary. Then 
 
 ^=20 '/{B'-^'ACJiix + Qr + jD'l; 
 
 and, since {x + QY + -D' is positive for every real value of x, 
 the value of F is real for every such value. Since Fdoes not 
 vanish for any real value of a;, the diameter DT 
 
 2Gy + £x + JE=0, 
 
 which bisect^ all chords parallel to the axis of y, does not 
 meet the curve; for the ordinate of the curve cannot be made 
 equal to that of the diameter. As before, any number of 
 points may be found equidistant from' the diameter DT, by 
 taking values of Y corresponding to different values of x. 
 The least values of Y will be when x = — Q, represented 
 by GP, CF in the figure; and on each side of this line 
 the values of Y increase indefinitely, forming an hyperbola 
 
270 GENERAL PBOPERTJfES OP CONIC SECTIONS. 
 
 witli four infinite branches, as in- the figurej As in. -the 
 ellipse, it may be shewn that G is the centre ; also GP is 
 
 parallel to the ordinates of the diameter' CZ>,. and is therefore 
 the diameter conjugate to it. 
 Example. Let the equation be 
 
 whence 22;-2a!-l±\/»^-2a! + 5 = 0, 
 
 or 2y- 2a; - 1 ± J{x - 1)!! + 4 = 0. 
 
 Since the quantity under the root can never =0, the curve does not meet 
 
 the diameter ^y -ix-l=0; and, since it can.never be negative, there are 
 
 points on the curve corresponding to everv abscissa. The smallest value of 
 
 i ■" ' ' -. 3'. 
 
 yis 1, corresponding to a:=l. The co-ordinates of the centre are a!=l, y=-g- 
 
 We shall be aided in trkcing' the figure of the hyperbola, 
 if we find the asymptotes, which we now proceed to do. 
 
 \ ^88. To find the equation to the ^asyimptote^ of a coni^ 
 section from, the general equation. ■ 
 
 Resuming the notation of AiU 285, and writing P for 
 ff—4<AG, we have for the equation to the curve 
 
 2Cy + Bx + E±,^P\{x+Qf+{R^Q'.}} = (1). 
 
fi:::f> ■■:: THE ASYMPTOTES. 271 
 
 Now, when x becomes very large, the term {x + Qf is 
 very large compared with R — Q', so that the ordinate of 
 the curve would then be approximately obtained from the 
 equation 
 
 2Cy + Bx + E±'^r(x+ Q)=0 (2J. 
 
 This equation represents two straight lines, which in- 
 tersect in the point where 
 
 2Ci/ + Bx+E=0, x+Q=0, 
 
 which is, by Art. 287, the centre G. Also, when x becomes 
 indefinitely large, the difference between the ordinates of (1) 
 and (2) hecomes indefinitely small ; for (1) may be written 
 
 2C2f + Bx + E± ^T{x + Q) {l +'^^Y\- 
 
 Expanding by the 'Binomial, Theorem, we have 
 
 2C^ + 5a;+^±VP(a=+,Q)|l+2^^,+ &c.J = 0, 
 
 or, = 2 Gy + Bx + E± VP {x + Q) + terms involving nega- 
 tive powers of 03 + ^. 
 
 Now, by increasing x, these terms may be made as small 
 as we please, and therefore the straight hnes represented 
 by (2) approach indefinitely near to the curve, and are the 
 asymptotes. 
 
 We see then, that, to find the asymptotes of an hyper- 
 bola, we must solve the equation as a quadratic in x or y, 
 arid replace the last term in the trinomial under the root 
 by such a quantity, as will render the trinomial a perfect 
 square. The equation so altered will be the equation to the 
 asymptotes. 
 
 R—Q? 
 ^ This is a true arithmetical expansion, when, -. — nff^^ ^^^^ than unity, 
 
 and this will be the ease when a is increased indefinitely] 
 
272 GENERAL PROPERTIES OF CONIC SECTIONS. 
 
 Example; Let the equation be 
 
 42/" - 8aa/ + 3a;' - 4?/ + 6a! - 4 = 0, 
 
 whence 22/-2ic-l± V*'''-2a! + 5 = 0. 
 
 Replacing +5 under the riot by +1, we obtain for the equation to the 
 
 > asymptotes 
 
 2y-2a:-l±(a!-l) = 0, 
 
 or 2v-a;-2 = 0, 2y-3x=0. 
 
 289. We will here collect together some results, at 
 which we have already arrived, leaving the student to deduce 
 them from equation (2) of the preceding article. 
 
 (i) The asymptotes of the ellipse are (Art. 184) ima- 
 ginary, (ii) If the centre is origin, and the equation to the 
 conic, referred to any axes, 
 
 Ax^ + Bxy+ Of = F (1), 
 
 the equation to the asymptotes is (Art. 186) 
 
 Ax^ ^- Bxy + Cy" = Q (2), 
 
 and, consequently, whatever he the origin or axes, we shall 
 obtain straight lines parallel to the asymptotes, by equating 
 to zero the terms of two dimensions in the equation to the 
 conic, (iii) If ^ = or (7= in the general equation, the axis 
 of a? or y is (Art. 186, Cor.) parallel to an asymptote, (iv) All 
 conies, obtained by varying the constant term in any equa- 
 tion, have (Arts. 141, Cor., 186, Cor.) the same asymptote^; 
 but, if the sign of le is different in the equations of two of 
 them, when written in the form 
 
 {ax + ly+c) {a'x + Vy + d) = Ic, ■ 
 
 the curves (Art. 242) lie in different angles of the asymp- 
 totes, (v) If the axes are rectangular, the angle between 
 the asymptotes is (Art. 67) determined by the equation 
 
 tim.6=- — -. — 7=-^ f3 
 
 a value which (Art. 147) does not change for any transforma- 
 tion, so long as the axes remain rectangular, (vi) The equa- 
 
THE ASYMPTOTES. 273 
 
 tion to the axes, when the centre is origin, is (Arts. 68, 
 146) 
 
 x'-2^^xy-f = Q (4), 
 
 since they bisect the angles between the asymptotes repre- 
 sented by equation (2) ; and (4) represents (Art. 68) real 
 lines, whether the asymptotes are real or imaginary. 
 
 *290. The above results have been obtained for central 
 conies only ; and the method of Art. 288 will give no recti- 
 linear asymptotes to the parabola. Here ^ — 4<A (7=0, and 
 the solution of Art. 284 may be written in the form 
 
 2Cy + Bx + i:±'^Q{x+B) = 0; 
 
 and, by writing V Qx outside the bracket and expanding as 
 above, we obtain for the equation to the asymptotes 
 
 which equation differs from that of the original curve in the 
 constant term only, and therefore represents a parabola 
 equal , to it (Art. 156 Cor.) and upon the same diameter. 
 Such parabolas then approach indefinitely near to one ano- 
 ther at infinity; a conclusion which may be easily veri- 
 fied, by writing the equations to the curves, y" = 4(?« and 
 y" = id(x + h), and applying the method of Ait. 171. 
 
 291. The foregoing results have been obtained on the 
 supposition that the equation can be solved as a quadratic in 
 y; that is, we have supposed that does not = 0. This must 
 always be the case with the loci of Class I., for, if A or 
 C were to vanish, B^ — 4<A C could not be negative. But if 
 B* — 4u4 is positive, or the locus belongs to Class II., either 
 A or G may vaiiish. If C vanishes, we have, solving 
 for X, 
 
 '2.Ax+By+D±'J{B^-^AG)y*+2{BD-2AE)y^D^-4iAF=0, 
 
 18 
 
274 GENERAL PROPERTIES OF CONIC SECTIONS. 
 
 which will give results similar to those obtained above, and 
 the locus may be traced in the same manner. 
 
 If both A and G vanish, the equation is of the form 
 Bxy + Dx + Ey+ F= ; 
 
 hence we cannot solve the equation as a quadratic, and 
 trace the locus by the method of the preceding articles. If 
 we multiply by B, the equation may be written 
 
 {Bx + E) {By + D) = DE- BF, 
 
 a form of equation which has been fully explained in 
 Ai-t. 242. 
 
 Example. Let the equation be 
 
 2a;y+6!C-y-8 = 0, 
 whence (2a!-l)(jr+3) = 5, 
 
 wHch represents an hyperbola, -whose asymptotes are 2a!- 1 = 0, 3^ + 3 = 0. 
 
 292. III. When £=-4^(7=0. the locus is (Art. 153) 
 a parabola, two parallel straight lines, two coincident straight 
 lines, or imaginary. In this case if we draw the diameter 
 JDT whose equation is 
 
 2Gy + Bx + E=0, 
 
 we shall have, for the quantity to be added and subtracted 
 from the ordinates of this line, in order to obtain the ordi- 
 nates of the curve, 
 
 or = ^, V2 {BE - 2 GD) (x - a), 
 
 where a may be either positive or negative. 
 
THE LOCI TRACED. 275 
 
 We shall then have three cases : 
 
 (i) BE—2GD positive. Then, as in the ellipse, if we 
 draw the line DB whose equation is a; — a = 0, it will be a 
 
 tangent to the curve at the point D. Also if a; is less than 
 OB (or a), the values of Y become imaginary, or no part of 
 the curve lies to the left of DB. For all values of x from OB 
 to infinity, Y is real, and the curve, which we see to be a 
 parabola, may be traced as before, a;nd wUl have two infinite 
 branches as in the figure. 
 
 (ii) BE— 2 CD negative. In this case Fis imaginary, 
 if a; — a is > 0, that is, if a; > OB ; hence the curve has no 
 part to the right of D. It will be a parabola, with the infinite 
 branches turned in the direction opposite to those of the 
 figure. 
 
 ilsAiffLS 1. Let the equation be 
 
 S;'-2xy+ar'-2y+3a;-l=0, 
 whence y-x-li^,J -{x-i)=0. 
 
 The curve cuts the diameter y- a -1=0 at a point where x=i, y=3, and 
 has no part lying to the right of the tangent a; -2=0. Its infinite branches 
 will be turned in the direction opposite to those of the figure. 
 
 18—2 
 
276 GENERAL PROPERTIES OF CONIC SECTIONS. 
 
 ExAMPUB 2. Let the equation he 
 
 y'-2xy+x*-2y+x + 3=0, 
 
 ■whence y-x-lJ=,Jx-2=0. 
 
 The curve cuts the diameter y-x-l = in the points x—2,y=3. No 
 part of the curve lies to the left of the line, a; - 2 = 0, parallel to the axis of y, 
 and this line is a tangent. The form ia that of the figure. 
 
 (iii) BE-2CD=0. In this case 
 
 and is real, zero, or imaginary, according as the quantity 
 under the root is positive, zero, or negative.. In the first 
 case, the locus is two straight lines, parallel to the diameter 
 DT \fhose equation is 
 
 2Ci/ + Bx + :E=0, 
 
 and equidistant from it ; in the second, these two lines coin-, 
 cide with the diameter ; in the third, the equation has no 
 real geometrical signification, hut may be said to represent 
 two imaginary straight lines parallel to BT. The locus may 
 be called a Rectilinear Parabola. 
 
 293. An equation belonging to Class III. is not always- 
 capable of being solved as a quadratic in both x and y. It 
 can, however, always be solved as a quadratic in one of them ; 
 for, if both A and were to vanish, since B^ — ^A G = 0, B 
 would also vanish, and the equation would not be of the 
 second degree. Hence the locus may always be traced by 
 the method above. 
 
 *294. The two following examples will shew how certain 
 results can be obtained by means of Art. 147, ■without re- 
 ducing the general equation. 
 
ECCENTRICITY — EQUI-CONJUGATE DIAMETERS. 277 
 
 (i) To find the eccentricity of the conic section <j> (xy)=0, the axes being 
 rectangular. 
 
 If the equation is reduced to the form 
 
 A'3?+0'y^+,^{xfy')=0 (1), 
 
 as in Art. 150, it is easily seen, from Arts. 158, 174, that 
 
 5 C-A' A'-C 
 " =-0^ ""^ =-17- (2), 
 
 according as the axis of x is the major or minor axis in the ellipse, or is the 
 transverse or conjugate in the hyperbola ; hence 
 
 Al , , C" 
 'O' 
 
 l-e»=^^, or l-e^=^,; 
 A 
 
 A'+C 
 
 therefore Vl^«+^^ = . ^^. ^j; = 
 
 ^, , 2-e' 2{A + Crj 
 
 therefore ,- - = =f j (3), 
 
 since A' and C are (Art. 148) determined from the equation 
 
 . ,, ^ S'-4AC „ 
 z^-(A + C)z j =0. 
 
 If the conic is an ellipse, since A' and C" are of the same signs, one value 
 of e' is evidently, from (2), negative; hence in this case we can find the 
 eccentricity from (3). If the conic is an hyperbola, A' and O' are of different 
 signs, and therefore both values of e' are, from (2), positive. This ought to 
 be the case; for (3) is independent of the value of <l>(x'y'), and, if this 
 quantity has a different sign for two hyperbolas represented by (1), the curves 
 will (Art. 242) Ue in different angles of the asymptotes, and wiU not have the 
 
 same eccentricity, since (Art. 174) e=sec-, where 6 is the angle in which 
 
 the curve Ues. 
 
 (ii) To find the length of the equi-conjugate diameters and the equation to 
 thtm, in the ellipse whose equation with rectangular axes is 
 
 Ax'+S!e>/ + Oy'+F=0 (1). 
 
 This equation (Art. 148) may be reduced to 
 
 A'x^ + Cy + F=0 (2), 
 
 where A'+0'=A + 0, -iA'C'=B^-iAC; 
 
 also the equation to the equi-conjugate diameters is, from Art. 218, 
 
 A'x^-CY=Q (3). 
 
278 GENERAL PEOPEKTIES OF CONIC SECTIONS. 
 
 At the intersection of (2) and (3), 
 
 24" ^^ 20"' 
 
 therefore x^xt F^A'+O _ 1F(A^a) 
 
 tnereiore x ^y - ^^,^, -;gi_^Q^ 
 
 which is therefore the sqnare of the length of the semi-diameter. 
 
 Now the equation to a circle, concentric with the ellipse, and with radins 
 eqnal to this length, is 
 
 ^'-^-s^.-fif^'^-^^v-o (,. 
 
 Adding (1) and (4) we obtain 
 
 'i{A + a)(Ax^ + nxy+Cy^ + [E'-iAC){x>+y-^ = (5), 
 
 which must (Art. 43) represent a locus which passes through the intersection 
 of the circle and ellipse. But (Art. 63) it represents two straight lines 
 through the centre, which are therefore the equi-oonjugate diameters. 
 
 295. To find the length of a straight line drawn from 
 any point {x'y') to meet a conic section. 
 
 Let the equations to tie line and conic, referred to any 
 axes ■whatsoever, be 
 
 x — x v — v' 
 
 =^—^ = l (1), 
 
 c s •■•\^J> 
 
 Aa? + Bxy-i- Cf + Dx + Ey + F^O (2), 
 
 Then, as in Art. 114, we have, to determine the distances 
 of {x'y') from the points of intersection of (1) and (2), the 
 equation 
 
 Ac' 
 + Bcs 
 
 + Cs' 
 
 r+{2Aa/ + By' + B)c 
 + {2Cy' + Bx' + i;)s 
 
 l + cf>{a^y')=0 (3), 
 
 where ^ (x'^) = Ax'" + Bxy + Cy'^ + Boc + Ey' + F. 
 
 The remarks on the corresponding equation, obtained in 
 Art. 176, may be applied without alteration to (3). We 
 leave the student to verify results already obtained about 
 
INTEESECTION OP A STRAIGHT LINE WITH A CONIC. 279 
 
 the asymptotes, by equating to zero the coefficients of P a,nd 
 I, as in Arts. 177, 178. 
 
 296. The rectangle on the segments of the chord will 
 be equal to the product of the roots of equation (3), and 
 therefore 
 
 ___¥ei1__. 
 
 ^Ac' + Bcs+Cs" 
 
 and, if another chord be drawn through the same point (x'y), 
 and its direction be determined by s' and c', quantities cor- 
 responding to s and c above, the rectangle on its segments" 
 
 Ac" + £cs' + Gs'^' 
 
 hence the ratio of the rectangles is 
 
 Ag" + JBc^ + Cs'" : Ac" + Bcs + Cs\ 
 
 a ratio which does not depend upon the point {x'y'), and 
 which remains the same, as long as the chords make, respec- 
 tively, the same angles with the axes; hence, if QQ', RR' he 
 two chords of a conic, and P their point of intersection, the 
 ratio PQ . PQ' : PR • PI^' *s not altered by moving each 
 chord parallel to itself, and so shifting the position of P in 
 any manner. The reader will observe that this conclusion 
 might have been deduced from the equations of Arts. 114, 
 176, 251. 
 
280 GENERAL PROPERTIES OF CONIC SECTIONS. 
 
 Cor. 1. Let OV, CS be semi-diameters parallel to QQ', 
 RE respectively; then 
 
 PQ.PQ' _CV* 
 FM . FE 08* ' 
 
 Cor. 2. Let QQ' . PP' move parallel to themselves, till 
 they become tangents at L and T; then PQ . PQ' becomes 
 DL', and PE . PE becomes DT"; hence 
 I)r_CV^ DL_CV 
 'DT'~G8'" °^ DT~ CS' 
 
 COE. 3. Let Q, Q', E, E be the four points where a 
 circle intersects the conic; then (Euc. III. 35, 36) 
 
 PQ.PQ' = PE.PE; .: CV= CS; 
 
 hence the diameters parallel to QQ' and EE are equal, 
 and therefore equally inclined to the axes of the conic; i.e. 
 if a conic cmd a circle intersect in four points, two chords 
 passing through four points are equally inclined to the axes of 
 the conic. 
 
 Cor. 4. A particular case of Cor. 3 is when three of the 
 points of intersection approach indefinitely near to one an- 
 other. The circle is then called the Circle of Curvature at 
 the point to which the three points have approached ; and it 
 will be observed, that one circle and one only can be drawn 
 through these points, since three points determine a circle. 
 If then QQ: and EE move parallel to themselves, till the 
 points Q, E, Q' approach indefinitely near to one another, 
 the result of Cor. 3 may be thus stated : The common chord 
 of a conic and the circle of curvature at any point, and the 
 tangent to the conic at that point, are equally inclined to the 
 axes of the conic. 
 
 *297. To find the equation to the tangent of a conic sec- 
 tion at the point (x'y). 
 
TANGENT AND POLAR. 281 
 
 Exactly as in Art. 180, if (ajy) is on the curve, 
 
 and the condition that equation (1) (Art. 295) should be 
 a tangent is 
 
 {^Ax+By'+D)c+{2Cy' + Bx' + E)s=0 (4). 
 
 From (1) and (4) we obtain the equation to the tangent 
 
 y-y'_ 2Ax' + By' + D 
 
 x~x' 2Cy' + Bx' + i: ^ ' ■ 
 
 Multiplying up, and remembering that (x^y") is a point 
 on (2), we get equation (5) in the form 
 
 {2Ax'+By'+I))x+{2Cy'+Bx'+ir)y+l)x'+Ey'+2F=0...(<5). 
 
 CoE. If the curve passes through the origin, F=0, and 
 the equation to the tangent at the origin (0, 0) is 
 
 l)x + i:y = 0; 
 
 hence, if a conic passes through the origin, the equation to 
 the tangent at the origin is obtained by equating to zero the 
 terms of lowest dimensions in the equation to the conic. 
 
 *298. Equation (6) of Art. 297 is not altered, if x 
 changes place with x' and y with y'; hence (Art. 131) all 
 the theories of poles and polars proved for the circle in 
 Arts. 123 — 127, 129, 130, can be proved for any conic by 
 means of this equation. 
 
 *299. If any quadrilateral ABFD (fig. Art. 104) be in^ 
 scribed in a conic, then each of the points E, C, G is the pole 
 of the straight lime joining the other two. 
 
 ' The reader of the Differential Calculus will observe that this equation is 
 
282 GENERAL PEOPERTIES OF CONIC SECTIONS. 
 
 Take EB, EF as the axes of x and y, and let 
 EA = a, EB = a', ED = h, EF^h'; 
 tlien the equations to CA and CB are 
 
 ?+|=i (1). 5+«=i (2); 
 
 a 
 
 h~ ^" a'^b' 
 
 and the equations to AF and BD are 
 
 M=i (3)' 5+1=1 w- 
 
 Now by adding (1) and (2), we obtain the equation 
 
 <hl)Ml^l)=' (^)' 
 
 which is therefore (Art. 43) the equation to a straight line 
 passing through C. But we obtain the same equation by 
 adding (3) and (4) ; therefore also (5) passes through G. 
 Hence (5) is the equation to CO. 
 
 Now, if the equation to the conic is 
 
 Ax' + Bxy+Cy^ + Bx + Ey + F^O (6), 
 
 a and a' are found by putting ^ = in (6), and are therefore 
 the roots of the equation Aar'+Bx+F=0; hence (Appendix) 
 
 11 D . ., , 1 1 E 
 
 _+_ = __; similarly ^+^ = -^. 
 
 and equation (5) becomes 
 
 I>x + Fh/ + 2F=0, 
 
 which is (Art. 298) the polar of the point (0, 0), i. e. the 
 origin; hence CG is the polar of E. Similarly GE is the 
 polar of G; and hence, by Art. 130, EG is the polar oi .G. 
 The points C, E, G are called a conjugate triad with respect 
 to any conic passing through A, B, F, D. 
 
DIAMETERS. 283 
 
 *300. To find the equation to the normal to a conic 
 section at a point {x'jj'), the axes being rectangular. 
 
 The equation to the normal will be, from equation (5) of 
 Art. 297, 
 
 y - y' _ 2Gi/ + Bx' + E 
 x-x'~'iAai+By' + D' 
 
 Cob. If the curve passes through the origin, the equa- 
 tion to the normal at the origin becomes 
 
 Ex-^By = 0. 
 
 *301. To find the equation to the diameter of a conic 
 section, bisecting chords drawn parallel to a given line. 
 
 In Art. 295 let (1) represent one of the chords, and let 
 
 the given line be y = mx, so that — = m. Then, if (a/y) 
 
 
 
 be the middle point of the chord, we have, as in Art. 132, 
 
 for the equation to the diameter, 
 
 2Ax + By -VB + {^Cy + Bx-\- E)m = 0. 
 
 *302. Tangents at the extremities of any chord intersect 
 in the diameter of which the chord is an ordinate. 
 
 *30S. The polar of any point on a dia/meter and the 
 tangents at the extremities of a diameter are parallel to the 
 ordinates, 
 
 "We leave the student to prove these two propositions for 
 any conic. As we have found the equation to the diameter 
 bisecting any system of parallel chords, and shewn that the 
 equation to the polar is of the same form as the equation 
 to the tangent, he will find no diflSculty in imitating the 
 proofs of Art. 220 and the latter part of Art. 236. 
 
284 GENERAL PROPERTIES OF CONIC SECTIONS. 
 
 304. Polar equation, the focus being pole. 
 
 It is proved in Arts. 209, 274, that the focal polar equa- 
 tion to any conic may be written 
 
 - = 1 — e cos d, 
 P 
 
 where I = half the latus rectum, and e = 1 for the parabola, 
 
 the left-hand focus being the pole in the ellipse, and the 
 
 right-hand in the hyperbola. 
 
 Ex. In any conic lection the semi-latas tectum is an harmonic mean 
 between the segments, made by the focus, of any focal chord. ^,__^ 
 
 Let PSp be the focal chord; then 
 
 ^=l-ecosfl, ^=l-ecos(x+e); 
 
 . , , 1 1 1 - e COS 6 1 + e ces 2 
 
 therefore ^ + ^ = — j— + — ^_ = -^ , 
 
 which proves the proposition. 
 
 305. To find the polar equation to the chord of a conic 
 section, the focus being the pole, and thence to deduce the 
 polar equation to the tangent. 
 
 Let the equation to the conic and the chord be 
 
 -=l-ecos^ ... (1), -=^cos^-|-5cos (5-a)...(2)'; 
 
 and let the angular co-ordinates of two points on the conic be 
 a-jg, a-|-/3; then, if (2) passes through a -/3, we have for 
 this point, from (1) and (2), 
 
 1 - e cos (a -yS) = - = -4 cos (a - ^e) -1- 5 cos jS ; 
 therefore {A+e) cos (a-/8) -l-5cos/3-l = (3). 
 
 1 The student should satisfy himself that any straight line can be repre- 
 sented by equation (2). 
 
, POLAR EQtTATIOK TO CHORB AND TANGENT. 285 
 
 Similarly, if (2) passes through a + /8, 
 
 (^ + e)cos(a + /S)+jBcos;8-l = (4). 
 
 Subtracting (4) from (3), we have 
 
 A + e = 0, and therefore B cos /8 — 1 = 0; 
 hence, if (2) is the chord joining the two points, it becomes 
 
 - = secyS cos {0 — oL) — e cos 9 
 
 ,(5). 
 
 If the chord becomes a tangent at the point a, /8 = 0, and 
 (5) becomes 
 
 - = coB(9 — a) — e cos 6. 
 
 306. In any conic section, if SP, SQ he two radii vectores, 
 and PT, QT tangents at P and Q, then ST bisects the angle 
 PSQ, unless PT, QT be drawn to different branches of the 
 hyperbola, in which case ST bisects the angle supplementary 
 to PSQ. 
 
 Take the left-hand focus in the ellipse and the right-hand 
 focus in the hyperbola, and let the vectorial angles of P and 
 
 ^ be a and /3; it will be seen from Art. 210 that these 
 
286 
 
 GENERAL PEOPERTIES OF CONIC SECTIONS, 
 
 angles are as in the figure ; for in the polar equation used in 
 Art. 304, the. outer branch of the hyperbola corresponds to 
 the negative values of the radii vectores, but the vectorial 
 angle is formed by the positive direction of the radius vector. 
 Then the equations to PT, QTare 
 
 - = cos (^ — a) — e cos 6, - = cos (d — B) — e cos 0; 
 P , P 
 
 and therefore at the point T we have from these two equa- 
 tions 
 
 cos (5 - a) = cos (0 - iS) ; 
 
 therefore e-a = ^-d, or e-0L = 2ir-{e -^), 
 
 for — a cannot = 0-/3, since a and yS are by hypothesis not 
 equal; hence 
 
 e = - 
 
 or = TT + - 
 
 2 ' 2 ' 
 
 the latter value belonging to the case drawn in the figure 
 for the hyperbola, where 8T bisects the supplemental angle 
 Q8P. It will be seen that in this case PT, QT subtend 
 supplemental angles at the focus ; for 
 
 Q8T = T8P' = 180" - T8P. 
 The figure for the parabola is similar to that drawn for the 
 ellipse. 
 
 Ex. The circle, which passes through the points of intersection of three 
 tangents to a parabola, will pass through the focus. 
 
 Let P, Q, B be the pointa of tangency, 
 and let the tangents at P, Q, intersect 
 in r ; at Q, iS, in jp ; at R, P, in 2 ; tlien 
 by Art. 269, Ex, 
 
 RSP SSQ+QSP 
 
 Tgr=- 
 
 2 
 
 =pSQ + QSr, 
 = pSr; 
 
 that is, pSr is the supplement of pqr, 
 which (Euc. III. 22) proves the proposition. 
 
TANGENT AND NOEMAl AS AXES. 287 
 
 307. To find the equation to a conic, when the tangent 
 and normal at any point are the axes ofj and x. 
 
 Since the axis of y is a taagent at the origin, when a; = 
 in the general equation 
 
 Ax'' + Bxy + Of +Dx + %+ F=0, 
 
 the values of y become each = ; hence R= 0^ F= 0,. and 
 the equation is reduced to 
 
 Aa? + Bxy + C/ + Dx = 0, 
 
 where the axis of x is any straight line drawn through the 
 point of contact. If the axes are rectangular, they are the 
 tangent and normal at the origin. 
 
 Ei. If, fhrough a given point on a conic, any two straight lines at right 
 angles to each other be drawn to meet the curve, the straight line joining their 
 extremities will pass through a fixed point on the normal of the given point. 
 
 The equation to the conic, referred to the tangent and normal at the 
 point, is 
 
 Ax^ + Bx^+Cy'+I)x=^0 (1). 
 
 Letthe equations to the two lines be 
 
 1 
 
 y^nuc, y=--x, 
 
 or as one locus 
 
 whence if-im — j xy-x^=Q (2). 
 
 Multiplying (2) hy C and subtracting it from (1), we have 
 
 {A + C)!>?+\B + c(m-^\xy+Dx=0, 
 
 as the equation to a locus which passes through (Art. 43) the intersections ol 
 (1) and (2). But this equation represents two straight lines, namely, the 
 tangent at the given point (x=0), and 
 
 {A. + O)x+\B + o(m-^\y+D=0, 
 
 which must represent the chord joining the extremities of the two lines. 
 
288 GENERAL PROPERTIES OF CONIC SECTIONS. 
 
 The point, where this line cuts the normal, which is the axis of x, is found 
 by making y=Om the equation; thus we obtain 
 
 D 
 
 hence the chord cuts the normal at an invariable distance from the origin. 
 In the circle the point of section is the centre, by Euc. iii. 31. 
 
 308. To find the equation to a conic section, when the 
 axes of co-ordinates are tangents to the curve. 
 
 Suppose the axes of x and y to touch the conic at dis- 
 tances a and b from the origin. Putting y = and x = 
 successively in the equation <f> {xy) = 0, we obtain 
 
 Aa? + Dx + F=0 (1), Cy''+Ey + F=Q (2), 
 
 where the roots of (1) are each = a, and the roots of (2) are 
 each = h ; hence (Appendix) 
 
 \^A 2^_:D. I _G 2__F 
 a" F' a~ F' ¥~ F' b~ F' 
 
 and the equation ^ (xy) = becomes, after dividing by i''and 
 substituting these values, 
 
 x\ B , v' 2a; 2« , ^ 
 
 or ('? + |_iy + Aa:^ = 0, where ifc = f-l. 
 
 \a b J •' F ah 
 
 309. If the conic belongs to the parabola class, 
 therefore Te = 0, ori = — - 
 
TWO TANGENTS AS AXES. 289 
 
 If k = 0, the equation represents two coincident straight 
 
 4 
 
 lines. IiJc = = , it becomes 
 
 ao 
 
 ('f + ^_lV = ^ or-+^ + 2 /^=1- 
 \a h J ab ' a b~ \/ ab . ' 
 
 therefore ^f ± ^/f =± L or^f + ^1 =1 ; 
 
 for the latter form is equivalent to the former, if we remem- 
 ber that the radicals may be positive or negative. 
 
 *310. Any straight line drawn through the intersection of 
 two tangents to a conic section, is harmonically divided by the 
 curve and the chord of contact. 
 
 Take the two tangents as axes ; then the equation to the 
 conic is 
 
 (^ + l-lj + Jcxy = (1), 
 
 and the equation to the chord of contact is 
 
 -+f = l (2). 
 
 a o 
 
 Let the equation to any straight line through the origin 
 be 
 
 ^ = y = l (3), 
 
 c s 
 
 where I is the distance of any point {xy) from the origin. 
 Hence, to find the distance from the origin of the points of 
 intersection of (1) and (3), we have 
 
 (f+I-^y + ^^^='' org + |-^)V^c« = 0...(4). 
 
 19 
 
290 GENEEAL PEOPEETIES OP CONIC SECTIONS. 
 
 Now, if ^,, jT, be the roots of this quadratic in -y, that 
 
 is, if I', I" are the values of I at the points where (1) and (3) 
 intersect, we have 
 
 ^r-KM) <^)- 
 
 Also forthe point of intersection of (2) and (3) we have 
 
 cl si ^ 1 c s , , 
 
 - + x = l» or7 = - + T (6); 
 
 a b I a ' 
 
 hence, denoting this value by l^, we have from (5) and (6) 
 2 = 1 + 1 
 
 or 7i is an harmonic mean between I' and I". 
 
 OoE. Any stfaight line drawn through the pole is har- 
 monically divided by the curve and the polar. 
 
 This has been proved for the case when the pole is 
 without the conic. If the pole is withiri, the conic, let A be 
 the pole, BC the polar (figw Art. 334), and let AB be any 
 straight line drawn through A, meeting the polar in B; 
 then, if two tangents are drawn from B, the chord of contact 
 will (Art. 129) pass througb A, and therefore the straight 
 line AB is harmonically divided, as above. 
 
 311. Conditions necessary to determine a locus of the 
 second degree. 
 
 We observed (Art. 24), that the general equation of the 
 first degi'ee has in reality only two independent constants, 
 .though apparently containing three, and that, consequently, 
 a straigbt line could be subjected to two independent con- 
 ditions only, since these would give two relations between 
 the constants, which would suffice to determine them. The 
 general equation of the second degree contains six coeffi- 
 
CONDITIONS NECESSARY TO DETERMINE A CONIC. 291 
 
 cients ; but, as -we may divide all the terms by one of these 
 coefficients, we see that the equation contains five inde- 
 pendent constants only. To determine a locus of the second 
 degree, we must give the values of these five constants, or 
 give five independent equations between them, by which 
 they may be determined ; but, in this case, it is necessary to 
 examine, whether the equations admit of a system of real 
 solutions, or of more than one, and whether the resulting 
 equation of the second degree represents a curve, or one 
 of those varieties, which have been explained above. The 
 simplest condition would be, that the locus should pass 
 through five points ; for the co-ordinates of these points, 
 substituted successively in the general equation, will give 
 five equations to determine the five constants. These equa- 
 tions, being of the first degree, will admit of one system of 
 real solutions, and one only, if they are consistent and inde- 
 pendent; but we have not shewn that this condition will 
 always be satisfied. We shall therefore prove the follow- 
 ing theorem. 
 
 312. Through five real points, no four of which are in the 
 same straight line, one conic section and one only can be drawn. 
 
 We will first consider the case, where no three are in 
 the same straight line. Let the axes be so chosen that two 
 of the points are on the axis of x, and two upon the axis 
 of y, and suppose the points on the axis of x and y, respec- 
 tively, to be at distances a, a' and 5, b' from the origin. Let 
 the equation to the conic be 
 
 Ax^ + Bxy+Cf + Dx + Ey + F=0 (1) ; 
 
 then the values of' a, a and b, V will be found by putting 
 y and X successively =0 in (1), and will therefore be the 
 roots of the equations 
 
 A^^Dx^F=(S (2), 
 
 Gf + Ey + F=0 ; (3); 
 
 19—2 
 
292 GENEEAL PKOPERTIES OF CONIC SECTIONS, 
 
 therefore 
 
 1 
 
 aa 
 
 A 
 "F' 
 
 1+1 — ^. 
 
 a^a'- F' 
 
 1 
 
 W 
 
 G 
 ~F' 
 
 I \ E 
 
 b^ V~ F' 
 
 Hence the general equation becomes, after dividing by F 
 and substituting, 
 
 X 
 
 aa 
 
 ?+f»*+&-(^l>-(^F)^+'=» <*'■ 
 
 Let {xy) be the fifth point ; then, by substituting x, y 
 
 D 
 
 in (4), we shall obtain a simple equation to determine -^. 
 
 Hence one conic and one only can be drawn through the 
 five points. 
 
 If three of the five points be in one straight line, we 
 shall have (xy) on one of the axes, and therefore x =0 or 
 
 ^' = ; in either case the value of -^ from (4) would be 
 
 infinite, and therefore equation (4) would, by dividing by -r, , 
 
 become xy = 0, and would represent the two axes of co-or- 
 dinates. We might have foreseen, that the locus in this 
 case could not be an ellipse, hyperbola, or parabola, since 
 these curves cannot be cut by a straight line in more than 
 two points. 
 
 If more than three of the points are in one straight line, 
 the coefficients of (1) cannot all be determined by the method 
 of this article; and it is obvious that this ought to be the 
 case, since more than one pair of straight lines can then be 
 made to pass through the five points. 
 
 Thus we have proved, that we can always find a real 
 equation of the second degree, and one only, which is satis- 
 fied by the co-ordinates of the five points. This will always 
 
CONDITIONS NECESSAKY TO DETERMINE A CONIC. 293 
 
 represent a real geometrical locus, since the imaginary loci of 
 the second degree (Arts. 150, 153) could not be satisfied by 
 the co-ordinates of these points. 
 
 313. We may say, generally, that a conic can be made 
 to fulfil five independent conditions, where each condition 
 enables us to eliminate one constant. Thus the position of 
 the centre must be counted as two conditions, for it gave us 
 (Art. 141) two relations between the coefficients, and enabled 
 us to eliminate two. Again, the position of the focus must 
 count as two; for the general equation to a conic with a 
 focus at a given point {x'y') may be written (Art. 208) 
 
 {x-xr+{y-y'Y={Px+qy+Rf (1), 
 
 involving three undetermined constants only. It must not 
 be assumed that five conditions such as the above will always 
 give us one conic and one only, as in Art. 312. 
 
 Ex. 1. How many parabolas can ie drawn through four points? 
 
 We have here five conditions, since 5^-44(7=0 is one. Suppose no 
 three of the points to be in one straight Une ; then we obtain equation (4) of 
 Art. 312 for a conic passing through the four points. If the conic is a 
 parabola, we have 
 
 \Fj aa'bb' 
 
 If the product aa%h' is positive, there will be two parabolas passing 
 through the four points ; it the product is negative, no real parabola can be 
 drawn through them. If more than two points are on the same straight 
 line, the parabola is rectilinear, or parallel straight lines. 
 
 Ex. 2. Mow many conies with a given focus can be described about a 
 triangle ! 
 
 Let the focus be {x'y'), and the angular points be {xiyi), (Xiy^), (x^ys); 
 then, if we write 5i' for the known quantity {xi - x!)^ + (yi- ^0^ ^° *^^* ^i is 
 the distance between (x^y^ and the focus, we have, from equation (1) 
 (Art. 313), 
 
 S,= ±(Pa!i+Qy,+i2), 5a=±(i'ai4Q2^2+ii:), 5a= ±(Px3+ Qy3+-K). 
 
 Every combination of signs gives a system of equations, to determine P, 
 Q, is, and there are eight combinations ; but evidently, if we change the signs 
 in the three equations, it is equivalent to changing the signs of P, Q, B, 
 
294 GENERAL PEOPERTIES OF CONIC SECTIONS. 
 
 ■which would leave equation (1) the same. Hence' there are four different 
 solutions, and four different conies can be' described about the triangle. 
 
 314. When a conic has been subjected to four conditions 
 only, there will remain one arbitrary constant in the equa- 
 tion, and there will be an infinity of curves fulfilling the 
 given conditions. We may then find the locus of any of 
 the remarkable points of the curve ; for, having introduced 
 the conditions, we may obtain the equations for determining 
 the point in question, and find the equation to the locus by 
 eliminating the arbitrary constant between them. 
 
 Ex. A conic is described, touching two straight lines at given distances 
 (a and b) from their point of intersection ; find the locus of its centre. 
 
 Taking the two lines as axes, the equation to the conic is, by Art. 308, 
 hence, the equations for the centre are (Art. 141) 
 
 Eliminating le, we have for the required locus, 
 
 ii,y-ix=0, 
 a straight line bisecting the chord of contact. 
 
 *315. Similar Conies. 
 
 Def. Two curves are said to be similar and similarly 
 placed, when, any point being taken in the plane of one 
 curve, another point (7 can be found in the plane of the 
 other, such that parallel radii, drawn from and 0' to the 
 first and second curve respectively, are to one another in a 
 fixed ratio. They are said to be similar when they can be 
 made to fulfil the above condition, by turning one of them 
 round a fixed point. 
 
 The points 0, 0' are called Centres of Similitude. 
 
SIMILAR CONICS. 295 
 
 316: If, in the planes of two curves, one such pair of 
 points, as and 0', can be found, an infinity of other pairs 
 can hefovmd. 
 
 For suppose OB, C/B to be parallel radii in the fixed 
 
 ratio of 1 : /c; and take in the first figure any point 0, and 
 draw OC parallel to OC, so that 
 
 O'C : 00=0'B' : OB=h : 1, 
 
 and join GB, O'B; then the two triangles OCB, O'G'B are 
 similar (Euc. VI. 6), and therefore G'E is parallel to GB; 
 also 
 
 G'B : GB=0'B : OB = k : 1; 
 
 therefore, since OB, O'B are any parallel radii, G and C' 
 are centres of similitude. Hence for every point in the first 
 figure there is a point in the' second, such that the pair are 
 centres of similitude. 
 
 COE. If be the centre of the first curve, 0' must be 
 the centre of the second; for every diameter BOA is bisected 
 in 0, therefore every parallel diameter B'O'A' through 0' is 
 bisected in 0' ; for otherwise the ratio OB : O'B would not 
 be equal to the ratio OA : O'A'. 
 
 *317. All conies, whose eccentricity is the same, are 
 similar figures. 
 
 Let two of such cohicSj which are evidently (Art. 174) of 
 the same class, be placed with their axes parallel, and a focus 
 in one coincident "with the corresponding focua in the other; 
 
296 GENERAL PROPERTIES OF CONIC SECTIONS. 
 
 then, the common focus being pole, their equations are 
 (Art. 804) 
 
 7 y 
 
 - = l-ecos^, -=l-ecos0; 
 
 P P 
 
 hence, when is the same in each, we have the parallel 
 radii vectores in the proportion of the latera recta. The 
 two conies then are similar, and the common focus is a centre 
 of similitude. 
 
 Cor. All parabolas are similar curves. 
 
 *318. To find the condition that the two conies 
 
 'Ax' + Bxy+ Cy^ + Dx-\- Ey + F=0 (1), 
 
 A'a;' + S'xy+Cy + I)'x + i:'y + F' = (2), 
 
 should he similar and similarly placed. 
 
 Let us suppose 0{x'y'), C/(x"i/") to be two points, such as 
 are described above, and suppose OB, O'B' to be parallel 
 radii vectores making an angle 6 with the axis of x, and let 
 their equations be 
 
 c s ' c s 
 
 Combining (3) and (4) with (1) and (2) respectively, we shall 
 obtain two equations (Art. 295), 
 
 .Fl' + Ql + B = 0...(5), PT+Q7 + iJ' = 0...(6), 
 
 the roots of which are the distances of and 0' respectively 
 from the curves. But by hypothesis, as long as OB is 
 parallel to O'B', their ratio is constant, or 
 
 0'F = k. OB, suppose. 
 
 Hence, if we write kl for I in equation (6), the values of I in 
 (5) and (6) wiU be the same, and therefore we shall have 
 
 Fk' Qk E ^>- 
 
SIMILAR CONICS. 297 
 
 Since (Alt. 295) B and B' depend only upon the position of 
 and 0', and do not change as the lines move about these 
 points, we may assume the ratio oi B : B a,s constant, and 
 
 we have, writing m for - , 
 c 
 
 P A + Bm+Cm^ FB ,„, 
 
 -F = A' + B'm+CW = ^^f''''^ ^^^' 
 
 or A-fj,A' + {B-fjiB')m+{C-fiC')7ri' = (9), 
 
 for all values of m; but this can only be the case when 
 
 A-fiA' = 0, B-,ji.B = Q, O~iJ,C'=0, 
 
 A_B_C^ 
 or A'~B'~ G" 
 
 Hence, if two conies are similar and similarly placed, the co- 
 efficients of the highest powers of the variables are the same 
 in both, or differ only by a constant multiplier. 
 
 *319. "We proceed to enquire, whether the above con- 
 ditions are sufficient to ensure similarity. Suppose them 
 
 fulfilled, and that we have multiplied equation (2) by -jj, 
 so as to bring it to the form 
 
 Ax' + Bxi,+ Cf + D,x + B^y + F,=0 (10). 
 
 (i) If B^—iAG= 0, the curves are in general parabolas, 
 and are similar by Art. 317. This conclusion does not apply 
 to the case where one equation represents straight lines or 
 an imaginary locus. 
 
 (ii) If B' — 'iAO is not =0, the curves have centres, 
 and (Art. 316, Cor.) we need only examine; whether the 
 centres of the curves are centres of similitude. Now we 
 may, by transferring the origin to the' centre of each, the 
 
298' GENERAL PROPERTIES OF CONIC SECTIONS. 
 
 axes remaining in each case parallel to their original direc-; 
 tion, reduce (1) and (10), as in Art. 141, to 
 
 Aa? + Bxy+Cf + <^{xy)^0 (11), 
 
 As? + Bxy+Cf+4>,t^^y,) = (12), 
 
 where <f>{xy)=0, ^^{xy) = represent the equations before 
 reduction, and (iy), (^,,y,) are the centres. Let the equa- 
 tion to any straight line through the centre of- (11) be 
 
 -=y-^l (13), 
 
 c s 
 
 where I is the distance of the centre from {xy) ; then for the 
 distance of the intersection of (11) and (13) from the centre, 
 we have, as in Art. 295, 
 
 „_ -(fjjxy) ■ ,,,> 
 
 Ac' + Bcs+CY ^ '' 
 
 Similarly, "with equation (12) we should have 
 
 7'2 _ ~yi('^iyi) /-i K\ . 
 
 ^ -Ad' + Bcs+Cs' ^^^''' 
 
 and, if c and s have the same values in (14) and (15), that 
 is, if the two diistances denoted by I and I' be parallel, we 
 shall have the ratio of I : I' constant. Hence the centre of 
 each curve is a point answering the required condition; 
 consequently any number of such points ma,y be found, and 
 the curves are similar. ' 
 
 *320. If fjiixy) and ^,(»iyi) be of different signs, the 
 ratio P : T" will be negative, so that the fixed ratio of 
 the radii is imaginary, and the condition is fulfilled alge- 
 braically only. If the curves be of the Ellipse class, we 
 see, by Axt. 150, that in this case one of them represents 
 an; impossible locus.. If they be of, the Hyperbola class. 
 
SIMILAB CONICS. 299 
 
 they -will (Art. 242) represent two hyperbolas with the 
 same asymptotes, but not lying in the same angles of the 
 asymptotes. 
 
 Again, one or both of the quantities <j>{xy), ^,(^1^1) 
 might = 0, in which case the fixed ratio would be zero, in- 
 finite, or indeterminate ; but in this case one or both of the 
 equations would (Art. 150) represent straight lines which in- 
 tersect, imaginary in the case of the ellipse, and real in the 
 case of the hyperbola. 
 
 *321. If two conies are similar and not similarly placed, 
 they can, by definition, be made to fulfil the conditions of 
 Art. 318, by turning one of them round a fixed point. 
 
 Let equations (1) and (2) of Art. 318, transformed to 
 rectangular axes, if necessary, represent two conies, similar 
 but not similarly placed, and let conic (1) be turned about 
 the origin through such an angle a, that it may be similarly 
 placed with (2). This will, of course, as far as regards the 
 change in its equation, be the same as if the axes were 
 turned through an angle a, and we shall have a new 
 equation, 
 
 ase' + bxy + cy^ + dx + ei/ +/= (3), 
 
 and, from Art. 147, we have 
 
 V-iac = B'-'iAG, a + c = A+ G. (4). 
 
 But, since (3) is similar and similarly placed to (2), we 
 have 
 
 a _ h _ c _ 
 A'~B'~ C'~'^^^^' 
 
 from which equations we have . 
 
 ' 6' - 4ac = /*' {B" - iA'C), a + c =/t {A' + C). 
 
 B"-iA'C' b'-^iac 
 
 Hence 
 
 {A' + .Cy {a + c)' 
 
300 GENERAL PROPERTIES OF CONIC SECTIONS. 
 
 tueretore (^a' + 0')' ~ IJ+W ' 
 
 is, from (4), the required condition. If this is satisfied, the 
 conies will be similar, with the same exceptions as in Art. 
 319. This is simply the condition (Art. 289) that the asymp- 
 totes should contain equal angles, or that (Art. 294) the 
 eccentricities should be determined by the same equation. 
 
 322. Sections of the Cone. 
 
 The surface described by an indefinite straight line, which 
 is carried round the perimeter of a given circle, always passing 
 through a given point, is called a cone. The circle is called 
 the base of the cone, the fixed point the vertex, and the line 
 joining the vertex and the centre of the base is called the axis. 
 A cone is said to be right if the axis is perpendicular to the 
 plane of the base, and ohlique if the axis is inclined at any 
 other angle to that plane. As the generating line is not 
 limited, the surface of the cone consists of two portions or 
 sheets (fig. Art. 325), perfectly similar, situated on opposite 
 sides of the vertex, and of indefinite extent. It is evident, 
 from the method in which the cone is generated, that every 
 plane parallel to the base will cut the cone in a circle, and 
 that every plane through the axis will cut it in two straight 
 lines, in both which cases the section will be represented by an 
 equation of the second degree. We shall now shew that the 
 same is the case, in whatever manner the plane cuts the cone. 
 We shall content ourselves with proving this property in the 
 case of right cones only, since a full investigation of this part 
 of the subject will be found in most Geometrical Treatises. 
 
 323'. Every section of a right cone hy a plane is a cwrve 
 of the second degree, 
 
 Obs. The generating line, in a right cone, will always 
 make the same angle with the axis. 
 
SECTIONS OF THE CONE. 
 
 301 
 
 Let HRKL be a plane ; AB a fixed line, the axis of a 
 cone, inclined at an angle a to the plane; ^ (7 a perpendicular 
 
 from A, the vertex, on the plane ; AP the generating line, 
 revolving round AB, inclined to it at a constant angle ^. 
 Then P, the extremity of AP, will evidently trace out some 
 section of the cone on the plane, whicli is supposed to be 
 intercepted between the vertex and the circle, round the 
 perimeter of which the generating line is carried. 
 
 Draw PJf perpendicular to 50, produced; join BP and 
 CP. Take G as origin, OM as axis of x, and a perpendicular 
 to it in the plane HBKL, as axis of y; let P be the point 
 [xy) and AB = a. 
 
 Then BP'= PIP + BAP 
 
 = y^+ {x+acosay (1), 
 
 since ^ ABG=a; also 
 
 BP" = a' + AP' - 2a AP cos /3, 
 = a" + {a" sW a + f + «^- 2a cos ySVa'sin^a + y + a;', . . (2), 
 since ^P'^ = ^0^+CP^ 
 
 Equating (1) and (2), we have for the locus of P 
 a' cos' a + 2ax cos oi. = a' + a' sin' a - 2a cos jSVa" sin' a + x' + y". 
 
 or cos ^ 'Ja' sin'a + af+y^=a sin' a - a; cos a, 
 
 a curve of the second degree. 
 
302 
 
 GENERAL PROPERTIES OF CONIC SECTIONS. 
 
 324. Comparing this equation with the general equation 
 of the second degree. 
 
 Ax' + Bxy + Of^-Bx + Ey + F= 0, 
 we see that in this case 
 
 ^ = cos'';S-cos''«, B = 0, C=cos*;S. 
 Now (Arts. 150, 153), the curve is an Hyperbola, Parabola, 
 or Ellipse, according as B' — 4<AG> = < 0, or as 
 
 — cos*" /S (cos'' ^ — cos" a) > = < 0, 
 as cos^a — cos'jS> = < 0, 
 
 as sin {fi + «) sin (/S — a) > = < 0, 
 
 as sin (/8 — a) > = < 0, 
 
 or as /9 > = < «, 
 
 since /3 + a is by construction less than ir, and therefore 
 sin (/3 + a) always positive. 
 
 325. "We may easily identify the above results with the 
 forms of the Curves that we 
 have already discovered; for, 
 let 8AR be a cone, AO the 
 axis, B the point where the 
 cutting plane EE' cuts the 
 axis; draw BP parallel to 
 AR, then we have by our 
 assumptions 
 
 AB=a, /.EAB = 0, 
 zEBA = a, ^8AR = 2^, 
 AAEB = ir-^-a., 
 and hence 
 
 z 8AR+ZAEB = -jr + /3 - a. 
 If /3 < a, these two an- 
 gles are less than two right 
 angles, and the point E will ' 
 lie below the point P, and 
 
EXAMPLES XII. 303 
 
 the section will evidently be limited in every direction, and 
 be an Ellipse. 
 
 If /3 = a, the two angles = two right angles ; the point E 
 coincides with P, and the cutting plane is parallel to AR. 
 The section will evidently be limited at P and unlimited in 
 the direction PB, and be a Parabola. 
 
 If /3 > a, the two angles are greater than two right angles; 
 the point E lies above P, and the cutting plane meets both 
 sheets of the cone. The section will be unlimited in every 
 direction, and be an Hyperbola. 
 
 EXAMPLES XII. 
 
 1. Employ the method of this chapter to verify the results 
 obtained in Examples VIII. , 
 
 2. Shew from the equation to the asymptotes, that the 
 general equation with rectangular axes represents an equilateral 
 hyperbola ii A+G=0. 
 
 3. Shew that the equation 
 
 y- 4a!y + 6k^- 6a;+ 2y- 11 = 
 
 represents an elli j)se, lying wholly between the lines a; = 3 and 
 a; = -2. 
 
 4. Shew that the equation 
 
 2/''-4a;y + 2a;'-2a! + 2y + 13 = 
 
 represents an hyperbola, no part of which lies between the lines 
 a; = 3 and a; = — 2. 
 
 5. Shew that the equation 
 
 y^ + 2xy + x' +2y + 4 = 
 represents a parabola, extending indefinitely in the positive direc- 
 tion of the axis of x. 
 
304 GENERAL PROPERTIES OF CONIC SECTIONS. 
 
 6. Shew that the equation 
 
 2/' + 2a;y-a!' + 2«/- 2a;-5 = 
 represents an hyperbola, whose ordinates are always real. 
 
 7. Shew that the equation 
 
 represents an imaginary locus. 
 
 8. Find the asymptotes and the eccentricity of each of the 
 hyperbolas 
 
 Axy-3af-2ai/ = 0, f -Ax' + 2i/-ix-9 = 0, 
 
 the axes being rectangular. 
 
 9. Trace the curve x' + y^= (ax + by + cf, and determine the 
 focus and directrix. 
 
 10. Find the axes of the hyperbolas 
 
 xy-ax + hy = Q (1), xy — ax + a^ = Q (2), 
 
 the axes of co-ordinates being inclined at an angle a. 
 
 11. If the equation to a conic is not altered when x and y 
 change places, the bisector of the angle between the positive 
 directions of the axes is an axis of the curve ; if it is not altered, 
 when X is changed to - y and yto-x, the bisector of the supple- 
 mentary angle is an axis. 
 
 12. Shew from Art. 284, or otherwise, that, if the axes of 
 co-ordinates are parallel to conjugate diameters, B = in the 
 general equation. 
 
 13. Determine /*, so that ^ (a^)+/t = may represent two 
 straight lines, and hence prove Art. 186 Ex. 
 
 . *14:. In the conic <^ (a!y) = 0, if a diameter parallel to y = 9Ka; 
 bisects a system of chords parallel to y = m'x, shew that 
 
 2Cmm' + B{m + m'} + 2As: 0. 
 
EXAMPLES XII. 305 
 
 * 1 5, Shew that the diameters parallel to y = mx and y = m'x, 
 in Ejs. 14, are conjugate ; and, supposing the axes rectangular, 
 find the direction of the axes of the conic. 
 
 *16. Find the equation to the tangents at the points where 
 the conic whose equation is 
 
 {ax + hy — \){ax + Vy — \) + cxy = 
 
 meets the axis of y ; and the equation to the straight line joining 
 their point of intersection with the point where the conic touches 
 the axis of x. 
 
 *17. From Art. 301 find the directions of the axes of the 
 conic, the axes of co-ordinates being supposed rectangular; (2) 
 find the equations of the centre, and shew that they represent 
 diameters bisecting chords parallel to the axes of co-ordinates; 
 (3) if these diameters are parallel, shew that the locus is a 
 parabola. 
 
 *18. In the conic <j>(xy)=0, shew that {Dx + Ey = f)) is a 
 chord bisected at the origin. 
 
 19. Given the length of a focal chord PSp in a conic, and the 
 rectangle contained by the segments SP, Sp ; find the latus- 
 rectum. 
 
 20. A circle is described concentric with a given conic 
 section ; find the equation to their common diameters, and the 
 condition that the curves should touch. 
 
 *21. If two hyperbolas have the same asymptotes, their 
 eccentricities are equal, or are connected by the equation 
 
 I 1 _. 
 
 *22. Find the equation of Art. 294 for the eccentricity, by 
 assuming (Art. 174) that e = the secant of half the angle between 
 the asymptotes. 
 
 *23. If any straight line OR be drawn through a ^oint 0, 
 and P, the pole of that line, be joined to 0, then the straight lines 
 OP, OB will form an harmonic pencil with the tangents from 0. 
 
 20 
 
306 GENERAL PROPERTIES OF CONIC SECTIONS. 
 
 24. Find the locus of the centres of the conies, obtained by 
 varying each of the constants in the general equation of the 
 second degree. 
 
 i 25. Find the locus of the centre of a conic passing through 
 four points, using equation (4) Art. 312, and shew that it is an 
 ellipse or hyperbola, according as aal and W are of different or 
 the same sign. 
 
 26. If a conic section and a circle have a common tangent 
 and a common chord, joining two points where the .curves inters 
 sect, the tangent and the chord are equally inclined to the axes 
 of the conic. 
 
 27. A conic section is cut in four points by a circle, and two 
 straight lines, each passing through two of the points of inter-, 
 section, are taken as axes of co-ordinates ; shew that the equation 
 to the conic may be written in the form 
 
 as' + hxy + y^ -v dx ■¥ ey +f= 0. 
 
 28. From a point two tangents OH, OK are drawn to a 
 conic section ; a straight line MQR is drawn, parallel to OK, to 
 intersect OH in M, and the curve in Q, R. Shew that 
 
 MH' : MQ.ME = OH' : OE'. 
 
 If QR meets the chord HK in T, then MQ . MR = MT'. 
 
 29. From the equation to the tangent in Art. 305, find the 
 angle that it makes with the axis of x. 
 
 30. In the parabola, if SP, SQ be two radii veotores, and 
 TT, TQ tangents at P and Q, then SP.SQ = ST'. 
 
 31. The locus of the intersection of tangents, at points which 
 subtend a given angle (<^) at the focus of a conic, is a conic which 
 
 has the same focus and directrix, and whose eccentricity = - 
 
 cos ^<j> ' 
 
 32. PSF, QSg are two focal chords of a conic ; shew that a 
 straight line through S bisecting the angle P,S^, will intersect the 
 chord QP produced in the directrix. 
 
EXAMPLES XII, > 307 
 
 33. If a chord: of a conic section, the eccentricity of which is 
 e, subtends at the focus a constant angle 2y, shew that it always 
 touches a conic section having the same focus, and of which the 
 eccentricity is e cos y. 
 
 34. The angular co-ordinates of the extremities of two chords 
 of a conic are, as in Art. 305, + ^, 6 — ^ and <I> + P, <^-)8; find 
 the locus of their intersection, if <ft — d = e, where j3 and e are 
 constants. 
 
 35.; A small arc of a conic being traced upon a plaije, shew 
 how to determine the nature of the conic to which it belongs. 
 
 36. ' If in a conic section a series of right-angled triangles is 
 described, having the right angles at a given point in the curve, 
 and tangents be drawn at the extremities of the hypotenuses, the 
 points of intersection of these tangents lie in one straight line. 
 
 37. A series of triangles is inscribed in a conic section, having 
 a common angular point, the angle at which is bisected by the 
 normal to the curve at the angular point ; shew that the sides, 
 opposite the common angular point, will all meefi in the point 
 of intersection of the tangents drawn at the two ends of the 
 normal. 
 
 38. How must Art. 307 Ex. be enunciated, if the conic is a 
 rectangular hyperbola 1 . 
 
 39. Shew that equation (4) of Art. 312. can be piit in the 
 form 
 
 Interpret this equation, and deduce the equation to a conic 
 refeiTcd to two tangents as axes. 
 
 40. If two conic sections have two points of contaqt, shew 
 that they will not meet in any other point. 
 
 41. Shew that two parabolas cannot touch each other in 
 more than one point, 
 
 20—2 
 
'308 GENERAL PROPERTIES OF CONIC SECTIONS. 
 
 42. If tte parallel chords of a conic are divided in any 
 proportion, the locus of the dividing point is a conic. 
 
 43. Find the equation to a circle referred to two tangents of 
 length a, containing an angle w. 
 
 *44. Shew that two conies of the same class, whose axes are 
 proportional in magnitude and parallel in direction, cannot have 
 more than two real finite points in common. 
 
 *45. A pair of tangents are drawn from a point to a conic, 
 and a straight line bisecting the angle between them cuts the 
 chord of contact in ; shew that the polar of is the external 
 bisector of the angle between the tangents. 
 
 46. If two tangents be taken for axes, the co-ordinates of the 
 centre of a conic are proportional to the lengths of the tangents; 
 hence shew that, if straight lines are drawn from the centre 
 parallel to the tangents, so as to form a parallelogram, the chord 
 joining the points where they cut the conic is parallel to the 
 diagonal, 
 
 47. In the sides AB, AO ot a. given triangle ABC take two 
 points M, N, and in the line joining them take a point P, 
 such that 
 
 BM : MA=:AN : NC=MP : PN ; 
 
 find the locus of P. 
 
 48. If a conic be iascribed in a triangle, and touch two sides 
 at the point of their bisection, shew that the straight line joining 
 the centre and the third point of contact will pass through the 
 opposite angular point. 
 
 *49. Shew that the equation to the equi-conjugate diameters 
 of the conic 
 
 5x' + 2j(;y + 5y' - 12a; - 12y = 
 
 is x^ + '^^xy+y'-l'2ix-l2y + l2 = 0. 
 
EXAMPLES XII. 309 
 
 *s 
 
 *50. If the equation to an ellipse is ^05° + Boey + Gy' + F=0, 
 stew that the equation to an hyperbola upon the same axes is of 
 the form 
 
 2 {A + G){Aoe' + Bxy+Gy') + {ff-4:AG) {!^ + f)+F' = 0, 
 
 and find the value of F'. 
 
 51. She-w that the equation 
 
 represents two ellipses. Find the principal axes, and shew that 
 the semi-diameter, drawn to the point where their common tangent 
 meets either of them, is equal to ^Za. 
 
 52. If the general equation of the second degree represents a 
 parabola, and the axes (rectangular) are tangents, find the equation 
 to the axis and directrix, and the co-ordinates of the focus. 
 
*CHAPTER XIII. 
 Airi^geA Notation and Trilinear Co-ordinates. 
 
 326. Bt reasoning precisely similar to that made use of 
 in the case of straight lines, we see that, if 
 
 8 = 0, 8' = (1), 
 
 be two conic sections {8 and 8' being of two dimensions in x 
 
 and y), then 
 
 '8+k8' = (2), 
 
 where k is some arbitrary constant, since it is also of two 
 dimensions, is also a conic section ; and, since the co-ordinates 
 of the points of intersection of equations (1) evidently satisfy 
 equation (2), therefore (2) represents a system of conic sec- 
 tions, passing through the four points of intersection of (1). 
 
 327. We sajfour, because the elimination of one of the 
 variables between two equations of the second degree pro- 
 duces, generally (Appendix), an equation of the fourth degree. 
 If the resulting equation should in particular cases fall below 
 the fourth degree, in consequence of the coefficients of one or 
 more of the higher powers vanishing, the curves may still be 
 said to intersect in four points, one or more of these points 
 being infinitely distant. If therefore we take into account 
 points which are infinitely distant or imaginary, or both, we 
 may say, that two Conic 8ections always intersect in four 
 points. In considering the infinitely distant points of inter- 
 
INTEESECTION OF TWO CONICS. 311 
 
 section of two conies, we must bear in mind (Arts, 42, 179^, 
 253) that, when algebraic results lead to the conclusion that 
 loci meet at infinity, the geometrical interpretation is, that 
 the loci tend to have the same direction, or, in other words, 
 to become parallel straight lines. 
 
 Ex. 1. If two hyperbolas have an asymptote of one parallel to an asymp- 
 tote of the other, they will meet in three finite points and one infimtely distant 
 point. 
 
 Ex. 2. If two hyperbolas have a common asymptote, or have the asymp' 
 totes of one parallel to the asymptotes of the other, they will meet in two 
 finite and in two infinitely distant points. 
 
 These propositions may be easily proved by the equation of Art. 240. 
 They may be illustrated with every variety of figure, by drawing on thin 
 paper two hyperbolas with their asymptotes, and placing them one over the 
 other, against the light. 
 
 328. Since three pairs of straight lines (Art. 104!, Ex. 1) 
 can be drawn through four points, two conies will have six 
 chords of intersection., The reader who is acquainted with 
 the Theory of Equations will observe, that the imaginary 
 points of intersection of two conies must occur in pairs 
 similar to those of Art. 127; and consequently the chord of 
 contact, joining a pair of such points, is real. Thus, if two 
 conies intersect in four imaginary points, and the pairs be 
 A, B and 0, D, the chords AB and CD are real. The rest 
 are imaginary; for, if AO, for example, were real, would 
 be the intersection of two real straight lines, and would 
 therefore be real. Again, if the conies intersect in two real 
 and two imaginary points, the chord joining the two real 
 J)oints, and that joining the two imaginary points, will be 
 real, and the rest imaginary. 
 
 Ex. Shew that for three values of 7c the equation S+iS'=0 will repre- 
 tent a pair of straight lines. 
 
 Let the equations, for which 5^=0, S'=0 stand, be 
 
 aa!'+bxy + (!y'+dx+ey+f=0, 
 
 . , a'x'' + i'!)!y + cY+d'x+efy+f'=0; 
 
312 ABRIDGED NOTATION, 
 
 then the equation S+ hS'=0 will be 
 
 where A-a+ha', B=h+W, &e. The condition that this equation should 
 represent two straight lines (Art. 62) becomes, after dividing by 4C, 
 i^H OS^ - BDE+ {B'-4:AC)F=0; 
 
 and, if A, B,... &c. are replaced by their values a+Jca', h + hb',... &c., we shall 
 obtain an equation of the third degree in h. The roots of this equation, 
 substituted in S+}eS'=Q, would give us the equations to three pairs of 
 straight lines. These are the three pairs of chords, real or imaginary, which 
 can be drawn through the four points of intersection of (5) and (5"). 
 
 829. Suppose that 8' = LM, where L and M are of one 
 dimension in x and 7/; then 8' represents two straight lines 
 and 
 
 S+kLM=0 (1), 
 
 will represent a conic, two of whose chords of intersection 
 with the conic 8 are 
 
 i = 0, M=0. 
 
 This can be instantly deduced from the previous article ; 
 for, since the conic {8') has become two straight lines, the 
 conic (1), which passes through the intersections of (8) and 
 these straight lines, must have the lines for two of its six 
 chords of intersection with {8). 
 
 It can be also shewn independently, as follows. Since 
 the co-ordinates of the points of intersection of (Z) and (8) 
 make these expressions vanish simultaneously, they also 
 make the expression 8+ kLM vanish, and therefore satisfy 
 equation (1). Hence the conic (1), passing through the inter- 
 sections of (i) and {8), has (L) for its chord of intersection 
 with {8). Similarly {M) may be shewn to be a chord of 
 intersection of (1) and {8). 
 
 330. The conies {8) and {8+hLM) wiU touch one 
 another, if two of their points of intersection coincide. This 
 
INTEBSECTION OF TWO CONICS. 313 
 
 will be the case, if either (i) or {M) touches (S), or if (i) 
 and {M) intersect in a point on (S). Hence, if T= is the 
 equation to the tangent to (S) at a given point {x'y'), then 
 
 8+kT{Ax+By+C) = (1) 
 
 is the equation to a conic touching {8) at the point (x'y). 
 
 If the straight line {Ax+JBi/+C=0) pass through the 
 point (x'l/'), which gives 0=- Ax — By , three of the points 
 of intersection will coincide ; and the equation to a conic, 
 having such a contact with {8), is 
 
 8 + hT{Ax+By-Ax'-By')=0 (2). 
 
 If we introduce the conditions (Arts. 106, 110) that equation 
 (2) should represent a circle, we shall have two equations, 
 by which A and B may be determined, and we may thus 
 obtain the equation (Art. 296) to the circle of curvature at 
 the point {x'y'). 
 
 If the line {Ax -^^ By + = Qi) coincides with (T), the 
 conies will have four points coincident, and the equation is 
 oftheform ^ + ^T= = 0. 
 
 The above equations may be written without the multi- 
 plier /<;; thus 8+hLM=0 may be written 8i-NM=0, 
 where N= kL, and consequently (Z) and {N) represent the 
 same straight line. 
 
 Ex. 1. To find the equation to the circle of curvature at any point {x'y') 
 of a central conic. 
 
 The equation will be of the form 
 
 i+t-'^+{'i+w-^)<'^''+^^-'^'''-^^=° w- 
 
 where we omit the multiplier i, since hA, TcB are not more general than A, 
 B; then from the conditions that the coefficients of a? and y^ should be 
 equal, and that the coefficient of xy should vauish, we obtain 
 
 X Aa^ _1 Sj/ ^/.^_n 
 
314 ABRIDGED NOTATION. 
 
 whence ^=L__2_. s= ^^; 
 
 Bubstituting these values in (1), we have 
 
 '"'^S'' ^ J^ -''>'' [<^ ^yjL iV^' ^^'_5!°x2^'^-n 
 
 The reduction of this equation is made easier, by expressing a;' and ^ in 
 terms of the eccentric angle of the point j[x'y'}- ^^^ result is 
 
 where a' is the semi-diameter through (it;V)i ii'^'l V the semi-conjugate. 
 
 Ex. 2. Shew that the equation to the circle of curvature at any point 
 {x'y') of a parabola is 
 
 id(d + x'){j/''-idz)+{yy'-id{x+x')}\6da^-2dx-j/y']^0. 
 
 331. If i = ilf, then ;Si' becomes of the form L^, and the 
 equation S' = 0, or X' = 0, represents two straight lines 
 which coincide. Then, by following out the same train of 
 reasoning as that used above, we see that, since the lines 
 (i) and (If) coincide, the conic 
 
 has one chord of contact instead of six chords of intersection 
 with {8), and that this chord of contact is the line (L = 0). 
 The conic represented by S+kL'=0 is said to have a 
 
 double contact with (S) along the line (L). The annexed 
 figures will sheW; the position of the conies of Arts. 329, 331. 
 
INTERSECTION OF TWO CONICS. S15 
 
 It is evident from an inspection of the figure, that two 
 out of the three pairs of chords common to {8+kLM) and 
 {S) have now come into coincidence with the chord of. contact 
 {L), and that the remaining pair have become tangents at 
 its extremities. If we had made use of these chords instead 
 of (Zr) and (M) in the equation, we should have proved that 
 the equation to a conic, having a double contact with (8) at 
 two given points, may be written in the {oim 8 + kTl" = 0, 
 where (T) and {T') represent the tangents at these points. 
 These results will be seen to agree with Art. 335. 
 
 332. The equation S + ]cL = may be considered as a 
 limiting form of the equation 8+kLM = 0; for it may be 
 written 
 
 8+{0.x + 0.y + Jc)L=0, 
 
 which indicates that one of the chords of contact has (Art. 
 99) become indefinitely distant. It will represent a series 
 of conies, having (L) for their common chord, similar (Art. 
 318) to (8), and similarly placed. Hence two similar and 
 similarly placed conies have two finite and two infinitely 
 distant points of intersection, a result which may easily be 
 verified by the equations of Arts. 318, 319. 
 
 If the curves are hyperbolas, their asymptotes are (Art, 
 186) parallel, and the infinite points of intersection are ex- 
 plained by the tendency of the curves to merge into two 
 pairs of parallel straight lines. If the curves are ellipses, 
 as the chord moves off, its intersections with {8) become, 
 necessarily imaginary,, and we can attach no geometrical 
 meaning to the infinitely distant points of intersection. If 
 the curves are parabolas, their axes are parallel, and the 
 curves tend to become four straight lines parallel to their' 
 axes, and infinitely distant from them. 
 
316 ABRIDGED NOTATIO:iSr. 
 
 333. The equation 8+Jc = may be written 
 
 8+k{0.x + 0.y+iy = 0, 
 and is therefore a particular case of the equation 8 + JcL' = 0. 
 It represents a conic, having a double contact with {8), the 
 chord of contact being infinitely distant. Now 8+Jc = 
 differs from yS'= in the constant term only ; hence the 
 conies are not only similar and similarly placed, but they are, 
 (Art. 141, Cor., 145, Cor.), if central conies, concentric, and 
 their axes and asymptotes are coincident. If the curves are 
 hyperbolas, they have their asymptotes as common tangents 
 at infinity, and therefore touch one another, and have a 
 common chord, at infinity. If they are ellipses, we can 
 attach no geometrical meaning to the infinitely distant, but 
 imaginary, points of contact. If they are parabolas, they 
 are equal, and their axes are (Art. 1.56, Cor.) coincident; and 
 we have shewn (Art. 290) that two such curves approach 
 indefinitely near to one another, or touch, at infinity. 
 
 334. If 8 also, as well as 8', can be split up into two 
 linear factors, then each of the two conies becomes two 
 straight lines, and our equation 8+ kS' = becomes of the 
 form 
 
 L]!r+7cME = 0, 
 
 where L, M, N, R are all of one dimension in x and y; and 
 this represents a conic section passing through the four 
 joints of intersection of 8 = and 8' =0, or, in other 
 words, circumscribing the quadrilateral formed by these four 
 straight lines. We shall, however, prove this proposition 
 independently ; it may be stated thus : 
 
 If L = 0, M= 0, N=0,R = Q,he the equations to the four 
 sides of a quadrilateral taken in order, then the equation 
 
 LN+kMB = (1) 
 
 represents a system of conies circwmscribing the quadrilateral. 
 
CONIC CIRCUMSCRIBING A QUADRILATERAL. 317 
 
 For the co-ordinates of the point {L, M) make L and M 
 simultaneously vanish, and therefore make the expression 
 LN+lcMR y&msh, and therefoi'e satisfy equation (1). This 
 corner of the quadrilateral lies therefore in (1). Similarly it 
 . may be shewn that the points (if, N), (N, B), {B, L), or the 
 other three corners, lie in (1), and therefore that (1) (which 
 being of two dimensions in x and y must be some conic) is 
 one of a system of conies circumscribing the quadrilateral. 
 We have noticed an equation of this form in Chap. Xll. 
 Ex. 39. 
 
 Since the expressions L, M, N, R are proportional to the - 
 perpendiculars from the point {xy) on the four sides of the 
 quadrilateral, we have the following geometrical interpreta-, 
 tion of equation (1). If any quadrilateral figure be inscribed 
 in a conic, the product of the perpendiculars, drawn from any 
 point on the curve to two opposite sides, is in a constant ratio 
 to the product of the perpendiculars on the other two sides. 
 
 335. If M=R in the equation LN'+7cMR = 0, it be- 
 comes 
 
 and we see that, owing to the two opposite sides of the quad- 
 rilateral approaching to and ultimately coinciding with each 
 other, instead of two chords of intersection, {M) and {R), of 
 the circumscribing conic, we have only one chord, which will 
 be one of contact; for the lines (L), (iV) will then each pass 
 through two consecutive points in the circumscribing conic, 
 and will therefore be tangents to it. "We have noticed in 
 Art. 308 the particular case, where the tangents are them- 
 selves the, axes of co-ordinates. 
 
 Ex. The general equation of the second degree may be written 
 Ai(? + £xy+Of + (Dx+JEy+F)(0,x + 0.y + l)=0, 
 
 and is therefore of the form LN+hME=0. The first three terms represent 
 two straight lines through the origin, parallel (Art. 186) to 'the asymptotes, 
 
S18 ABRIDGED NOTATIOIT. 
 
 each meeting the curve in one finite and one infinite point. The line 
 {I>x+Ei/+P=0) is the chord joining the two finite points, and the line at 
 infinity joins the two infinite points, 
 
 If F=0, or the origin is on the curve, the two finite points coincide, and 
 {Dx+Ey=0) is the tangent at the origin, as in Art. 297, Cor. 
 
 Ex. 1. The ordinary equation to central conies may be written 
 
 (ai-o)(a!+tt)+^/=0 (1), 
 
 where a:-a=0 and a! + a=0 are two tangents, and y=0 is their chord of 
 contact. 
 
 Ex. 2. If the vertex is origin, the equation of Ex. 1 is 
 
 («-2a)x+l°2,«=0, 
 
 or, as in Art. 249, 
 
 Now, when a becomes infinite, or the curve becomes a parabola, this 
 equation becomes 
 
 2(i(0.ie-2)a; + y==0, 
 
 where one tangent (0 . a; - 2 = 0) has (Art. 99) become infinitely distant. 
 
 Ex. 3. The equation to an hyperbola referred to its asymptotes may be 
 written 
 
 xy=(0.x+0.y + };)', 
 
 the lines x=0, y=0 being tangents, whose chord of contact is infinitely 
 distant. 
 
 Similarly the ordinary equation to the hyperbola may be written 
 
 with a like interpretation, 
 
 Ex. 4. The general equation to the parabola may be written 
 
 {a(ii;+cyy + (l)x+lh/+F){0.x+0.y+Tj=0, 
 
 where (Art. 27S) the line (aa! + cy=0) is a diameter, and {Dx+Ey+F^^O) the 
 tangent at its vertex, the tangent at the other end of the diameter being 
 infinitely distant. 
 
TWO TANGENTS AND CHORD. 319 
 
 336. If we write the equation of Art. 335 in the form 
 LM= R^, whicli is equally general, any point on the locus 
 may be defined by the equations 
 
 L^R_M 
 1 /* ,." 
 
 the equations . B == jjlL, M= fiB, 
 
 where /j, is arbitrary, representing a pair of straight lines, 
 drawn through {L, E) and (M, B) and intersecting at a point 
 on the curve, which, we may call ' the point fj,.' 
 
 337. To find the equation to the chord Joining two points 
 on the conic LM=B^, and to deduce the equation to the 
 tangent 
 
 Let /*„ fj.^ (Art. 336) be the two points ; then at these 
 points we have 
 
 ^=^^^ (1) L_R_M 
 
 Suppose the equation of the chord to be 
 
 lL + rB + mM=0 (3), 
 
 then, since /tj, fi^ both he on this line, we have, from (1) and 
 (2), the equations 
 
 l+rn^ + mii^ = Q (4), Z + r/i, + m/^/ = (5); ~ 
 
 hence, obtaining the values of y , -^ from (4) and (5), and 
 
 substituting them in (3), we have for the equation to the 
 chord, 
 
 HjiJj-{ji^ + ^^B + M=Q (6). 
 
 If we make (^1 = /^^ in equation (6), we obtain, for the 
 equation to the tangent at /i^, 
 
 li^L~2ii,B + M=0, (7), 
 
320 ABRIDGED NOTATION. 
 
 338. Conversely, if a linear equation contains an arbi- 
 trary constant of the second degree, the straight line which 
 it represents will touch a fixed conic. For its most general 
 form is 
 
 fj.^L + fjiM+Ii = (1), 
 
 where fi is arbitrary, and L, M, E are linear. 
 
 Now (1) may be written 
 
 M\ 
 
 f.'L-2p,(-j')+B = 0, 
 
 which represents a tangent to the conic 
 
 4 
 
 339. It will be observed, that we have considered in the 
 present chapter equations representing conies passing through 
 some four fixed points. Each equation has involved an un- 
 determined constant h, which may receive different values, 
 distinguishing the different conies which can be drawn 
 through the same four points. By giving a suitable value to 
 h, we may make the equation represent any conic whatsoever 
 passing through the four points. For let S = represent 
 any such conic, and let {x'y') be any other point in it ; then, 
 by substituting x, y in the equation, we shall obtain the 
 appropriate value of h, when the locus passes through {x'y'). 
 If this value of h is substituted in the equation, the conic 
 represented by it will pass through the same five points as 
 2 = 0, and therefore (Art. 312) be identical with it. 
 
 Ex. 1. Find the equation to the conic section that passes through the 
 intersection of the circles 
 
 3fi+y*-ix-6y=23 (I), 
 
 st5+j,s= 4 (2), 
 
 and through the centre of (1). 
 
EXAMPLES. 321 
 
 Equation (1) may be written 
 
 (a;-2)=+(j,-4)s=48 
 and its centre is therefore the point (2, 4). 
 
 The equation to the required conic -will be of the form 
 
 and since it passes through the point (2, 4), we have, 
 
 -48+S.16=0, 
 therefore k=3, 
 
 and the equation becomes 
 
 (a;-2)= + (j/-4)'-48+3(a;2+3,a-4)=0,. 
 or 4x^+4:y^-4x-8y=^, 
 
 or x' + y'-x-2y=10; 
 
 a circle, the co-ordinates of whose centre are (J, 1), and whose radius 
 is 4 s/5. 
 
 Ex. 2. To find the equation to a pair of tangents drawn from the point 
 (o^) to the conic 
 
 The tangents are a conic (i.e. locus of the second degree), having a double 
 contact with the given conic, along the hne 
 
 a" +6= ^~"' 
 which is the polar of {a^ ; hence their equation is 
 
 and, since the locus passes through the point (ajS), 
 
 1 
 
 i = - 
 
 
 Hence. (^> + f5-l) (p +F- ^ =(^ + f "V 
 
 is the only conic answering the conditions, and must therefore be the 
 
 21 
 
322 
 
 ABEIDGEB NOTATION. 
 
 340. By means of Art. 335 we may interpret the equa- 
 
 tion 
 
 rU-rri'M^-n^N\=0'. 
 
 for it may be written in either of the 
 forms 
 
 {IL + mM) {IL -. mM) - n'N' = 0, 
 
 [IL + nN) {IL - nN) .- rn'M" = 0, 
 
 and therefore represents a conic, so 
 
 situated, that AB {N) is the chord of 
 
 contact of (IL+mM), {IL-mM), the ' 
 
 pair of tangents thi-ough G{L, M), and 
 
 CA (M) is the chord of contact of (IL+nN), {IL-nN), 
 
 the pair of tangents through B (L, N). In other words 
 
 (L, M) is the pole of AB (N), and B (L, N) is the pole. 
 
 of GA (M); and consequently (Art. 130) A (if, N) is the 
 
 pole of BC (L). It will be seen from the equation that, 
 
 although A {M, N) is the pole of jBC (Z), the tangents 
 
 from A are imaginary. 
 
 The points of intersection of the lines (Z), [M), {N) form 
 (Art. 299) a conjugate triad, and the triangle formed by (i), 
 [M), {N) is (Art. 130) self-conjugate. 
 
 341. Suppose the equation to the conic of the preceding 
 article to be (Art. 88) 
 
 a' + ^» = eV, ' (1), 
 
 and that the lines (a), {fi) are at right angles. Then this 
 equation asserts, that the. square of the distance of a point on 
 the locus from the point (a, ;S) has constant ratio to the 
 square of its distance from the line (7); hence the locus is a 
 conic with (a, ^) for focus, (7) for directrix, and e for eccen- 
 tricity. Now (i) may be written 
 
 (e7-a)(e74-a) = /8»; , , 
 
pascal's theorem. 323 
 
 hence (Art. 335) the lines (67 — a), (ey + a) are tangents, and 
 the focal chord (j8) is their chord of contact. But (ey — a), 
 (ey + a) meet in (7) ; hence tangents at the extremities of any 
 focal chor^ intersect in the directrix. Also they meet in (a), 
 which is perpendicular to (/3) ; hence every focal chord is 
 perpendicular to the straight line joining the pole with the 
 focus. The lines (a), (/3), (7) occupy the positions of HK, 
 PQ, KX in the fig. of Art. 205. 
 
 342. Pascal's Theorem, The three pairs of opposite 
 sides of a hexagon inscribed in a conic intersect in points 
 which all lie in one straight line. 
 
 Let L = 0, M=0, iV=0, -B = 0, S=0, T = 0, be the 
 equations to the six sides of the hexagon, and let Q = be 
 the equation to the diagonal joining (L, T) and [N, R) ; then, 
 since the conic is circumscribed about the quadrilateral whose 
 sides are {L), {M), {N), (Q), its equation (Art. 334) can be 
 written in the form 
 
 LW+kMQ^O (1); 
 
 also, since it is circumscribed about the quadrilateral whose 
 sides are (E), (S), {T), (Q), its equation can be written in 
 the form 
 
 BT+k'SQ=0 (2); 
 
 then, since (1) and (2) represent the same locus, one of them, 
 must be derived from the other by the introduction of some 
 constant factor X ; that is, 
 
 L]Sr+kMQ = X{BT+h'8Q) 
 
 identically; hence 
 
 LN-\BT-(Xk'8-hM)Q 
 
 identically; but LN—\RT=Q represents a locus passing 
 through the points (L, B), {L, T), (N. B), {If, T), hence 
 (\k'8—kM) Q=0, which evidently represents two straight 
 
 21—2 
 
324 TEILINEAE CO-ORDINATES. 
 
 lines, represents the same locus; but Q = passes through 
 the points {L, T), {N, B), hence \k'S-kM=0 must pass 
 through the other two .points {L, E), (N, T) ; but it evi- 
 dently passes through {M, 8), hence (Z, B), [M, 8), (N, T) 
 are in one straight line. 
 
 343. Brianchon's Theorem. If tangents he drawn at the 
 six angular points A, B, C, D, E, F, of a hexagon inscribed 
 in a conic, so that the tangents at A, B meet in c, those at B, C 
 in d, &c:'; then the three straight lines of, da, eb m£et in 
 a point. 
 
 This foUows at once from Pascal's Theorem, or vice versd. 
 For c is ijhe pole of AB and / is the pole of JDB ; hence 
 (Art. 130) the intersection of AB, DB is the pole of cf 
 
 Similarly BG,EF. da, 
 
 and CD, FA eb. 
 
 But these three intersections lie on one straight line ; there- 
 fore the three polars meet in a point. Conversely, if the 
 three polars meet in a point, the three poles will lie on one 
 straight lifife. 
 
 344. Trilinear Co-ordinates. 
 
 Since (Art. 96) every equation with these co-ordinates 
 can be made hbmofgtfneous, the general equation of the 
 .second degree is 
 
 ^a" + B^' + C7' + 2A'^y + 2B'rya + 2 C'a0 = 0, 
 
 where the constants are written in this form, in order to 
 obtain greater symmetry in results hereafter obtained. We 
 shall call the equation ^ {a^y) = 0. 
 
 Precisely as in Art. 97, it may be seen that the Cartesian 
 equation to any conic can be transfoilned into an equation' 
 of the above form ; that is to say, the general equation 
 
CIRCUMSCEIBED CONIC. 325 
 
 of the second degree in Trilinear Oo-ordinates can be made 
 to represent any conic, by giving suitable values to the 
 constants. 
 
 COE. If i: = 0, if=0, iV=0 be the equations to the 
 sides of the triangle of reference, it is evident, as in Arts. 92, 
 98 Cor., that the above statements are true, if we write the 
 abbreviations L, M, N, instead of the trilinear co-ordinates 
 a, /3, 7, The same remark will apply to Arts. 345, 346, 348, 
 349. 
 
 345. To find the equation to a conio section which cir- 
 cumscribes the triangle of reference. 
 
 Let the equation to any such conic be 
 
 Aa'' + Bfi" + Crf + 2A'^y + 2B'ya. + 2 C'aB= 0; 
 
 then, since the conic passes through the point C(0, 0, — ) , 
 
 the equation must be satisfied by these values; but this 
 gives ■ 
 
 hence we must have 0=0. Similarly it may be shewn 
 that ^ = 0, and B=0, and the equation becomes 
 
 A'^y + B'yaL+C'a^=0. 
 
 This equation may be written in the form 
 
 a fi y 
 and evidently involves two independent arbitrary constants 
 only, the conic having already fulfilled three conditions. 
 
 346. By writing the equation l^y + mya. + na^ =0 in 
 the form 
 
 Z;87 + a(M7 + 7iy3) = (1), 
 
326 TRILINEAE CO-ORDINATES. 
 
 we see that it is reduced to ZyS^ = 0, if we put mrj + n/3 = 0. 
 Hence the line wiy + w/8 = meets the conic in the points in 
 which it meets the lines /8 = 0, 7 = 0; but these two points 
 coincide, since the line mr/ + reyS = passes through the in- 
 tersection of jS = 0, 7 = ; hence the straight line and the 
 conic meet one another in two coincident points, that is, 
 they touch one another at the point A. We may obtain this 
 result by putting |S' = 0, 7' = in equation (2) of Art. 354, 
 Ex. 1. Hence the tangents at A, B, are 
 
 ^+1=0, y-+^=o, u^=o. 
 
 m n n i cm 
 
 347. To find the condition that the conic circumscribing 
 the triangle of reference should he a circle. 
 
 , Let DE be the tangent at the point A ; then the angles 
 DAB, EA G are equal to the angles C and B respectively. 
 Also, since DE lies in the H-t- and — + compartments formed 
 by the lines (^), (7), we have for its equation (Art. 57), 
 
 p_ sinEAO _^,_7 
 
 7 sin DAB °^ sinB'^smC~ ' 
 
 but (Art. 346) its equation is — + 5' = o- hence 
 
 m n ' 
 
 -=«i^. Similarly l = !l54 
 w sin c; -^ m sm B ' 
 
INSCEIBED CONIC. 327 
 
 and the equation to the circle required is 
 
 /37 sin A + ya.sinB+ a^ sin (7=0, 
 or a/37 + 57a + ca/3=0. 
 
 Ex. If one focus of a conic which touches the three sides of the triangle 
 of reference lies on a fixed straight line, shew that the other foeixt will lie on a 
 conic circumscribing the triangle ; and hence deduce Art. 306, Ex. 
 
 If (0^7), (o'/S'-yO be the foci, we have (Art. 204), 
 
 oa'=;8/3'=77'=52. 
 
 If therefore the co-ordinates of one focus of the conic satisfy any homo- 
 geneous equation /(o^7)=0, the co-ordinates of the other focus will satisfy 
 the equation 
 
 Thus, if one focus of a conic lies on the Une 
 
 la + mp+ny=0, (1), 
 
 the other focus will lie on the looua 
 
 I m n „ 
 
 «+^+7 = °' (')' 
 
 that is, on a conic circumscribing the triangle of reference. 
 
 If the ratio ot l.:m : n approaches indefinitely near to the ratio a : i : c, 
 the line (1), and consequently the focus, become indefinitely distant, and the 
 conic is a parabola. In this case (2) becomes the circle circumscribing the 
 triangle. 
 
 348. To find the equation to a conic, which touches the 
 three sides of the triangle of reference. 
 
 Let the side BG{a) be a tangent to the conic represented 
 by the equation ^ (ay97) = 0; then, making a = 0, we have 
 
 5^^+C7' + 2^'/37 = 0, 
 
 which is the equation to a locus passing through the inter- 
 section of (a) and the. conic ; but, as in Art. 63, we may see 
 that it represents two straight lines 
 
 ;8 — my = 0, ^ — to'7 = 0, 
 
328 TEILINEAE CO-ORDINATES. 
 
 TfrhicL. pass througli A (/S, 7), and therefore join A viith the 
 intersection of BG{ix) and the conic. If these two straight 
 lines coincide, that is, if A'^ = BC, (a) will pass through two 
 coincident points, and be a tangent. Hence ii BC, GA, AB 
 are aU tangents, we have 
 
 A" = BC, B'^=CA, C" = AB. 
 
 These conditions shew that A, B, C have the same sign, 
 since the product of anjr two of them is positive. We shall 
 suppose that the general equation has been so written, as to 
 make A, B, (7 positive, and shall assume, therefore, that . 
 
 A = P, B = m\ G = if; 
 
 therefore A! =± mn, B" =±nl, C" = + Im, 
 
 and the equation ^ (0/87) = becomes 
 
 Pa' + m^^" + mV + 2mn^r^ + 2wJya + ilma^ = 0. 
 
 The signs of A', B', C may be taken in eight different 
 ways, thus 
 
 + + +,--+,- + -,+ ^ . 
 
 , + + -, +-+, -++. 
 
 The upper line, where all are positive, or only one, will 
 give the equations 
 
 (Zx + TO/3 + nrff = 0, {h + mp - 'nr/y.= (fee, 
 
 and will represent coincident straight lines, which equally 
 weU fulfil the conditions of the problem, since they meet (a), 
 (^)' (7) each in two coincident points. 
 
 The lower line, where all are negative or only one, will 
 give four forms of the equation, which represent curves; they 
 are equivalent to 
 
 V?a + V^ + Vi^=0...(l), VZa + V^ + vT^=0.,.(2), 
 Via + V-m;S + Vwy = 0...(3), V^^ + V^ + V^ = 0...(4), 
 
INSCRIBED CONIC. 329 
 
 as may be seen by bringing tbese equations to a rational 
 form. We shall use the first of these forms, -which cor- 
 responds to 
 
 ZV + m''/3'.+ wy-2m«/37-2wZ7a-2ZOTa^ = (5), 
 
 as the genersi- equation to the conic which touches the three 
 lines (a), (/3), (7), with the understanding that the signs' of I, 
 m, n are undetermined. 
 
 349. Equation (5) of the preceding article may be writ- 
 ten in the form 
 
 ny (ny - 2h -2m0) + {la. - m^f = 0, 
 
 which is (Art. 335) the equation to a conic referred to two 
 tangents (7) and (ny—2la—2m^), which have (h — m^) for 
 their chord of contact; but Qx — m^) passes through (a, i8), 
 and is therefore, the straight line joining with the point 
 where the conic touches (7) or AB. Hence the three straight 
 lines which join the angles with the points of contact on the 
 opposite sides, are 
 
 Za— m/3 = 0, nl^ — ny=0, ny — h = 0, 
 
 and these lines (Art. 93) meet in a point. Again* the three 
 tangents to the conic, at the points where these lines meet 
 the conic again, are 
 
 ny - 21a. - 2mfi = 0, la- 2m^ - 2ny = 0, wijS - 2*17 - 2h = 0. 
 
 For the points where these straight lines intersect the 
 opposite sides (7), («). (^)> respectively, we have 
 
 7 = 0, Za + »n/3 = 0; a = 0, m0 + ny = O; '^ = 6, tv^ + h = 0; 
 
 all which points lie on the line 
 
 la, + wij8 + "7 = 0. 
 
330 
 
 TRILINEAR CO-OEDINATES. 
 
 350. To find the equation to the circle inscribed in tlie 
 triangle of reference. 
 
 The equation must be of the form 
 
 '/Ia + '/m^+'^my = (1). 
 
 Then at the point A', where BG meets the circle, we 
 have a = 0, and therefore from (1) 
 
 V ?n/8 + Vm7 = 0, or m^ = ny (2). 
 
 Again, from the figure, if r be the radius of the circle, we 
 have at the point A' 
 
 C 
 
 a A' n • rt J-cot-rrsin (7 cos" — 
 p _A Usm V 2 2 ; 
 
 y~A'BsmB~ ,.B~~I""TB 
 r cot -g- sm B cos -^ 
 
 B C 
 
 hence, from (2), m : n = cos" -^ : cos" - ; 
 
 similarly. 
 
 = cos"— : cos - ; 
 
 therefore I : m : n = cos" -^ : cos" -^ : cos" -^ , 
 
 and the required equation is 
 
 A 
 
 2 "" ' ""2 
 
 cos -^ Va + cos -^ ^^^+ cos ^ V7 = 0. 
 
INSCRIBED CIRCLK. 
 
 331 
 
 351. To find the equation to the circle which touches on£ 
 side of the triangle of reference and the two others produced. 
 
 Let the circle touch BO, and AB, ^(7 produced; then, as 
 in the tast article, we have at the point A', m/3 = ny ; and 
 from the figure 
 
 ■JT-O 
 
 13 A'CamO """"^"^ 
 
 sin C 
 
 7 ^'J5sin^ ,iT-B . ' 
 
 r cot — ^ — sin B 
 
 hence 
 
 . .C . ^B 
 
 n : 9ra = sin -X : sin -^r. 
 
 Also, at the point B' we have la = ny; and" from the figure, 
 since a is negative at that point (Art. 94), 
 
 „,„ . ^ root — 5 — smO' 
 — a B Osm C 2 
 
 7 ~SA sin A 
 
 rcot-^sinJ[ 
 
 hence 
 
 , . ,0 ^A 
 
 n : 1 = sm" ^ : — cos" ^ ; 
 
 therefore I : m : ws-cos'-^- : sin'^ : sm ^, 
 
332 TRILINEAR OO-OEDINATES. 
 
 S-nd the tequired equation is 
 
 A,— . . B,^. _.__0 j- 
 
 cos- 
 
 ■ V— a + sin -jr^ V/3 -f-sin o- ^^7 = 0. 
 
 Similarly the equations to the other escribed circles may 
 be written down. It may be observed, that, as a is negative 
 for every point of the circle, V— a is ah impossible quantity 
 in appearance only. 
 
 352. To find the length of a straight line drawn from 
 the point (a'/SV) to meet the conic 
 
 Acl' + B/3'+ Gy' + 2A'^y + 2B'jx+2C'a^=0 (1). 
 
 Let the equation to the straight line be (Art, 101) 
 
 ^' = ^^=a: = z (2); 
 
 then, for the distance of (a'/SV) from the curve, we must 
 write in (1) 
 
 a=pl+a', /8 = 2? + j8', y=rl + y. 
 Making this substitution, and arranging, we have 
 
 Z + <^(a'^V) = 0...(3). 
 
 ipqr) Z?+ 2{Ap+C'q + B'r) a' 
 + 2{O'p + Bq + A'r)0 
 + 2lB'p + A'q + Gr)y' 
 
 To this equation the remarks of Art. 176 may be applied 
 without alteration. 
 
 Ex. To find tlie condition that the equation 4>{a^)=0 should represent 
 a circle. 
 
 If (a'^'Y) be the centre in Art. 352, the coefficient ot I ia equation (3) 
 must vanish, and we have, to determine the semi-diameters, 
 
 *(p2r)i^ + *(a'/3'y)=0. ■ 
 
THE TANGENT. ' 333 
 
 Now suppose the line (2), Ait. 352, to be drawn parallel to the side £C; 
 then for p, q, r we may write 0, sin C, - sin B, and thus obtain, for the semi- 
 diameters parallel to BG, 
 
 (0, sin C7, - sin 5) 3 + ^ (a'/S'V) = 0, 
 
 or, B sm^ + so? B -iA' sm B smC =- ^^^^i^ . 
 
 Similar expressions may be obtained for the semi-diameters parallel to 
 CA, AB ; but, if the curve is a circle, these are all equal ; henoe the required 
 condition is, after substituting for sin j4, sin B, sin 0, the proportionals a, b, c, 
 
 Bc'- + OV'-'iA%c = Oa^ + Ac'^-'iB'ca^A}fi+Ba^-'iO'al. 
 
 353. To find the eqiuition to the tangent to the conic 
 <jj (0/67) =0 at the point (a'/8'Y). 
 
 Let equation (2) of the last article I'epresent the tangent ; 
 then since {o.'^'y) is on the curve, ^ {a'^y) = 0, and from 
 equation (3) we obtain for the condition of tangency, as in 
 Art. 117, &c. 
 
 {Ap + C'q + Br) d -f (C> + -B2 4- A!t) jS' 
 
 + (^p-l-^'2-f Cr)7'=0, 
 or (^a + C'j8' + 5'7')jp+ (C^V-f-B/r-f^'ryO^ 
 
 then, since ^, g-, r are, from (2), proportional to a -a', ^-/3', 
 N — r^' for every point in the tangent, we may substitute these 
 values for p, q, r in the latter form ; hence we obtain 
 
 {Ad +C'^ + B'y) a + {G'd + B^ + A'y') ^ 
 
 + {B'a' + A'^'+Cy')y 
 = Ad' + B^" + Cy'' + 2^ W + SB'vV + 2 O'd^ = 0. 
 
 Hence the tangenl; at (a'jSV) is represented by the 
 equation 
 
 {Ad + C'0 + By') <x+{C'd + B0 + A'y') /8 
 
 + {i;d + A'^'+Cy')y = 0. 
 
334 TRILINEAR CO-ORDINATES. 
 
 354. The reader who is acquainted with the Differential 
 Calculus will observe that the above equation to the tan- 
 gent is 
 
 Those not acquainted with that subject may regard -^ 
 
 as a symbol used to denote the result of the following opera- 
 tion : Take those terms only in the equation to the conic, 
 which contain a, multiply each term by the index of a in it, 
 
 and diminish that index by unity. For j^, perform the same 
 
 operation, and then write a'^'y for a, ^, y. The symbols 
 
 ^ , -^, &c. must be understood in a similar manner. The 
 dp dp 
 
 result, when divided by two, will be the equation to the 
 tangent obtained above. Arts. 353, 354 are true for homo- 
 geneous equations only. 
 
 Ex. 1. To find the equation to the tangent at the jpoint (a'^'y') of the 
 conic. 
 
 lpy+n!r^a + nap=0 (1), 
 
 and the eciuatiou to the tangent is 
 
 {np'+my)a + {ly'+na')^ + (:ma'+lp')y=0 (2).' 
 
 This may be written in another form; for, since (a'/3y) is on the 
 curve, 
 
 „^'+^=_?^. lY+na'^.^, ^„'+;^'=_!^'. 
 
 therefore, substituting in (2) and dividing by o'jSy, we have 
 
 la mp ''7 n 
 
 ^1 + p't+yi (3)' 
 
THE TANGENT. 335 
 
 Con. In ordes: that any straight line 
 
 \a,+ii^+vy=0 (4) 
 
 should touch the conic, we have, by comparing (3) with (4), 
 
 I m n ' ' ' 
 
 hence, since a', j3', y' satisfy (1), we hare for the condition of tangenqy 
 ijlx+ ijmfi.+ s/nv=0. 
 
 Ex. 2. To find the equation to the tangent at the point (a'/S'vO «/ *'»« 
 conic. 
 
 Pa''+m''p' + n^'Y'-2m.nPy-2nlya~2lmap=:0 (1), 
 
 or tjla+isjmp+ ijn/y=0 (2). 
 
 Here we may obtain 
 
 dtp dip dip 
 da" d^' cP/ 
 
 . from form (1) ; then, since (a'p'y') is on (2), we have, by transposing and 
 Sj^naringl 
 
 la'-mp'-7ir/'=2 Jmn^'y', -la'+mp'-nr^'=2 sjnlr^a.', -la'-m^'+nY^^ Jlma'p'; 
 
 and the equation to the tangent becomes, after substitution and reduction. 
 
 ^/hW?W^'y-' (^)- 
 
 We have used the rational form of the equation, in order to imitate 
 exactly the method of Art. 353. By following the rule derived from thai 
 article, we may deduce (B) at once from the form (2). 
 
 CoR. In order that any straight line 
 
 \a+mp+vy=0 (4) 
 
 should be a tangent, we have, by comparing (3) with (4), the condition 
 
 ^\/l=''\/m='' \/i- 
 
 Hence, from (2), the required condition is 
 
 I m n ^ 
 r+- + -=.0. 
 
336 TRILINEAB CO-ORDINATES. 
 
 355. Since the equation to the tangent is not altered 
 when we write a', a for a, a', or 0, /8 for yS, ^, or 7', 7 for 7, 7', 
 we see, as in Art. 131, that the equation to the polar of the 
 point (a'/SV) is identical in form with the equation to the 
 tangent. 
 
 356. The Nine Points Circle. In any triangle ABC, 
 the feet of the perpendiculars from the angles {which meet in 
 P), the bisections of the straight lines AP, BP, CP, and the 
 middle points of the sides, are nine points which lie on the 
 same circle; and this circle touches the inscribed^ and the 
 three escribed circles of the triangle. 
 
 As the above remarkable theorem affords a good illus- 
 tration of the use of Trilinear Co-ordinates, we shall devote 
 to it the rest of this chapter. 
 
 357. The equation to any two circles, with Cartesian co« 
 ordinates, can be written (Art. 110) so as to have the terms 
 of the second degree the same ; that is to say, if ;S'= repre- 
 sents a circle, any other circle can be represented by 
 
 8+Ax + Bi/+ C=0. 
 
 Hence it follows, as in Art. 344, that, if 8=0 is the equa- 
 tion to a circle, with Trilinear Co-ordinates, 
 
 8+h + m0+ny = O 
 
 will represent any other circle ; or we may use (Art. 96) the 
 homogeneous form 
 
 yS'+(k+m/3 + W7) (aa + 6/8 + C7) = 0. 
 
 If the two circles are concentric, their equations will differ 
 only by a constant ; that is, any circle concentric with 8=0 
 will be represented by 8+c=-0, or, in a homogeneous form, 
 
 8+7c{a(i+h0+cyy=O. 
 
NINE POINTS CIRCLE. 337 
 
 COE. 1. If (8) is the circle circumscribing the triangle 
 of reference, the equation to any other circle may be written 
 in the form 
 
 a/37 + Sya + ca/3 + {la + m^ + ny) {wx + h^ + cy) = ; 
 
 and if the general equation 
 
 ^a" + B^ + G'f + lA^y + iBya. + 2C"a/3 = 0. . . (2) 
 
 be the equation to a circle, it must be identical with 
 
 Ic (a/37 + 67a+ ca/3)+ (4^+x+t') (««+^/3+C7) = 0. .. (3). 
 
 By equating the coef3Bcients of (2) and (3), we may 
 obtain the conditions of Art, 352, Ex., that (2) should 
 represent a circle. 
 
 CoE. 2. The radical axis of two circles, whose equations 
 are written in the above form, will be found by subtracting 
 one equation from the other, which leaves a linear equation, 
 as in Art. 133. 
 
 358. To find the equation to the circle, which passes 
 throv^h the feet of the perpendiculars from the angles of the 
 triangle ABC upon the opposite sides. 
 
 Let the equation be (Arts. 357, Cor. 1) 
 
 a/37 + 57a + ca;8 = (Za + in/3 + ny) (aa + 5/3 + cy), 
 
 and let the feet of the perpendiculars (fig. Art. 360) be A', B, 
 C'; then since it passes through A', (0, AA cos C, ^ J.'cos B), 
 
 the proportionals to these, 0, ^^^.3^^' satisfy (Art. 95) 
 
 the homogeneous equation to the circle ; hence 
 
 a ^ ( ^ . n \ / b _c\ .. 
 <;os B cos G \cos B cos Cj \cos B cos Cj '■ 
 
 22 
 
338 triLiSear co-ordinates. 
 
 or, since h cos + c cos B = a, 
 
 " \-^,-l (1). 
 
 cos B cos 
 Similarly 
 
 7^ + ^=1 (2), -.+ 5 = 1 (3). 
 
 cos 6 cos A cos ^ cos ij 
 
 Adding (2} and (3) and subtracting (1), we obtain 
 
 2? 
 
 = 1, and therefore, from symmetry, 
 
 cos -4 
 
 I 
 
 cos A cos B cos C 2 ' 
 hence the required equation is 
 
 a/Sy + 67a + ca/3= A (acDS^+/3cos5 + 7Cos G) (aoi+bfi+cy). 
 
 359. To find the equation to the circle, which passes 
 through the middle points of the sides of the triangle ABC. 
 
 Let the equation be 
 
 a^y + lyi + C2j8 = (Za + mj8 + ny) (aa + 6/3 + 07) ; 
 then, since this passes through the middle point of BC, 
 
 [O, I sin (7,|sin5j, 
 
 we may substitute in the equation the proportionals 0, t , 
 
 - ; this gives 
 
 ; ,..■ abo _ \b cj' 
 
 and similar equations may be obtained from the other middle 
 points; hence 
 
 m n _ c? **■ ^_-^* ^ m_ c' 
 & c~2a&c' c'^a~2a&c' ^'^h^2abc'' 
 
NINE POINTS CIRCLE. 
 
 339 
 
 aciding the two last equations and subtracting the first, we 
 have 
 
 a 2abo ~ a ' 
 
 therefore, by symmetry, 
 
 Z_ _ m _ n _ 1 
 
 cos A~ cosB~ COS G~ 2' 
 
 and, substituting these values in the equation to the circle, 
 we have 
 
 a^y+hyx+C3j3= ^ (a cos A+^cosB + y cos (7) (aa + J/3 + 07). 
 
 This equation is the same as that obtained in Art. 358 ; 
 hence the same circle passes through the six points. 
 
 360. Suppose the three perpendiculars AA', BB', CO', 
 to meet in P; then A', B', C 
 will be the feet of the per- 
 pendiculars dropped from B, 
 A, P, on the sides of the tri- 
 angle BAP; hence the circle 
 which passes through A', B', 
 G' passes through the middle 
 points of -dPand BP, and for 
 a similar reason must pass 
 through the middle point of 
 GP. Hence the bisections of 
 
 the sides, the feet of the perpendiculars, and the bisections of 
 AP, BP, GP all lie on the same circle. 
 
 We shall now proceed to find the equations to the 
 inscribed and escribed circles in the form of Art. 357, iu order 
 to prove that they are touched by the Nine Points Circle. 
 
 22—2 
 
^340 TEILINEAR CO-ORDINATES. 
 
 361. The equation to the circle inscribed in the triangle 
 of reference may (Art. 350) be written 
 
 a'cos*:^+^cos*|+7'cos*^-2,87COs''|cos»-^-&c. = 0. 
 
 Now this equation must (Art. 357, Cor. 1) be identical 
 with 
 
 h (a^y + byx + cajS) 
 
 Equating the coeflScients of ^y in these two equations, we 
 obtain 
 
 — ahcJc = c'cos* g + 6" cos* -^ + 25c cos"-^ cos' ^ 
 
 — ( c cos -g + cos* g- 1 
 
 = s', 
 
 where 2s = a + b + c. Hence the equation to the circle 
 
 becomes 
 
 a/87 + byx + c j^/8 
 
 abcfa iA B 4jB , 7 ^G\ , , ,„ , , 
 
 " ^ ^"^ ^* ~ "^''^ ^^ ^* ~ ^^''^ '^ (* ~ '')"J («« + ^'^ + <'7)- 
 
 362. Similarly it may be shewn that the equation to 
 the escribed circle opposite to A may be written 
 0^87 + byj. + ctJ^ 
 
 ab fa. tA ^ . .B y . ,G\ , 
 = (Ji:^t*=°^ 2 ^r''^ 2 +^^^ 2j(«^ + ^/5 + <'7), 
 
 " Wa ^*"*' + ^^ (« - ")* + C7 (s - bfl (aa + J/3 + 07), 
 
NINE POINTS CIECLK . 341 
 
 from which the equations to the other escribed circles may 
 be written down by symmetry. 
 
 363. By subtracting the equation to the inscribed circle 
 from the equation to the Nine Points Circle, we obtain for, 
 their radical axis 
 
 or ^(c_.a)(a-6)+£(a-6)(6-c)+^(6-c)(c^a)=0, 
 
 ax h^ cy 
 
 or J H + — S:=" 
 
 b — c c — a a — 
 
 and (Art. 354, Ex. 2) this is a tangent to the inscribed 
 circle, if 
 
 ABC 
 
 cos'-jr- (J - c) cos'-g- (c - a) cos'-g (a - h) 
 
 +— ^T + —^7 = 0. 
 
 a be 
 
 an identity which may easily, be proved. Similarly the' 
 radical axis of the Nine Points Circle and any of the escribed' 
 circles may be shewn to be a tangent to the two- circles. 
 Hence the Nine Points Circle touches the inscribed and 
 escribed circles. 
 
342 ABRIDGED NOTATION AND TKILINEAE CO-ORDINATES. 
 
 •EXAMPLES XIII. 
 
 1. If two conies have each a double contact with a third 
 conic, the chords of intersection of the first two, as well as the 
 chords of contact of the first two with the third, all pass through 
 one point. 
 
 2. Find the equation tct an ellipse referred to any pair of 
 conjugate diameters, by considering it as circumscribed about the 
 quadrilateral formed by joining the extremities of these diameters. 
 
 3. Shew from Art. 346 that iPy + mya + nafi = is the equa- 
 tion to a conic circumscribed about a quadrilateral, when two of 
 its angular 'points coincide. 
 
 4. Interpret the equation (Art. 329) S+kLM=0, (i) when 
 one of the asymptotes of (S) is parallel to (L), and (ii) when one 
 is parallel to (L) and the other to (M). 
 
 5. If three conies have each a double contact with a fourth, 
 six of their chords of intersection will pass, three by three, through 
 the same points. 
 
 6. ABC is a triangle, and P any point, such that the squares 
 of the three areas PA B, PBG, PC A is equal to the square of the 
 triangle ABC ; prove that the locus of P is an ellipse. 
 
 7. If Z = 0, if = 0, N= meet in a point, can any conic be 
 represented by ^(LMN) = ? 
 
 8. Interpret the following equations, where S is of two 
 dimensions, L, M, N linear, and k constant. 
 
 (i) LM+kN=0, (ii) LM+k^=0, (iii) L' + kM^d. 
 
 9. The equations to two conies are a^ = ^'y* and oy = X'/S' ; 
 shew that a + 4X/t (\/3 + /Ay) = is- the equation to their common 
 tangent. 
 
EXAMPLES XIII. . 343 
 
 10. AB and AC are two tangents to a conic, and the bisector 
 of the angle BAG meets BG in D ; prove that the segments of 
 any chord through D subtend equal angles at A. 
 
 11. If the asymptotes of an hyperbola (xr/—^m,') coincide 
 ■with the conjugate diameters of an ellipse which has a double 
 contact with the hyperbola, then ia^V = m*, where a, h are the 
 semi-conjugate diameters. 
 
 1 2. If three conies have a common chord, the other three 
 common chords meet in a point. 
 
 13. Shew that the asymptotes of an hyperbola and a pai* of 
 conjugate diameters form an harmonic pencil. 
 
 14. The bisectors of the angles A, B, G meet the conic round 
 ABC in D, E,F; shew that the equations to BD, CD, DE are, 
 with the notation of Art. 345, 
 
 m + n . 7n + n „„ 
 
 7 + — ^0 = 0, — ^a + )8 = 0, 
 
 {7n + n)a+{n + l)P — ny = (i. 
 
 Hence shew that DE, EF, FD will meet AB, BG, CA, respec- 
 tively, in points which lie on one straight line. 
 
 15. Shew that, if (a ^'y') is the centre, <^ (a;8y) = .^ (a'yS'y' ) is 
 the equation to the asymptotes. 
 
 16. Shew that a pair of tangents through any angle of a self- 
 conjugate triangle form an harmonic pencil with the sides which 
 meet in that angle. 
 
 17. From the equation to the circle circumscribed about a 
 triangle shew that the feet of the perpendiculars dropped upon 
 the sides of the triangle from any point in the circumference lie 
 in one straight line. 
 
 18. With the angular points of a triangle ABC as centres, 
 and the sides as asymptotes, three hyperbolas are described, so 
 that the perpendicular distance of the vertex of each from its 
 
344 ABEIDGED NOTATION AND TEILINEAR CO-OKDINATES. 
 
 asymptotes is the same ; shew that the intersection of each pair 
 lies on the axis of the third. 
 
 19. The asymptotes to an hyperbola are tangents to an 
 ellipse ; shew that the chords which join the points of intersection 
 of the two curves are parallel. 
 
 20. If L = 0, M= 0, N= are parallel straight lines, the 
 equation <f> (LMN) = can only represent parabolas and parallel 
 straight lines. 
 
 21. Shew that the tangents to a conic, drawn at the angular 
 points of an inscribed triangle, will meet the opposite sides in 
 points which all lie in one straight line. 
 
 22. A conic is inscribed in the triangle of reference ; shew that 
 the equations to the straight lines joining the points of contact are 
 
 la + m/3 - my = 0, la + ny — mfi = 0, mfi + ny-la = 0. 
 
 23. li ABG be a triangle, and AP, CP be so drawn, that the 
 sines of the angles GAP, PAB, PGB, and PGA are the consecutive 
 terms of a proportion ; shew that the locus of P is a oonic section, 
 to which two sides of the triangle are tangents. 
 
 24. A fixed conic circumscribes a triangle ABG, and is 
 touched at B and C by a variable conic, which meets GA, AB 
 again in D and E ; shew that BD, GE intersect in a fixed line 
 passing through A, and that DE passes through a fixed point. 
 
 25. Tkree circles, which touch each other mutually, are 
 described with the angular points of a triangle as centres ; shew 
 that the three straight lines, which join the centre of one circle to 
 the point of contact of the other two, meet in a point. 
 
 26. If the tangents to a circumscribed oonic at the angular 
 points of the triangle of reference are parallel to the opposite sides, 
 the equation to the conic is 
 
 a c 
 
EXAMPLES XIII. 345 
 
 27. The equation to the circle which passes through the 
 centres of the three escribed circles is 
 
 apy + bya + cafi + (a + ^ + y)(aa + 5)8 + cy) = 0. 
 
 .28. The equation to the circle, which passes through the 
 centres of the inscribed circle and two escribed circles which touch 
 CA and AB, is 
 
 a^y + hya + cafi + {a- P -y){aa + bp + ay) = 0. 
 
 29. A conic circumscribes a triangle ; and any conic is 
 described having a double contact with this, and such that the 
 bisector of the angle is the chord of contact. Prove that the 
 straight line joining the points, in which the latter conic cuts CB 
 and CA, meets AB in & fixed point. 
 
 30. Interpret the equations 
 
 iJa^oA + J p sin B + J y sinC = 0, 
 ,JacosA.+JpcosB + J^yc^G=0, 
 
 ^^ + ^/ya+^/r=o. 
 
 31. If a conic touch the sides of the triangle of reference in 
 A', B', C, then (i) any one of the lines AA', Bff, GC, drawn 
 from the opposite angles, passes through the intersection of tan- 
 gents drawn to the conic at the points where the other two cut it ; 
 and (ii) if^, q, r be perpendiculars drawn from the point wherp 
 these three lines meet (Art. 349), the equation to the conic is 
 
 JvJVJi-"- 
 
 32. Find the diameter of a circle described about a semi- 
 ellipse bounded by its axis minor, 
 
 33. If a conic be inscribed in a quadrilateral, the line joining 
 its points of contact with two opposite sides passes through the 
 intersection of the diagonals. 
 
34 J ABRIDGED NOTATION AND TRILINEAR CO-ORDINATES. 
 
 34. From the equation to a conic circumscribing a quadri- 
 lateral, shew that, if the conic be a circle, the opposite angles of 
 the quadrilateral are supplementary. 
 
 35. Shew that the three conies 
 
 Tm n' ' mn t' ' nl m? ' 
 meet in a single point, and that, if the constants I, m, n are 
 
 connected by the equation -^ + — + - = 0, where X, /*, v are fixed 
 
 quantities, the locus of that point is a conic circumscribing the 
 .triangle of reference. Explain this by a geometrical figure. 
 
 36. The cuiTe ^a' + mP' + ny" = will be a circle, if 
 
 I _ m n 
 
 sin 2il~ sin 25^ sin 2C' 
 
 37. Find the condition. that the equation l^y + mya + na^ = 
 may represent a rectangular hyperbola; hence prove that every 
 rectangular hyperbola described about a triangle passes through 
 the point of intersection of the perpendiculars from the angles on 
 the opposite sides. 
 
 38. Shew that la" + m^ + ny' = will represent a rectangular 
 hyperbola, i£l + m + n = 0. 
 
 39. An equilateral hyperbola is described, with regard to 
 which a given triangle is self-conjugate; shew that the curve 
 passes through the centres of the inscribed and escribed circles of 
 the triangle. 
 
 40. Find the conditions that the equations 
 
 (i) lfiy + mya+nap=0, (ii) Ju + Jmfi -i^ J^ = 6, 
 should represent ellipses, parabolas, or hyperbolas. 
 
APPENDIX. 
 
 PROPERTIES OF QUADRATIC EQUATIONS. 
 
 I. The most general form of a quadratic is 
 
 ase' + bx + c = (1). 
 
 Solving this in the ordinary manner, we obtain 
 
 -h+ ^l)' - 4!ac ,„, 
 
 "'^-—'^a •••■^'^- 
 
 The roots of (1) are, therefore, 
 
 — h+'JV — ^sac —b — 'Jif — Aac ,_. 
 
 hence, (i) if V > 4ac, we shall have b^ — 4ac a positive quan- 
 tity, and therefore V'6''' — 4ac a possible quantity, and the two 
 roots will be real and different, (ii) If b" = iac, VS" — 4ac=0, 
 and therefore the two roots will be real and equal, (iii) If 
 b^<4iaG, P — 4!ac is a negative quantity, and V6'' — 4ac is 
 imaginary, and therefore the two roots are imaginary. 
 
 Hence, the roots of equation (1) are real and different, 
 real and equal, or imaginary, ficcording as 
 
 b'> = <4!ac. 
 
 II. If a, jQ represent the roots of asi^ + bx+c=0, given 
 in (3), we have 
 
 ^ b „ b'-ib'-iac) c 
 , a 4a a 
 
 This property is expressed by saying that, if any quad- 
 ratic be so written, that the coefficient of x* is unity, then the 
 
34!8 APPENDIX. 
 
 coefficient of x is equal to minus the sum of the roots, and the 
 last term is equal to the product of the roots. 
 
 III. If we write equation (1) in the form 
 
 c4 + 5- + a = (4), 
 
 or X ^ ' 
 
 that is, as a quadratic in -', so that the roots are - and -7, , 
 ^ X a. p 
 
 we have in the same way 
 
 1 1^_J l^^a 
 
 IV. (i) If c = in (1), the quadratic is divisible by x, 
 
 so that one root is x = 0, and the other x = . If 6=0 
 
 a 
 
 as well as c = 0, the quadratic is divisible by a?, so that each 
 
 of the roots is = 0. This is also evident from (3). 
 
 (ii) If b only = 0, the sum of the roots = ; that is, the 
 roots are equal and of opposite signs. 
 
 (iii) If a = 0, then one of the roots is a: = 00 ; for, if we 
 
 write the equation in the form (4), the roots are by (i) - = 0, 
 
 ac 
 
 - = — , which gives a; = 00 , x = — j. This may be seen 
 
 from the form of solution given above, for, if we multiply the' 
 numerator of (2) by — 6 + VS' - 4ac, we obtain the roots in 
 the form 1 
 
 ^c 2« ... 
 
 _6_V6»_4ac* -j + Vi'-^oc ^^^' 
 
 and, by putting a = in (5), we obtain the same result as 
 above. If J = as well as a = 0, it will be seen in the same 
 way that each root = 00 . 
 
PROPERTIES OF QUADRATIC EQUATIONS. 349 
 
 V. If there he two eguations given of the second degree 
 between two unknown quantities, the elimination of either 
 of them will produce, generally, an equation of the fourth 
 degree. 
 
 Since tlie general equation of the second degree may be 
 written 
 
 two equations of this form may be written 
 
 «'+ Qx + R^O, a?+Q'x + R^(i (1). 
 
 By subtraction we obtain 
 
 {Q-Q)x + B-K = (2), 
 
 " and, eliminating x between (2) and either of equations (1), 
 we have 
 
 {R-By+iQB'-BQ')iQ-Q^) = (3). 
 
 A consideration of the values of Q, R, Sec, will make it 
 evident that (3) will contain y*, but no higher power of y. 
 
ANSWERS TO THE EXAMPLES. 
 
 '■ ('.-I). (-i.»). (f-D- '• '-I"''- 
 
 4. 5. ^37. 5. 3^2, S^S. 6. y . y • 
 
 7. PQ = j2l. 8. (1) p=2,tf=30"j. (2) p = 2, 61 = 150°; 
 
 (3) p=V2;e=135°. 9. (l)a!=-^ = y; (2)a; = -2, 
 
 2/ = -^-^; (3)a:=-^, y = -A. 10. pcos(e-a)=a, 
 
 p^(2 + sm26l) = 2&'. 11. (x^ + i/'y = a'{a^-tf). 
 
 12. -,l;--,-l. 13. (i)^, p (")-^-. 
 
 Sccos5 ,..., 6c ,, ,.'c+b cos A hsinA 
 
 ---; ,H«'=y=j— ,- 14. W— 2— > -2-; 
 
 /••\ • 2D D-D /— \ bcBinA ^A 
 
 (ii) c sin jBj c cos Bsm B ; (m) y = — ^ , x = y cot — . 
 
 II. 
 
 2. y-6a!+7 = 0. 3. (22/-y,-2/,)(a;3-a;,) 
 
 = (2a; - a;, - aj^) (yj - y^) is one equation. 4. 2/-*-7 = 0. 
 
 5. (i) y+a;-6V2 = 0; (ii) y + a!+ 6^ = 0. 
 
 3 1 
 
 6. a;=^, 2^ = = . 7. 3a; + 4y - Set = 0. 
 
 8. y = 7na; + c. 9. i{x + y)-5a = 0. 
 
 10. (a=H.6')i,(?^. 11. 2,-a;-l=0. 
 
 12. x + y = a + h. 13. (a - a) y - (6' - 6) a; = a'6 - a6', 
 
 (a-a')?/ + (5-6')a; = a6-a6'. U. x=JSy. 
 
ANSWERS TO THE EXAMPLES. 351 
 
 15. x^2, i/=3, x = S, y = 4:. 16. ^ 
 
 ^13" 
 
 17. y + 2a!=0. 18. a; + 2/ = 2a, or = 4fls, according to 
 
 the side on which the line is drawn. 20. m = 3. 
 
 21,22. Use Art. 43. 24. cos=a = i^-^JW 
 
 25. J. 26. . = y = lj-,. . 
 
 is. Co-ordinates of vertex being o, y', the intersection is 
 
 o, ^y'. 29. p = 2a, 6 = -; the angle = ^. 
 
 ^ 3 
 
 31. Take the general polar equation (Art. 44), and proceed 
 
 as in Art. 30. One triangle is equal in area to the sum of two 
 
 others. 
 
 III. 
 
 1. Ihj-{8^5j3)x-5(5=^j3) = 0. _ 2. (i) x = c, 
 
 y=0; (ii) 135". 3. (a±6)y = (6=Fa)(a!-c). 
 
 OK 
 
 4. 5w-8a;-40 = 0. 5. -~=^ . 6. 90°. 
 
 s/629 
 
 ' t. y = (l±.y2)(a; + 2). 9. 99a;-27y = 79, 
 
 21aj + 772/=l. 13. Any angle. 14. y=Q,y + x = a. 
 
 15. 45° or 135°. 16. '2x = y,2y = x. 
 
 ^^r i=^. 18. ± , 
 
 ^^- 11^26 ^^' + ^^ 
 
 19. (1, 2) on the origin side ; (3, — 4) on the side remote 
 from the origin. 20. The side remote from the origin. 
 
 22. (i) and (ii) Straight lines inclined at an angle a to the 
 initial line, (iii) A circle whose radius = a; a circle whose equa- 
 tion is p + a cos — 0; and a straight line pcosO—a=Q. 23. 45°. 
 
 24. (i) The initial line and a perpendicular to it through the 
 pble. (ii) A perpendicular to the initial line through the pole, 
 and two straight lines drawn through the pole, making angles of 
 
352 ANSWEES TO THE EXAMPLES. 
 
 30" and 150° with the initial line, (iii) Three straight lines, real 
 
 .. ab 
 or imaginary, passing through the pole. 25. (i) y > 
 
 (ii) 4«'i (iii) ± I («V -«>/)• 26. -. 
 
 IV. 
 
 \. x' + '!f = c'. 2. x' cos 2a - xy sin 2a = a^ 
 
 3. J2x = c, xi/ = 0. 5. y" sia' a = iax. 
 
 6. 272/'-£c'=12. 7. 3a;^ + 10/-773«y=6. 
 
 4. Take Ox, 00 as axes. 10. A straight line through 
 
 the intersection of the perpendicular lines. 11. A straight 
 
 line perpendicular to the base. 
 
 VI. 
 
 1. 6/3 + cy = 0. 2. -aa + 6/3 + Cy = 0. 
 
 3. ;(a-a')+m(/8-/8') + w(y-y') = 0- 
 
 4. a cos -i + j8 cos 5 - y cos (7=0. 6. — -, r- 
 
 mn — mn 
 
 P y 2A 
 
 - ; see Art. 95, Ex. 
 
 nl' — rCl Im — I'm a (mn' — m'n) + &c. 
 7. a (/3'V" - j8"'y") + P (7"» " - 7"'«") + 7 («"j8"' - «"'i8") = 0- 
 „ 2A 2A 2A in 7 1 
 
 ^- 3^' 36' ^- 1^- ^"''* = -l- 
 
 11. It is parallel to (|8 cos 5 — y cos (7), Art. 93, Ex. 4j hence, 
 using Ai't. 86 and the condition, that it passes through the 
 
 bisection of BC, where ^ = ^ sin C, y = ^ sin B, we obtain 
 
 j3 cos .B - y cos C = g sin (t7 - .5). 
 
 13, )8+ocosC=0. 14. If for o, &c. we write 
 
 p - a; cos a — y sin a, &o., then by Art. 62, the distance is 
 
ANSWERS TO THE EXAMPLES. 353 
 
 la + mff + wv' , i 7 a 
 
 t - , -where — j1 = f cos a + m cos S + w cos y^ 
 
 JA' + B' ' 
 
 and —B = lsma + msval3 + nsiay, 
 
 whence A' + B^=l' + m'' + n' — 2mn cos A — 2nl cos 5 — 2lm cos C: 
 
 16. (bi-c)l = (m + n)a. 17. ^-M"= ., " , AN'=j-^. 
 
 20. A straight line passing through the points where the- 
 external bisectors of the angles of the triangle meet the opposite 
 sides. The three sides of a triangle formed hy joinLag the points, 
 where the bisectors of the angles meet the opposite sides.. 
 
 VII. 
 
 1. Co-ordinates of centre are 3 and - 2 j rad. = 3. 
 
 2. A circle whose rad. = ri and a tangent to it. 3. x + y = S. 
 
 4. x + y + l = 0. 5. {l + m,'){x^ + y')-2r{x + mi/) = 0. 
 
 a" 
 6. x^-ax + y'^r"--^ . 7. An imaginary locus. 
 
 8. {^B^=^.AF, (ii> .&^ = 44i?'. 9. See Art. 121. 
 
 10. 4 = 4 + i- 11- a=' + 2/' = l- 12- ^^. -^■ 
 
 r ah "^ cos a sin a 
 
 13. a; cos (a' + a") + y sin (a + a") = r. cos (a" — o'). 
 
 14. Use Art. 129 and Cor. 
 
 16. x'' + y'+xy + x + y—l = (). 17. a; = a + c. 
 
 18. A circle, whoSe diameter = the radius of the' given circle, 
 
 20. The segment of a circle on the base, which contains the 
 
 24. y-6 = ni(a;-a)±r(l +OT')i 25. 4y-3a! = 0. 
 
 26. {i^-2(a-Vf}k 27. Area = '!r. 
 
 28 2/'r''--^,l*. 29. Take C/S as axis of a; : 
 
 \ a' + J 
 
 a' — 2a5 cos (0 + 6' 
 the locus is a straight line. 31. r = j-^t— • 
 
 "x aXlX ill 
 
 23 
 
354 ANSWERS TO THE EXAMPLES. 
 
 33. x' + 8ax + y'— 6asy = 0. 34. Take the given point 
 
 ds pole, and a straight line through the centre as initial line; 
 then the radius vector of the locus is half the sum of the roots in 
 the polar equation to the circle. The result is a circle described 
 on the line joining the point and the centre, as diameter. 
 
 35. A circle ■whose centre is on the line joining the given 
 point with the given circle. 41. The triangle must be 
 
 isosceles, and the ratio one of equality (Art. 106, Cor.). 
 
 42. The triangle must bs equilateral. 43. A circle. 
 
 44. The centre is on the base. 
 
 45. (x-ad=h':^Vf+{y-h-l'f = {h'-h)\ 47. y^x. 
 48. a^ + f-2,r{x + y)+r^=0. 49. Take the bounding 
 
 radii as axes, and use the equation of Ex. 48 ; then the circle 
 passes through {hk) on the quadrant, ■where h' + T(^ = E", and the 
 roots of the equation obtained from this condition are r and r^ &c. 
 
 50. The tangent to the first circle is x cos a + y sin o = r, and 
 to the second (cc -a) cos^ + y sin ^ = r' ; if these represent the* 
 same line, the equation is (r d^r^x^ {a^ — (r ± r'Yy'y = ar. 
 
 51. "Writer = - ^ —in the result of Ex. 48. SeeArt. 
 
 120, Cor. 1. 52. a;'+2/=-2{o+6=t(2a6)^}(a!+2/)+{a+6±(2a6)*}==0. 
 See Ex. 49. 53. If the circle touches BC in L, then 
 
 BL = s — c, &c. ; take t^wo sides as axes. 54. It AE be 
 
 initial line, AP = p, iPAE — 6, the polar equation to the locus Ls 
 p sin ^ = o sin {0 + a), -where a = AE, a. = ^ EAB. 
 
 55. -= r\ A ,\if • 56. The polar co-ordinates of the 
 
 y OA — OA' ^ 
 
 centre are p = 2, ^ = q) aid rad. = 3. 57. A circle ■whose 
 
 centre is the intersection of the diagonals. 58. Take A as 
 
 pole and the diameter through A as initial line j if GP = n. AG, 
 the locus is a circle whose radius = (w + 1) r. 
 
 59. If a be the given angle, centre origin, the equation is 
 
 y = tan(^a-ya; + -J=sec^a-^^. 
 
ANSWERS TO THE EXAMPLES. 355 
 
 60. A circle whose centre is the fixed extremity of one of 
 the lines. 61. Take the common tangent and common 
 
 diameter as axes, and use Art. 43. 62. A straight line 
 
 perpendicular to the line joining the fixed point and the centre. 
 
 63. Take AB, AG as axes; then the equation to the circle 
 is (Art. 110) af + y'+2cosm.xy + Dx + £y=0; then assume 
 MB : AM=AN : NG = n : \, and thus find AM and AN ; 
 D and E may be found from the condition of the circle passing 
 
 through M and iV. For the fixed point ^ = - = ^i — s — si 7 • 
 
 YIII. 
 14. y-x = 0; y-x = 0, y + x-2 = 0. 15. Use Art. 73. 
 
 IX. 
 
 , . 5J5-3J3 _ . ^5^17 
 
 4. cos5 = -> — ._^ . 7. cos6 = ^-~—. 
 
 475 27 
 
 8. (i) cos~'e; (ii) tan"'^. H- In any triangle j4-BC, 
 
 ^ A. B s-c . 
 
 tan ^tan -77 = , &c. 
 
 2 2 s 
 
 14 iL__ = l. 17. The extremities of the latus 
 
 ■ m' n 
 
 rectum; tan~'±e. IS. The equation ^3y^,j5x-9a=0 
 
 •will represent the four tangents. 21. r = eb. 
 
 22. The extremity of the latus rectum. 25. FQ is a 
 
 tangent to the inner ellipse at (o/t/), and the polar of R (hk) with 
 
 regard to the outer; by comparing these equations we get x', y' in 
 
 terms of A, k, &c. 28 and 29. Use Art. 197. 
 
 30. {x' + y'y^aV-by. 31. ^cos^ +|sin^=l. 
 
 33 tan"'^^' ^^- ^®® equation Art. 34, F being 
 
 (afj/), and combine with equation to asymptotes as one locus, &c. 
 35. See Art. 188. 
 
356 ANSWERS TO THE EXAMPLES. 
 
 1 
 37. —j^, 3. 38. y = 3x, 3i/ = x; tan"' 2^6. 
 
 39. See Art. 192. Pour, if the points on the major and 
 
 B 13 2 12 
 
 minor axes are not further from the centre than and — , — 
 
 a 
 
 respectively ; from other points two. 
 
 41. ( — j +( — j ={m + nf. 42. An ellipse whose 
 
 x' hV 
 equation is -^ H — |- = 1. 44. Use the polar equation of 
 
 Art. 209; the eccentricity is . /r . 45. See Art. 187. 
 
 46. The equations to the ellipses are — + '73 = 1, t + -is = Ij 
 where a' — b" = 0? — ^' ; take equations to tangents in form of 
 Art 182. 48. The centre {xr/), P (xY) ; then — ^ 
 
 {ae + xf 
 
 ^ ,PSH l-.cosFSH ... , , . ^ 
 
 = tan —^ — = -= ttntt j which may be expressed in terms 
 
 2 1 + cos FSH •' '^ 
 
 of as'. Similarly -; — - — rs = tan^ —-. — &c. Dividing the resulting 
 •^ (ae — tc) 2 
 
 equations, we obtain as = a for the equation. 
 
 10. Use equation of Art. 34, and ao obtain a polar equation 
 
 to the locus with the fixed point as pole. 14. See Ai-t. 163. 
 
 /fl 
 15. ^^. 17. Use Art. 223 to form the equations to 
 
 ,2. 2 
 
 CD and PF. The equation to the locus is -; + -; = ( -= \ . 
 
 t/ US' W + j/'J 
 
 18. Use Art. 223. 20. Art. 155 will suggest the solu- 
 
 tion. 21. The perpendicular from the centre (a;', 0) on the 
 
 asymptote = a - a/, &c. 23. Draw a diameter by Art. 214, 
 
 and the axes by Art. 234 ; then BS= a, hence the focus and latus 
 
AIJSWERS TO/THE EXAMPLES. 357 
 
 rectum ; draw a tangent at the extremity of tlie latus rectum by 
 Art. 214, Cor. J tliis gives the directrix, Examples ix. 17. 
 
 24. tan-'L 26. x=3, y = 2; x = 0, y + 2x=0. 
 
 27. Use Art. 223 to form the equations to the tangents at 
 P and Z>. The equation is a'f + b'x'=2aV. 28. If 2p 
 
 be the length of the chord, the equation is ^ + f^ + »" —r-^ — rr-5=l- 
 32. See Arts. 218, 225 ; a? = 2(? cos^ %, V= 2c' sin^ % , 
 " ^ ""^ " 34. TJse equation of Art. 34, and Art. 219. 
 
 1 + cos a 
 
 37. If (x'y') is the point of contact, and m, c constants, we 
 
 6V 
 have 5—. = m, a^ — b' = c; from these and the equation to the 
 
 ay 
 
 ellipse we must eliminate a and 6. 
 
 39. Use Arts. 132, 182; the equation is (a;' + yy= aV + hY- 
 
 40. Use Art. 223; the normal at B is perpendicular to GP. 
 
 41. A rectangular hyperbola, of which AB is an asymptote. 
 
 42. Take the sides as axes ; then, when a; = 0, the diflference 
 of the roots = one of the lines, and similarly when y = 0, &c. ' 
 
 43. An hyperbola whose asymptotes are parallel to the lines 
 containing the given angle. 49. Use equation of Art. 34 
 for any one of the chords, (a//) being middle point, and combine 
 it with the equations to the two hyperbolas ; then, if \ satisfies 
 one equation, - X will satisfy the other ; substituting and elimi- 
 nating X, we have for the required equation 
 
 50. Prove the ordinates of the points proportional to the 
 conjugate axes, &c. 
 
 XI. 
 
 1. id {2^ J?,). 2. (0,0), (4,-8). 
 
 3. y, = d{^t-2)\y, = d{Jl + 2)K 5. Art. 261. Cor, 
 
338 ANSWERS TO THE EXAMPLES. 
 
 8. Taking the centre of the circle as origin, and the given 
 diameter as axis of x, the latus rectum = twice the abscissa of the 
 
 point of contact. 11. x= — j — jy— , y = |- (6 + 6'). 
 
 14. See Art. 136. 15. y=^,cc + %±^. See 
 
 a + a Jsaa 
 
 Arts. 283, 257. 16. Parabolas whose latera recta = half 
 
 that of the original curve. Use. Art. 34. 
 
 17. See Art. 304, Ex. 19. T)ista,nce = 2j2cl. 
 
 20. See Art. 129, Cor. 21. «' = --^. 
 
 ' ^ 2d-x 
 
 22. See Art. 259. 24. Jx + Jy = Jd2^2. 
 
 25. 2a* (a!* -8*). 26. 2d. 27. Art. 256 for 
 
 . <^oos'<^ d 
 
 equations to pides. 28. — ; — —^ , -i — - . 
 
 sin (f> sin (p 
 
 29. A parabola whose focus is at the centre of the given 
 
 circle. 30. d{'!/ + x)'={y' + x'){y-^x). 31. If the focus 
 
 n 
 
 be pole, the axis initial line, the polar equation is p = 2d cot -^ . 
 
 QO . n 5 r, (Jid'+7^-r\k 
 
 65. y=»ia!+r;^'l +m , where m = ± I ^^^ = 1. 
 
 35. Find a diameter by Art. 276, and draw a tangent at its 
 vertex by Art. 278; draw a focal chord by Art. 269, the diameter 
 of which wiU intersect the tangent in the directrix j then find the 
 focus by Art. 272. 36. Equal tangents from {x'y) would 
 
 make equal angles with the polar of {x'l/). Use Art. 257. 
 
 38. Latus rectum = 4p sin*^, if p and ^ are the given 
 quantities. 43. y' = h{x — h), where h is any constant. 
 
 44. {y-x)''-MxJ2 = Q. 45. Take equation Art. 
 
 261, Cor., m°+ — -3 — m + ^=0, and suppose the roots to be 
 
 fi, — , /*'; then the sum =/«, i-/i' = 0; the sum of the products, 
 
 two and two, =-l+/iM/i — j = — -j~ : the product = .-/*' = - ^ j 
 eliminating /x and fi, we have r^=d{x — 3c?). 
 
ANSWERS TO THE EXAMPLES. 359 
 
 46. x = 0, x = 2{c-2d), 2/ = *(^^)^a:- 
 
 49. See Art. 271. 50. Take equation (2) of Art. 251, 
 
 and divide by P, so as to get a quadratic in -j ; then follow the 
 
 method of Art. 310. 51. A parabola whose equation is 
 
 y^ = d(x— d). 53. An extremity of the latus rectum. 
 
 55. a'=2b\ 56. Let the tangent of one of the 
 
 parabolas at the given point be inclined at an angle 6 to the 
 diamfiter; take the diameter and a perpendicular through the 
 point as axes> and find the co-ordinates of the vertex (Art. 280) 
 in terms of 6 and the parameter ; then eliminate 9. 
 
 57. Take y' = idx as the equation to the parabola in one of 
 its positions ; then the equations to the rectangular axes are 
 
 y = mx H — , y = md) the perpendiculars on these lines from 
 
 the focus or vertex will be the x and y of the locus, if the lines 
 are axes of co-ordinates ; writing these equations, and eliminating 
 m, we obtain (i) afy' = (P{af + y'), (ii) x^y^(x^ + y^) = d\ 
 
 XII. 
 8. ix-a=0, 8y-6x-3a=Q, e=-^; y-2x=0, y + 2x+2=0, 
 
 e=-^. 10. 2\/ai6sin5, 2va6cos^; 2asin^, 2acc)%-^. 
 
 2 M 2i A Ji 
 
 10. Semi-transverse (1) 2^a6sin^, (2) 2a sin ^j Semi- 
 conjugate (1) 2^*6 cos ^, (2) 2a cos 2 . See end of Art. 241. 
 
 16. Transfer the origin to the point of tangency; then use 
 Art. 297, Cor., and retransfer. The result is 
 (i2/_l)(6'_6)-Ha;{c+a(6'-6)}=0; (6V-l)(6'-5)-a!{c-ffl(5'-6)}=0; 
 2aa; + (6 + 6')y = 2. 
 
360 ANSWERS TO THE EXAMPLES. 
 
 19. 1^^. 20. {A-k)x' + Bxy+ {0-^^ = -J 
 
 Jrlip 
 
 7}^ = 4:{A —k)(G — k); where the conic and circle are 
 
 Ax' + Bxy + Gy' = 1, and k {x' + y*) = 1. 
 
 21. See Art. 294 (1). 23. See Art. 310. 
 
 24. For A or D, 2Gy+Bx+E=Q; for B, 2Gy'+Ey='2Aa?+Dx; 
 for G or E, 2Ax+By+D=0 ; for E, the conies are concentric. 
 
 aa oh \a aj \b hj" 
 26. In Art. 296, Cor. 3, suppose points Q, Q' to coincide. 
 
 28. Use Arts. 296, 308. 29. tan B = . . 
 
 sin a 
 
 30 — 34. Use Art. 305. 34. The locus is a conic, whose 
 
 eccentricity is e cos ^ sec {ym + ^ ) . 35. Draw two diameters, &o. 
 
 36. See Art. 307, Ex. 38. The hypotenuse is parallel 
 
 to the normal. 40, 41. Use Art. 308. 42. Use Art. 295. 
 
 43. {x + y-ay = i:xy sin=^ . 44. See Arts. 31 7, 318. 
 
 46. Art. 314, Ex. and Art. 296, Cor. 2. 47. A parabola 
 to which AB, AG are tangents. 48. Take the two sides as 
 
 axes, and use Art. 314, Ex. 50. See Arts. 218 and 294 
 
 i^ii). If the equation to the ellipse referred to its axes is 
 A'x''+G'y' + F=Q, the equation to the hyperbola referred to its 
 axes must be A'x' - G'y^ + F = 0, where A' and C" have the values 
 given in Art. 148 ; hence, referring the hyperbola to its axes, we 
 6]idF'=2(A'-G')E. 51. The ellipses are equal, and if 
 
 2a, 2j3 are their axes, a?=a'j2{j2 + 1), P'=a,'j2 (J2- 1). 
 
 52. The directrix is -Dy + ^a;= 0; for the focus, 
 x_y__ 2E 
 1>~ E~ B' + E' ' 
 the axis is (Z»= + E') (Dx -Ey) + 2 {D' -E'JE^O. See Arts. 259, 
 263, 272, 283. 
 
ANSWEBS TO THE EXAMPLES. 361 
 
 XIII. 
 
 ■4. (i) Conies having three points of intersection finite, and 
 one infinite, (ii) Conies having two points of intersection finite, 
 and two infinite. 7. See Art. 86. 8. (i) Similar and 
 
 similarly placed hyperbolas, having that portion of (i\^) which 
 is intercepted between {L) and {M) as common chord, (ii) 
 Hyperbolas having (i) and {M) for asymptotes. (iii) Parabolas, 
 of which (i) is a diameter, and {M) the tangent at its vertex. 
 
 9. Compare the equation la + m^ + «.y= with the equation 
 to the tangent (Art. 337) of each curve, and eliminate ft^. 
 
 10. Use equation ^y + ka^ = 0; then the chord through J) is 
 a+A'(;8 — y) = ; eliminate a, and use Art. 89. 11. From the 
 symmetry of the figs, the common chord must pass through the 
 centre. 12. See Art. 332. 13. See Examples x. 22. 
 
 15. See Examples xn. 13. 17. The equation asserts 
 
 that two triangles are together equal in area to a third. 
 
 18. The three hyperbolas will be py=^V, ya = m', aP = n', 
 and their axes j8 — y=Qj y — a=0, a — /3 = 0; then the condition 
 gives l=m=n; hence the first may be written ya— Z"— (a— y3)y=0, 
 &c. 19. Take the asymptotes as two sides of the triangle 
 
 of reference. 27. See Art. 357. 30. Conies touching 
 
 the sides of the triangle of reference j (i) at their middle points ; 
 (ii) at the feet of the perpendiculars from the angles ; (iii) where 
 the bisectors of the angles meet the sides. 31. To find the 
 
 equations to the tangents, use Art. 354, Ex. 2, Cor. 
 
 32. The equation to a circlOj having two common chords 
 (i), (M) with the ellipse {S), would be included in the equation 
 S — kLM=0 (Art. 329) ; hence the equation is 
 
 ^ + |j-l-Aa;(a!-a) = 0, 
 
 where the chord x~a — has become a tangent, <fec. then use 
 Art. 106, Cor. 33. Take equation (Art. 347) to the conic 
 
 touching three sides; then use Art. 354, Ex. 2, Cor. for the 
 fourth side. 34. Take the equation ay—k^S=0, and write 
 
 for a, p — x cos a-j/ sin o, &c.j then use Art. 106, Cor. 
 
 24 
 
362 ANSWERS TO THE EXAMPLES. 
 
 35. The point is defined by ^=-=^. 36. SeeAi-t.352,Ex. 
 
 37. See Ex. 34, and Art. 170. The condition is 
 lciOsA+mco&B + nGosG= 0. 39. See Art. 340. 
 
 40. Eliminate y by the relation aa + 6;8 + cy = 2A = ah sin G ; 
 then take CA and C£ as axes of x and y, as in Art. 97, and use 
 the test of Chap. Tin. The conditions are 
 
 (i) W + w»V + wV - 2lmab — 2mnbc - 2nlca < = > 0. 
 
 THE END. 
 
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