I ^8 la. COKNELL UNIVERSITY LI/^RARIES Mathematics Library Wmte Hall CORNELL UNIVERSITY LIBRARY 3 1924 064 186 905 DATE DUE , i ADD 9.P , \m ^^^ *^ ; i \ CAYLORD rniNTEOINU.S.A. The original of tliis book is in tlie Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924064186905 Production Note Cornell University Library pro- duced this volume to replace the irreparably deteriorated original. It was scanned using Xerox soft- ware and equipment at 600 dots per inch resolution and com- pressed prior to storage using CCITT Group 4 compression. The digital data were used to create Cornell's replacement volume on paper that meets the ANSI Stand- ard Z39. 48-1984. The production of this volume was supported in part by the Commission on Pres- ervation and Access and the Xerox Corporation. 1991. THE EVAN WILHELM EVANS MATHEMATICAL SEMINARY LIBRARY THE GIFT OF LUCIEN AUGUSTUS WAIT M...v.^3 msmfmsk ^^K 10 7348-1 (.02, A TBEATISE ON PLANE CO-OEDINATE GEOMETEY. A TREATISE ON %/% PLANE CO-OEDINATE GEOMETEY AS APPUKD TO THE STRAIGHT LINE CONIC SECTIONS. By I. TODHUNTER, M.A., F.R.S., HONOBABY FELLOW OF ST JOHN'S COLLEGE, CAMBBIDaE. SEVENTH EDITION. Hontion: MACMILLAN AND CO. 1881 [The right of trarulation U reierved.'] ■':&>e: FRINTED ET C. J. CIiAT, U.A. AT IBE ONITEBSITT FBESS. PREFACE. I HAVE endeavoured in the following Treatise to exhibit the subject in a simple manner for the benefit of beginners, and at the same time to include in one volume all that students usually require. In addition, therefore, to the propositions which have always appeared in such treatises, I have intro- duced the methods of abridged notation, which are of more recent origin; these methods which are of a less elementary character than the rest of the work, are placed in separate Chapters, and may be omitted by the student at first. The Examples at the end of each Chapter, will, it is hoped, furnish sufficient exercise on the principles of the subject, as they have been carefully selected with the view of illustrating the most important points, and have been tested by repeated experience with pupils. At the end of the volume will be found the results of the Examples, together with hints for the solution of some which appear difficult. The properties of the parabola, ellipse, and hyperbola, have been separately considered before the discussion of the general equation of the second degree, firom the belief that the subject is thus presented in its most accessible form to students in the early stages of their progress. L TODHUNTER. St Johu'b Ooluoz, July, 1855. vi PREFACE. In the fourth edition the work has been carefully revised, and a large amount of new matter has been introduced, chiefly relating to the more recent methods of investigating the properties of the conic sections. The work was origi- nally designed for early students, and in the additions which have been made this object has been constantly regarded. Accordingly great attention has been given to the explanation and illustration of the principles of the methods which are employed ; so that it will be easy for a student hereafter to develope these principles to any required extent. The favour with which the work has been received in- dicates that it has been found adapted for the purpose of elementary instruction ; and the hope may be expressed that the improvements now eflfected will increase its utility. May, 1867. CONTENTS. PLANE CO-ORDINATE GEOMETRY. CHAP. PAGE I. Co-ordinate3 of a Point ..... i II. On the Straight Line 11 III. Problems on the Straight Line 25 IV. Straight Line continued 55 v. Transformation of Co-ordinates ... 82 VI. The Circle 89 VII. Radical Aiis. Pole and Polar . . . . 106 Vin. The Parabola 116 IX. The Ellipse 143 X. The Ellipse continued 168 XL The Hyperbola 188 XII. The Hyperbola continued 203 XIII. General Equation of the Second Degree . . . 224 XIV. Miscellaneous Propositions 244 XV. Abridged Notation 272 XVI. Sections of a Cone. Anharmonic Ratio and Harmonic Pencil 308 XVII. Projections 322 ANSWERS TO EXAMPLES 335 Students reading thia work for the first time may omit Chapters IV, VII, XIV, XV, XVL PLANE CO-OEDINATE GEOMETRY. CHAPTER I. CO-ORDINATES OF A POINT. 1. In Plane Co-ordinate Geometry we investigate the properties of straight lines and curves lying in one plane by means of co-ordinates ; we commence by explaining what we mean by the co-ordinates of a point. o Y N F >f X' s / o X Let be a fixed point in a plane through which the straight lines X'OX, Y'OY, are drawn at right angles. Let Pbe any other point in the plane; draw PM parallel to OF meeting OX at M, and PN parallel to OX meeting OF at N. The position of P is evidently known if OM and ON are known; for if through if and ilf straight lines be drawn parallel to OX and OF respectively, they will intersect at P. The point is called the origin; the straight lines OX and OY are called axes; OM is called the abscissa of the point P; and ON, or its equal MP, is called the ordinate of the point P. Also OM and MP are together called co-ordi- nates of the point P. 2. Let OM = a, and 0N= b, then according to our defi- nitions we may say that the point P has its abscissa equal to a, and its ordinate equal to h ; or, more briefly, the co-ordinates T. c. s. 1 2 CO-OllDXXATES OF A POINT. of tJie point P are a and b. We shall often speak of the point which has a for its abscissa and b for its ordinate, as the.point {a, b). 3. A distance measured along the axis OX is however most frequently denoted by the symbol x, and a distance measured along the axis OY by the symbol y. Hence OX is called the axis of x, and Y the axis of y. Thus x and y are symbols to which we may ascribe different numerical values corresponding to the different points we consider, and we may express the statement that the co-ordinates of the point P are a and b, thus : for the point P, x = a and y = b. M X' 4. The straight lines X'OX, TOY, being indefinitely produced divide the plane in which they lie into four com- partments. It becomes therefore necessary to distinguish points in one compartment from points in the others. For this purpose the following convention is adopted, which the reader has already seen in works on Trigonometry; straight lines measured along OX are considered positive and along OX negative ; straight lines measured along OY are con- sidered positive, and along Y' negative. (See Trigonometry, Chap. IV.) If then we produce PN to a point Q such that N,Q=NP, we have for the point Q,x = — a, y = b. If we produce PMto R so that MR = MP, we have for the point R, x = a,y= — b. Finally, if we produce PO to S so that OS = OP, we have for the point S, x = — a, y = — b. POLAH CO-OBDIKATES OF A POINT. 3 5. In the figure in Art. 1 we have taken the angle TOX a right angle ; the axes are then called reidangular. If the angle YOX be not a right angle, the axes are called ohUque. All that has been hitherto said applies whether the axes are rectangular or oblique. We shall always suppose the axes rectangular unless the contrary be stated; this remark applies both to our investigations and to the examples which are given for the exercise of the student. G. Another method of determining the position of a point in a plane is by means oi polar co-ordinates. Let be a fixed point, and OX a fixed straight line. Let P be any other point ; join OP ; then the position of F is determined if we know the an- gle XOP and the distance OP. 0< The angle is usually denoted by ^^"■~~-~,y 0, and the distance by r. is called the pole, OX the initial line ; OP the radius vector of the point P, and POX the vectorial angle. 7. The position of any point might be expressed by positive values of the polar co-ordinates and r, since there is here no ambiguity corresponding to that arising from the four compartments of the figure in Art. 4. It is however found convenient to use a similar convention to that in Art. 4 ; angles measured in one direction from OX are con- sidered positive and in the other negative. Thus if in the figure XOP be a positive angle, XOQ will be a negative angle; if the angle XOQ be a quarter of a right angle, we may say that for XOQ, d = — ■^. It is, as we have stated, not absolutely necessary to introduce negative angles, but con- venient; the position of the straight line OQ, for instance, might be determined by measuring from OX in the positive direction an angle =27r — -^j as well as by measuring in the negative direction an angle = r . 1—2 4 LENGTH OF THE STRAIGHT LINE Also positive and negative values of the radius vector are admitted. Thus, suppose the co-ordinates of P to be -j- and 4 a, that is, let XOP= t anci y-^^^r^ x OP=a; produce PO to P', so that OP'=OP, then F may be determined by saying its co-ordinates are t and — a. Thus when the radius vector is 4 a negative quantity, we measure it on the same straight line as if it had been a positive quantity but in the opposite direc- tion from 0. Hence if ^ represent any angle and c any length, the same point is determined by the polar co-ordinates /3 and — c as by the polar co-ordinates ir + yS and c. 8. Let X, y denote the co-ordinates of P referred to OX as the axis of x, and to a straight line through at right angles to OX as the axis of y. Also let 6 and r be the polar co-ordinates of P. If we draw from P a perpendicular on OX, we see that x = r cos 6, and y = r sin 6. These equations connect the rectangular and polar co-ordi- nates of a point. From them, or from the figure, we may deduce a;»+«« = r', ?^ = tan^. X We proceed to investigate expressions for some geome- trical quantities in terms of co-ordinates. 9. To find an expression for the length of the straight line joining two points. Let P and Q be the two points ; to the inclination of the axes OX, OT. Draw PM, (JN parallel to OF; let a;,, y^ be the co-ordinates of P, and a;,, y, those of Q. Draw PR parallel to OX. Then, by Trigonometry, PQ' = PIl' + QE'-2PR.QRcos PRQ = Pfi* -I- QiJ' -(- 2PB . QR cos Q>. JOINING TWO POINTS. 5 But P^ = a;, — a;,, and QR = y^ — i/^; therefore -P<2' = (^.-*.)'+(y,-yx)' + 2(a',-a:.)(2/.-y.)cosa,...(l), and til us the distance PQ is determined. If the axes axe rectangular, we have FQ^ = {a>^-o.,f + (y,-ff,y (2). The student should draw figures placing P and Q in the different compartments and in different positions ; the equa- tions (1) and (2) will he found universally true. From the equation (2) we have P<2'=a;.' + y.'+<+y/-2(*,^,+y.y,) (3). The following particular cases may he noted. If P he at then a, = and 5^, = ; thus PQ' = x^' + y*. KP he on the axis of x and Q on the axis of y, then y, = and a;, = ; thus PQ' = a;,' + y*. . Let 6^, r, he the polar co-ordinates of P, and 6^, r, those of Q ; then, hy Art. 8, . Find the area of the triangle formed by joining the first three points in Example 1. 5. A is a, point on the axis of x, and B a point on the axis of y : express the co-ordinates of the middle point of AB in terms of the abscissa of -4 and the ordinate of .$; shew also that the distance of this point from the origin = J AB. 10 EXAMPLES. CHAPTER 1. 6. Transform equation (2) of Art. 11 so as to give an expression for the area of a triangle in terms of the polar co-ordinates of its angular points. Also obtain the result directly from the figure. 7. A and B are two points and is the origin: express the area of the triangle AX)B in terms of the co-ordinates of A and B, and also in terms of the polar co-ordinates of A and B. 8. A, B, are three points-the co-ordinates of which are expressed as in Art. 11 ; suppose D the middle point oi AB; join CB and divide it at (r so that CG = 2 6D : find the co-ordinates of G. 9. Shew that each of the triangles GAB, GBC, GAC, formed by joining the point G in the preceding Example to the points A, B, C, is equal in area to one-third of the triangle ABC. See Art. 11. 10. A and B are two points ; the polar co-ordinates of A are 0^, r, ; and those oiB are d^,r^. A straight line is drawn from the origin bisecting the angle A OB; if C be the point where this straight line meets AB shew that the polar co- ordinates of C are 5 = i (6^ + 6^ and r = 2^.^» c"s |(g, - g.) _ ^ ' ' '■i + »=s 11. Find the value of CL^ and AD"* in Example 8 in terms of the co-ordinates there used ; and shew that AC + BC^ = 2CD' + 2AI)''. 12. Find the value of GA\ GB", and GC, in Example 9 in terms of the co-ordinates there used ; and shew that 3 ( GA'' + GE' + GC) = AB' + BC + GA\ ( 11 ) CHAPTER II. ON THE STRAIGHT LINE. 1 4. To find the equation to a straight line. Y We shall first suppose the straight line not parallel to either axis. Let ABD be a straight line meeting the axis of y at B. Draw a straight line 0^ through the origin parallel to ABD. In ABD take any point P; draw PJf parallel to Y, meet- ing OX at M, and OE at Q. Suppose 0B= c, and the tangent of EOX=m ; and let «, y be the co-ordinates of P ; then y = PM=PQ+ QM= 0B+ QM = c+ Oif tan QOM=c + mx. Hence the required equation is y = ma; + c. OB is called the intercept on the axis of y; if the straight line crosses the axis of y on the negative side of 0, then c will be negative. 12 EQUATION TO A STRAIGHT LINE. We denote by m the tangent of the angle QOM or £A 0, that is, the tangent of the angle which that part of the straight line which is above the axis of a; makes with the axis of X produced in the positive direction. Hence if the straight line through the origin parallel to the given straight line falls between Y and OX, m is the tangent of an acute angle and is positive; if between 01^ and OX produced to the left, m is the tangent of an obtuse angle and is negative. So long as we consider the same straight fine m and c remain unchangeable, they are therefore called constant quantities or constants. But x and y may have an indefinite number of values since we may ascribe to one of them, as x, any value we please, and find the corresponding value of y from the equation y = mx + c; x and y are therefore called variable quantities or variables. If the straight line pass through the origin, c = 0, and the equation becomes y = mx. 15. We have now to consider the cases in which the straight line is parallel to one of the axes. If the straight line be parallel to the axis of a;, m = 0, and the equation becomes y = c. If the straight line be parallel to the axis of y, m becomes the tangent of a right angle and is infinite ; the preceding investigation is then no longer applicable. We shall now give separate investigations of these two cases. To investigate the -equation to a straight line parallel to one of the axes. Y D A M X. \x' First suppose the straight line parallel to the axis ( Let BGhe the straight line meeting the axis oiy&tB; sup- pose OB = b. the axis of x. EQUATION OF THK FIRST DEGBEE. 13 Since the straight line is paxallel to the axis of x, the or- dinate PMoi any point of it is equal to OB. Hence calling y the ordinate of any point P, we have for the equation to the straight line y = h. Next suppose the straight line parallel to the axis of y. Let AD he the straight line meeting the axis of a; at ^ ; sup- pose OA = a. Since the straight line is parallel to the axis of y, the abscissa of any point of it is OA. Hence calling x the abscissa of any point, we have for the equation to the straight line x = a. 16. We have thus shewn that any straight line whatsoever is represented by an equation of the first degree; we shall now shew that any equation of the first degree with two variables represents a straight line. The general equation of the first degree with two variables is of the form Ax+By+C=0 (1), A, B, C being finite or zero. First suppose B not zero; divide by B, then from (1) y^-B-B"" <^^- Now we have seen in Art. 14, that if a straight line be (J drawn meeting the axis of y at a distance — = fi:om the origin and making with the axis of x an angle of which the tangent is — -p , then (2) will be the equation to this straight line. Hence (2), and therefore also (1), represents a straight line. If .4 = 0, then by Art. 15 the straight line represented by (1) is parallel to the axis of x. If £ = 0, then (1) becomes Ax+C=Q, G or ^="2' and from Art. 15 we know that this equation represents a straight line parallel to the axis of y. Hence the equation Ax + By + 0=0 always represents a straight line. 14 EQUATION IN TERMS OF THE INTEECEPTS. 17. Eqiiation in terms of the intercepts. The equation to a straight fine may also be expressed in terms of its intercepts on the two axes. Let A and B he the points where the straight line meets the axes of a; and y respectively. Suppose OA = a, OB = b. Let P be any point in the straight line; x, y its co-ordinates; draw PM parallel to OY. Then by similar triangles, PM _AM OB AO' that is therefore y _a — X X y , a 1 8. It will be a useful exercise for the student to draw the straight lines corresponding to some given equations. Thus suppose the equation 2y + 3x = 7 proposed; since a straight line is determined when two of its points are known, we may find in any manner we please two points that lie on the straight line, and by joining them obtain the straight line. Suppose then a; = 1, it follows from the equation that y = 2; hence the point which has its abscissa = 1, and its ordinate = 2, is on the straight line. Again, suppose x = 2, then .y = i; the point which has its abscissa = 2 and its ordinate = | is there- lore on the straight line. Join the two points thus deter- mined, and the straight line so formed, produced indefinitely both ways, is the locus of the given equation. The two points EXAMPLES OF STRAIGHT LINES. 15 that will be most easily determined are generally those at which the required straight line cuts the axes. Suppose x = in the given equation, then y = ^, that is, the straight line passes through a point on the axis ofyatsi distance f from the origin. Again, suppose y = 0, then a; = |, that is, the straight line passes through a point on the axis of x &t a, dis- tance |- from the origin. Join the two points thus deter- mined, and the straight line so formed, produced indefinitelj' both ways, is the locus of the given equation. What we have here ascertained as to the points where the straight line cuts the axes, may be obtained immediately from the equation ; for if we write it in the form 7 + 7-1, and compare it with the-equation in Art. 17, a^b ' we see that a = f and b = ^. Again, suppose the equation y = x proposed. Since this equation can be satisfied by supposing a; = and y = 0, the origin is a point of the straight line which the equation repre- sents ; therefore we need only determine one other point in it. Suppose 3: = 1, then y = l; here another point is determined and the straight line can be drawn. The straight line may also be constructed by comparing the given equation with the form in Art. 14, y = mx. This we know represents a straight line passing through the origin and making with the axis of x an angle of which the tangent is m. Hence y = x represents a straight line passing through the origin and inclined at an angle of 45° to the axis of X. Similarly the equation y = — x represents a straight line inclined to the axis of x at an angle of which the tangent is — 1 ; that is, at an angle of 135". Hence this equation repre- sents a straight line through bisecting the angle between Y and OX produced to the left in the figure to Art. 14. IG EQUATION IN TERMS OF THE PERPENDICULAE. 19. The student is recommended to make himself tho- roughly acquainted with the previous Articles before proceed- ing with the subject. In Algebra the theory of indeterminate equations does not usually attract much attention, and the student is sometimes perplexed on commencing a subject in which he has to consider one equation between two unknown quantities, which generally has an infinite number of solutions. Our principal result up to the present point is, that a straight line corresponds to an equation of the first degree, and the student must accustom himself to perceive the appropriate straight line as soon as any equation is presented to him. The straight line can be determined by ascertaining two points through which it passes, that is, by finding two points such that the co-ordinates of each satisfy the given equation ; and the straight line being thus determined, the co-ordinates of way point of it will satisfy the given equation. 20. Equation to a straight line in terms of the perpendicular from the origin, and the inclination of this perpendicular to one of the axes. Let OQ he the perpendicular from the origin on a straight line AB. Take any point P in the straight line ; draw PM perpendicular to OA, MN perpendicular to OQ, and PR perpendicular to MN. Suppose OQ=p, and the angle QOA = a. Let x, y be the co-ordinates of P; then OQ=ON + NQ= ON+PB = OM cos QOA+ PM sin PMR = X cos a -(- y sin a. RELATIONS BETWEEN THE CONSTANTS. 17 Therefore the equation to the straight line is X cos a + ^ sin a =^. 21. We have given separate investigations of the dif- ferent forms of the equation to a straight line in Articles 14, 17, 20 ; any one of these forms may however he readily de- duced from any other hy making use of the relations which exist between the constant quantities. The quantity which we have denoted by 6 in Art. 17, that is OB, is denoted by c in Art. 14 ; therefore 6 = c (1). In Art. 17, 6 a - = tan 5^0 = tan (tt - £^Z) = - tan 5AX ; in Art. 14 we have denoted the tangent of BAX by m, therefore - = — m (2). a In Art. 20, OA cosa = OQ, and OB suia= OQ; that is, 2) = acosa = 6sina (3); therefore from (2) and (3), m=-cota (4). Also if the equation Ax + By+ C=0 represent the straight line under consideration, then by Art. 16, therefore -g=cota, and ;g = -^i~ (6)- And, by comparing Arts. 16 and 17, we have ° = -2' ^^~B ^^^• By means of these relations we may shew the agreement of the equations in Arts. 14, 17, 20, or from one of them deduce the others,' T. C. s. 2 18 OBLIQUE CO-ORDINATES. 22. The student may exercise himself by varying the figures which we have used in investigating the equations. Thus, for example, in the figure to Art. 17, suppose the point P to be in BA produced, so that it falb below the axis of x. We shall still have PM_AM PM x-a OB~AO ' °^ b ~ a ' Now since P is below the axis of x, its ordinate y is a negative quantity, hence we must not put PM = y but PM'= — y, because by PM we mean a certain length esti- mated positively. Thus — T = , and therefore, as before, - + ^ = 1. a a Oblique Co-ordinates. 23. Eqiiation to a straight line. We shall denote the inclinatioji of the axes by w. Suppose first, that the straight line is not parallel to either axis. Let ABB be a straight line meeting the axis of y at B, Draw a straight line OE through the origin parallel to ABD. In ABB take any point P; draw PM parallel to Y, meeting OX at M, and OE at Q. Suppose OB = c, and the angle QOM = a. Let X, yhe the co-ordinates of P; then y=PM=PQ + QM= OB + QM. EQUATION TO A STRAIGHT LINE. 19 _, , QM sin a ., , _,. a; sin a But -^Tr>= -: — 1 r; therefore QM= -. OM sin (w — a) ' "» — gj^ (» — «) ' Hence the required equation is _ X sin a 2'~sin(a)-a)"'"*'- If we put m for -: — -. r- the equation becomes ^ sin (co — a) ^ y = mx + c, as in Art. 14. The meaning of c is the same as before ; m, is the ratio of the sine of the inclination of the straight line to the asis of x to the sine of its inclination to the axis of y. Since sin a is always positive, m will be positive or negative according as sin (w — a] is positive or negative; thus, as before, m will be positive or negative according as the straight line through the origin parallel to the given straight line falls between OY and OX, or between OY&ndi OX'. The mean- TT ing of m coincides with that in Art. 14 when <» = „ . for then m = tan a. „. sin a. 24. Smce m = - sin (&) — «)' therefore m (sin a> cos o — cos to sin a) = sin a ; therefore m (sin a> — cos oi tan a) = tan a ; ^, . . msinw \ therefore tan a = Hence sin a — 1 + rni cos «a " msinm cos a = - + V(l + 2m cos to + to") ' 1 + m cos r sm a) rr > thus —1— = - =»"• ( n For the equation to the straight line it is sufficient to put — = — = , but it is useful to remember that each of b n these quantities is equal to r. The constants I and n are connected by a certain con- dition. For, by Art. 9, {x-hy+ (y- A)* 4- 2 (a! -A) {y - k) cos a> = r' ; substitute £oi x—h and y — k: thus I' + r? + 2ln cos a>=l. If the axes are rectangular, I and n become respectively cos a and sin a, that is, the cosines of the inclinations of the straight line to the axes of x and y respectively. In the preceding figure P falls to the right of Q and x — h is positive. If P were to the left of Q then x — h would be negative. Thus since x — h = lr, the product Ir must be capable of changing its sign ; this leads us to consider r as positive or negative according to circumstances. When there- fore we write the equation to a straight line under the form x—h_y—k and ascribe to I and n the values given above, we conclude that each of the expressions — j— and ^ is nvmerically equal to the distance between the point {h, k) and the point (x, y), but that the sign of each expression will depend upon the relative positions of the two points. 22 POLAK EQUATION TO A STRAIGHT LINE. Polar Co-ordinates. 28. Polar equation to a straight line. Let AB be a straight line, OQ the perpendicular on it from the origin, OX the initial line, P any point in the straight line. Suppose OQ=p, and the angle QOX=a. Let r, 6 be the polar co-ordinates of P ; then OQ = OP cos POQ; that is, p = r cos {d — a). This is the polar equation to the straight line. 20. The polar equation may also be derived from the equation referred to rectangular co-ordinates. Let Ax + By + G=0 be the equation to a straight line referred to rectangular co- ordinates. Put r cos 6 for x, and r sin d for y, Art. 8 ; thus Ar cose + Br sai.e+C=0 (1) is the polar equation. This equation may be shewn to agree with p=r cos {6 — a) (2). For by Art. 21 we have -ji = cot a and -^ = — r*-— . B .0 sma FORMS OF THE EQUATION TO A STRAIGHT LINE. 23 Hence (1) becomes cot a r cos ^ + r sin ^ — P— = 0. sin a which agrees with (2). 30. The equation to a straight line passing through the origin is, by Art. 14, Put r cos 6 for x and r sin 6 for y; the equation then becomes r sin 5 = mr cos 6 ; therefore tan = m; therefore 6 = a, constant ; this is therefore the polar equation to a straight line passing through the origin. 31. We will collect here the different forms of the equa- tion to a straight line which have been investigated, y = mx + c. Arts. 14 and 23. X = constant, or, y = constant. Arts. 15 and 25. -+^-1 = 0, Arts. 17 and 25. a X cos a + ysma. —p = 0, Art. 20. y = -7—f r x+c. Art. 23. ^ sm ( = tan ( C4X - CDX) _ tan CAX— tan CDX _ m^ — m^ ~ 1 + tan GAXtAn. CDX ~ 1 + m^m^ From this we may deduce 1 + m^^ cos A CD = sin j4 CD =-7T7i jT-n ST, . V{(1 +0(1+01 42. To yired the form of the equation to a straight line which is perpendic/uiar to a given straight Une. Let y = mx + c be the equation to the given straight line, and y = m'x + c' the equation to another straight line. Then the tangent of -the angle between these straight lines is 1 + mm ' If these straight lines are perpendicular to each other, 1 + mm' = 0: therefore m' = . m Hence « = f- c' ^ m represents a straight line perpendicular to the straight line y = mx + c. PEBPENDICULAB TO A GIVEN STRAIGHT LINE. 31 thus, 43. The result of the last Article may also be obtained Let AB be the given straight line, so that tan BAX=m. Let CD be a straight line perpendicular to AB; then tan I>CX= - tan DCO = - cot £.4 =-- . m Hence the equation to CB is where ^ m c'=OD. 44. To find the equation to the straight line which passes through a given point, and is perpendicular to a given straight Let «,, y^ be the co-ordinates of the given point, and y = 7na!+c (1) the equation to the given straight line. The form of the equation to a straight line through {ai^, y^ is y-y^ = m'{x-x;). (2). If (2) is perpendicular to (1), we have mm + 1 = 0. Hence the required equation is 32 STRAIGHT LINES WHICH MAKE 45. To find the equations to the straight lines which pass throutgh a given point a/nd make a given angle with a given straight line. Let AB be the given straight line ; C the given point ; h, h its co-ordinates ; yS the given angle. Let the equation to ^5 be Suppose CD and CE the two straight lines which can be drawn through C, each making an angle /3 with AB. Then tan ODX= tan (BAX +B)= "' + ^f^^ 1— mtan/3 tan GEZ= - tan CEA = - tan O - BAX) = "^-^°^ . ' l+mtan/3 Hence the equation to CD is , m + tan iS , , . "^ 1— mtanyS^ " and the equation to CE is , m — tan fi , , . •^ l + mtan/8^ ' 46. The following particular cases of the preceding results may be noted. (1) Suppose m = 0; then the given straight line is parallel to the axis of a;. The required equations then are y — k = tan /8(a; — h), and y — k = —iaxi ^{x — h). A GIVEN ANGLE WITH A STRAIGHT LINE. 33 (2) Suppose m = 00 ; then the given straight line is parsJlel to the axis of y. And since , . - l + -tan/3 m + tan p m 1— mtan/8 1 ^ _' tan a m we have when «t = oo , and therefore — =0, for the equation to CD, y -h = - ^^^{x-h) = - cot p(a>-h). Similarly the equation to CE hecomes y—k = coty9 {x —h). (3) Suppose m = tan /3. In this case the equation to CD hecomes y — k = =-—7 — Vo {""— A), that is, y—k= tan 2^{x—h). The equation to CE hecomes y — k = 0, so that CE is parallel to the axis of x. (4) Suppose m = cotfi. The equation to CD may he written in the form (y —k)(l — m tan JS) = (»H- tan ^) (x — h), and we see that when m — cot/S the left-hand side is zero ; thus the required equation is then x — h = 0. r^ ■ ^-rr, T cotiS — tanfi , ,, The equation to CE becomes y — k = „ {x — n) cos'/S — sin'/8 . ,, .oof i\ = s — „ ■ — ^ (x—h)= cot 28 (x — h). 2cos/3sm/8 ^ ' a- v / (5) Suppose m = — tan j8. Then the equation to CD becomes y—k = 0; and the equation to CE becomes 2^ - A =f|^ (^ - A) = - tan 2yS (a; - A). (6) Suppose TO = — cot /3. Then the equation to CD becomes y-k= ^^P~^" ^ (a; - A) = - cot 2^ (a; - h). The equation to CE may be written in the form (y — fc) (1 + m tanyS) = (m - tan )8)(a; — A), T. c. s. 3 S4 EQUATIONS TO CERTAIN STRAIGHT LINES, and we see that when m — — cot/8 the left-hand member is zero ; thus the required equation is then x — h = 0. (7) Suppose /3 = ^ . The equation to CD may be written , wcot/8+1 , ,, y-k = — —pi (X — h). When /3 = 5 we have cot /8 = ; thus the equation becomes y — k=^ (x — h). Similarly the equation to CE takes the same form ; and thus the result agrees with that of Art. 44. We have discussed these particular cases as an example of the manner in which the student should test his comprehen- sion of the subject by applying the general formulae to special examples. He will find it useful to illustrate these cases by figures. 47. To find the length of the perpendicular drawn from, a given point on a given straight line. Let AB be the given straight line ; D the given point ; h, le its co-ordinates. Let the equation to A£ be y = mx + c. .(1). LENGTH OF A PEBPENDICULAB. 35 The equation to the straight line through D perpendicular to J. £ is, by Art. 44, y-k = -^{x-h) (2). Let iTj, y^ be the co-ordinates of ^; then, by Art. 9, I)E' = (x,-hy + (i,,-kf (3). It remains then to substitute for a;, and y^ their values in (3). Now, since x^ , y are the co-ordinates of E, which is the point where (1) ana (2) meet, we have y, = «wc,-|-c, and yi-h=- — {x^-h); therefore mx, + c = & (x.—h), and x, = — z = — : , mk — m?h. — mc m ,, , . thus x, — h= =— — 3 = r- — „(k — mh—c). Also 2/1— *= — ::r K~^)=-^r~; — s— 5 •'I m ^ ' 1 + m therefore by (3) „™ m' ., 7. N2 , {h-mh-cf (k-mh-cY „ ^ „ A; — wiA. — c Hence x/A = ..., , — ^t- . V (1 + mr) The radical in the denominator may be taken with the positive or negative sign, according as the numerator is posi- tive or negative, so as to give for DE a positive value. We may also obtain the value of DE thus : draw the ordi- nate BM meeting the straight line AB at H ; then DE = DE sin DEE = DE cos EAM. Now OM = h ; therefore EM =mh + c, and DM = k ; therefore DE = k — mh — c. Also t&nEAM = m: therefore coaEAM = —-:r-, — ^; V (1 + »* ) .1. c T\r> k — mh — c therefore DE=-rpr-^ — ^. 3—2 36 LENGTH OF A CERTAIN STRAIGHT LINE. Hence if on the straight line y — 'mx — c = a perpen- dicular be drawn from the point (Aj, A,) and also a per- pendicular from the point (A„ k^, the ratio of the length of the first perpendicular to the length of the second is equal to the numerical ratio of k^ — mh^ — c\x>k^ — mh^ — c, 48. Tojvnd the length of the straight line drawn from, a given point in a given direction to meet a given straight line. Let (h, k) be the given point ; and suppose a straight line drawn from this point at an inclination a to the axis of x to meet the straight line Ax + By+C=0 (1). Let r be the required length ; x^, y, the co-ordinates of the point where the straight line drawn from (A, k) meets (1) ; then, by Art. 27, x^ — h = rcosa, y, — i = r sin a (2). But (x,, y,) is on (1), therefore A (A + rcosa) + .B(& + rsina) +C7= 0; Ah + Bk + G therefore r= — j1 cos a +5 sin a" 49. In this Chapter we have used equations of the form y = mx + c to represent straight lines. The student may exercise himself by solving the problems by means of the more symmetrical forms of the equation to a straight Une, Ax + By + = 0, - + ^—1=0, .r cos a + y sin a — p = 0. The results of course can be easily compared with those we have obtained. For example, if in Art. 47 we represent the given straight line by the equation Ax+By+G=0, the result obtained should coincide with the value of ,.- , — jr when we V(H-wi') A G write — o for m, and — „ for c ; that is, the result must be JO Jo Ah + Bk+C RXTLE FOR TBANSFOSMINQ AS EQUATION. S7 Similarly, if the given straight line be represented by the equation a; cos a + y sin a —p = 0, we shall find for the length of the perpendicular on it from {h, k) ± {h cos a 4- A sin a —p). Thus if the equation to a straight line be in the form a; cos a+y siaa.—p = 0, the length of the perpendicular drawn from a point on this straight hne is the numerical valiie of the eccpressian on the left-hand side of this equation, when for x and y are substituted the co-ordinates of the given point. This result is of such great importance that we shall proceed to examine it more closely. 50. We may however previously observe that if the equation to a straight line be given in any form, we can immediately transform it so that it may be expressed in terms of the length of the perpendicular from the origin and the inclination of this perpendicular to the axis of x. For ex- ample, suppose the equation to be 2a: + 3y + 4 = 0. Change the sign of every term so that the last term may be negative ; thus the equation becomes - 2a; - 32^ - 4 = 0. Divide by VIS" +3»); thus _-^__^ ^-0 V13 713 Vi3 ~ This is of the form a; cos a + y sin o — |) = 0, andcosa = -^, sina = -^^3, p = ^^. In this example a. is an angle lying between tt and -^ . Any other example may be treated in a similar manner, the rule being the following. Collect the terms on one side, and, if necessary, change the signs so that the equation may 38 EULE FOR TRANSFOEMma AS EQUATION. be in the form Aa; + By—C = 0, where G is positive ; then divide by ,t/{A' + B'); thus the equation becomes ^^ I % _o. ^{A' + B')^ ^/(A' + B') V(^° + B') " ' this is of the required form, and A . B G ''*'^"~vp^+^)' ^™"~v(3Mrs^)' ^"iji^n^y Thus every equation representing a straight line may be brought to the form a; cos a + y sin a— ^ = 0, where p is a positive quantity, unless the straight line passes through the origin, and then ^ = 0. When we use the equation x cos a + y sina — p = we shall always suppose p positive. 51. The straight line whose equation is X cos a + y sin a—p = might be called " the straight line ( p, a)," since the constants p and a. determine the straight line ; but when there is no risk of confounding it with another straight line, it may be more shortly called "the straight line a," and the equation may be expressed shortly, thus, "a = 0." We shall now give another investigation of the expression for the length of the perpendicular from a given point on the straight line {p, a). X Let AB represent the straight line (p, a), the origin, P the point {x, y), so that P and are on opposite sides of AB. LENGTH OF A PEftPENDICULAR. 39 Draw OQ, PZ perpendicular to AB, and PM parallel to OF; through M draw a straight line parallel to AB, meeting OQ and PZ, produced if necessary, at Q' and Z' respectively. Then OQ' = OM cosa = a; cosa ; PZ' = PM sina = y sina ; PZ=OQ' + PZ'-OQ = xcosci + i/sma-p. If P and be on the same side of AB we shall obtain for the length of the perpendicular p — X cos a — ysiaa. It will be found that these results will hold for aU varieties of the figure. 52. Or we may proceed thus. Let X cosa + y sina— ^ = (1) be the equation to a straight line, and let x, y be the co- ordinates of the point from which a perpendicular is drawn on the straight line : it is required to find the length of this perpendicular. The equation to any straight line which is parallel to (1) and on the same side of the origin, may be written thus X cosa+y sina— j)' = (2), where p is the perpendicular from the origin on this straight line. If this straight line pass through the point (a;', y), we must have sd cosa + y sina — ^' = ; therefore "p = a;' cos a + ^ sin a. The length of the perpendicular from («', y') on (1) will be p'—pi{ the point and the origin are on different sides of the straight line, and p—p' if they are on the same side; that is, x' cos a+y'sina—p in the former case, and in the latter case p — x' cos a—y sin a. K the straight line parallel to (1) be on the opposite side of the origin, its equation will be X cos {ir + a)+y sin {ir + a)—p'= 0, 40 LENGTH OF A PERPENDICULAR. where 'p' is the length of the perpendicular from the origin on it. If this straight line pass through the point {x, y) we must have X cos a + y' sin a +y = ; therefore p = — x' cos a — y sin at. The length of the perpendicular from {x, y) on (1) will be the sum of js' and p, that is, p — x' cos a — y' sin or. We may now suppress the accents on x and y, and we have the same conclusion as before. 53. Thus the length of the perpendicular from the point {x, y) on the straight Gne X cos T-r , „ ^ , . 1 4- TO, cos « 1 + TO, cos O) Hence tan a or tan (a, — a.) = ■ t •^ \ 2 1/ mm. sin' m (1 + TO, cos + m^m^ ' Hence - the condition that the straight lines may be at right angles is 1 + (»n, + m^ cos w + to,TOj, = 0. 57. To find ike length of the perpendicular dra/um from a given poirH on a given straight line. We shall proceed as in the latter part of Art. 47; the student may also obtain the result by the method in the former part of that Article. Let AB be the given straight line; D the given point; h, k its co-ordinates. Let the equation to AB be y = mx + c. Draw DHM parallel to Y, and DE perpendicular to AB; then DE=DHwi.DHB. POLAR CO-OHDINATES. 43 Now BH = DM-HM = k-{mh + c)=h-mh-c. Let BAX = a, then DEE or AHM= ca-a, and-: — -. r = m (Art. 24): sin (q) — a) ^ ' therefore sin(6> — o) = = -77^ — ^ ; — s-, (Art. 24) ; ^ ' m V(l + 2»n cos ca + m) ^ therefore D^= jf " ^^ " '^^ ^^ " ,, . V(l + 2m cos a + m) If a straight line be drawn from J) to meet ^5 at an angle ;8, its length will be DE cosecfi, and wiU therefore be known since DE is known. If the equation to a straight line be in the form given in Art. 26, namely, x cos a + y cos ^ —p = 0, the length of the perpendicular on it from the point (x', y) will be + [x cos a+y cos /3 —p). This may be deduced from the preceding expression, or it may be obtained in the manner of Art. 51. Polar Co-ordinates. 58. To find tJie polar equation to the straight line which passes through two given points. Let r,, ^, be the co-ordinates of one point; and r,, 0^ those of the other ; and suppose the equation to the straight line rcos(^ — a)=_p, that is, rcos^cos a + r sin 0sin a=p (1). Since this straight line passes through the two points, we have r, cos ^j cos a + »*i sin 0^sma=p (2), Tj cos ^j cos a + T-j sin 0j sin a =p (3). From (1) and (2) (r cos — r^ cos 0^) cos a + (r sin ^ — r, sin 0^) sin a = 0...(4). From (2) and (3) (r^ cos ^j-r,cos0,) cos a + (r,sin 0^— r^sin 5,) sin a = 0. . .(5), 44 LOCI CONSISTINO OF STRAIGHT LINES. , , r COS ^ — r, COS ^, _ r sin ^ — r, sin 6^ r, cos U^ — r^ cos 6', r^ sm o^ — r^ sm p, After reduction we obtain rr, sin (^, - 0) + r,r, sin {6^ - 0.) + r,r sin (0 - ^,) = 0. . . (6). This equation has a simple geometrical interpretation ; for if we draw a figure and take for the origin and A, B, P for the points (r,, ^,), (r-j, 6^, (r, ^), respectively, we see that equation (6) is the expression of the fact that one of the tri- angles OAP, OBP, OAB, is equal in area to the sum of the other two. 59. We have seen that a straight line is the locus of an equation of the first degree; as we proceed it will appear that if an equation be of a degree higher than the first, the cor- responding locus will be generally some curve; we may notice here some exceptional cases. Suppose the equation a;' — 4aa; -1- 4a' + y" = be proposed ; this equation may be written (j;-2a)'-l-y' = 0. Hence we see that the only solution is y = 0, x = 2a. Thus the corresponding locus consists only of a single point on the axis of fc at a distance 2a from the origin. Again, suppose the equation to be a;'-f-y + l = 0. No real values of x and y will satisfy this equation ; in this case then there is no corresponding locus, or as it is usually expressed, the locus is impossible. Thus, the locus corresponding to a given equation may reduce to a single point, or it may be impossible. 60. We have seen that the equation to a single straight line is always of the first degree ; an equation of a higher degree than the first may however represent a locus consist- ing of two or more straight lines. For examjde, suppose ^'-^ = (1); therefore y = x (2), or y = — x (3). LOCI CONSISTING OF STRAIGHT LINES. 45 If the co-ordinates of a point satisfy either (2) or (3), they ■will satisfy (1) ; that is, every point which is comprised in the locus (2) is comprised in (1), and every point which is comprised in (3) is also comprised in (1). Hence (1) repre- sents two straight lines which pass through the origin, and make respectively angles of 45° and 135" with the axis of x. Again take the general equation of the second degree be- tween two variables aa^ + bxy + cy^ + da;+ey+f=0; and let us determine when it represents two straight lines. We haxe cy' + {bx+ e) 7/ + ax' + cb;+f=0. Hence con- sidering this as a quadratic equation in y, and solving in the usual way we obtain bx + e ^{(bx + ef -^cjaaP + dx +/)} ^ 2c - 2c The expression under the radical sign is {b* - 4ac) a;* H- 2 {be - 2cd) x + e"- 4c/; if this expression is an exact square with respect to x it is obvious that the proposed equation of the second degree breaks up into two equations of the first degree between X and y, and so represents two straight lines. The condition which is necessary and sufiScient to ensure that the expression under the radical sign is a perfect square with respect to x is, by Algebra, Chapter xxii, (be - 2cdf = (6* - 4ac) (e' - 4c/), 61. An equation which involves only one of the varia- bles, represents a series of straight lines parallel to one of the axes. Thus, if there be an equation f(x) = 0, where /(a;) denotes any expression which involves x and known quantities, we obtain by solving it a series of values for x, as x = a^, x=a^, and each of these equations represents a straight line parallel to the axis of y. Similarly f(y) = Q represents a series of straight lines paraJlel to the axis of x. An equation of the form Jr (-] = represents a series of straight lines passing through the origin ; for by solving the 46 LOCI CONSISTING OF STRAIGHT LINES. V V V equation we obtain a series of values for - , as - = to, , - = TOj, . . . and each of these equations represents a straight line passing through the origin. Of course if an equation / {x) = 0, f{y) = 0, or / (-) = have no real roots, the corresponding locus is impossible. The equation Ay' + Bxy + Co? = may be put in the form Since this is a quadratic in - we obtain two vjJues for it, a; suppose — = m, and - = m^; hence the equation generally represents two straight lines passing through the origin. If B' be less than 4-4(7, then «», and m, are impossible, and the only solution of the given equation is as = 0, ^ = ; that is, the locus is a single point, namely, the origin. 62. It is obvious that if the locus represented by an equation f{x, y) = passes through the origin, the values a; = 0, y = must satisfy the equation. We can thus imme- diately determine by inspection whether a proposed locus passes through the origin or not. 63. In Art. 39 we determined the co-ordinates of the point of intersection of two given straight Lines : the pro- position may obviously be generalised. Let /, («, y) = 0, ft (*i y) = 0, represent two curves, then the co-ordinates of the points where they meet wiU be determined by solving these simultaneous equations. It may be shewn that if one equation be of the m" degree and the other of the n"" degree, the number of common points cannot exceed mn. (See Theory of Equations, Chapter xx.) 64. We win exemplify the Articles of this Chapter by applying them to demonstrate some properties of a triangle. The straight lines drawn, from the angles of a triangle to the middle points of the opposite sides meet at a point. PBOFEBTIES OF A TBIANGLK 47 Let ABC he a triangle, JD, E, F the middle points of the sides ; take A for the origin, AB for the direction of the axis A F B X of X, and a straight line through A at right angles to AB for the axis of y. Let AB = a, and let «', y be the co-ordinates of C. Since D is the middle point of GB, the abscissa of D is J («' + a) and its ordinate ^ (Art. 10) ; since E is the middle x' v' pomt of ^(7, the abscissa of .E" is-^ and its ordinate ^; since J" is the middle point of AB, its abscissa is ^ and its ordi- nate zero. Hence by Art. 35, n§ erf the equation to AD is y = -r- — (1); the equation to BE is y = ^ , ~ (2) ; bC ^ I2CK the equation to CF is v= ^ } , ~ (3). ^ " za; — a ^ ' To find the point of intersection of (2) and (3) we put y'(x-a) ^ y'(2x-a) X — 2a 2x' — a ' therefore (x — a) {2x' — a) = (2a; — a) (x' — 2a) ; therefore 3aa; = a{x' + a); therefore x = ^(x' + a). Substitute this value of a; in (2) and we find y =^ . o We have thus determined the co-ordinates of the point of intersection of (2) and (3) ; moreover we see that these values satisfy (1) ; hence the straight liue represented by (1) passes through the intersection of the straight lines repre- sented by (2) and (3), which demonstrates the proposition. 48 PBOPEKTIBS OF A TRIANGLE. The straight lines drawn from the angles of a triangle ■perpendicular to the opposite sides meet at a point. The equation to BG is (Art. 35) y' / X hence the equation to the straight line through A perpen- dicular to BG is (Art. 44) y — "^- W- v' The equation io AG is y = -,x; hence the equation to the straight line through B perpendicular to .4(7 is y = -^(«-a) (5). if The straight line through G perpendicular to AB will be parallel to the axis of y, and its equation will be (Art. 15) x = x' (6). Now at the point of intersection of (5) and (6) we have and as these values satisfy (4), the straight line represented by (4) passes through the intersection of the straight lines represented by (5) and (6). The straight lines drawn through the middle points of the sides of a triangle respectively at right angles to those sides meet at a point. The equation to the straight line through D at right anarles to BG is -i-'^i^-H^) w to the straight line through E at rigl ^-i-fH) <»)• The equation to the straight line through E at right angles to GA is PEOPEETIES OF A TRUNGLE. 49 The equation to the straight line through F2A, right angles to AB is -=i (9)- Now at the point of intersection of (8) and (9) we have ."' ..-V ' _« /a so\ these values satisfy (7) ; hence the straight lines represented by (7), (8), and (9), meet at a point. Let us denote by P the point of intersection of the three straight lines in the first proposition, by Q the point of inter- section of the three straight lines in the second proposition, and by R the point of intersection of the three straight lines in the third proposition ; we will now shew that P, Q, and R lie on one straight line. The co-ordinates of P are flj = ^ {x +a), y = '^; x' of Q are x = x', y=—,[a — x) ; ofiJarea;=^, y = =__^_^^ Hence the equation to the straight line passing through P and Q is y' y-(^-^')-3 / X-^dX y 3-a;'-J(a!'+a)r 3 ) ^^"^• In this equation put a; = ^ , then ^' / '\ y' y-|= 'H2.--a) (6-3j = -^{F^°-"^-|r , x'{a-x') , y' , y y' x'{a-x') therefore y ^-^ + | + |=| ^7— Hence the point R is on the straight line represented by (10), for the co-ordinates of iJ satisfy (10). T.C.S. .. 4 ( 50 ) EXAMPLES. 1. Find the equations to the straight lines which pass through the following pairs of points: (1) (0,1), and (1,-1). (2) (2, 3), and (2, 4). (3) (1, 1), and (- 2, - 2). (4) (0, - a), and (0, - b). 2. Find the equations to the straight lines which pass through the point (4, 4) and are inclined at an angle of 45° to the straight line 7/ = 2a!. 3. Find the equations to the straight lines which pass through the point (0, 1) and are inclined at an angle of 30" to the straight hne y + x=2. 4. Find the equations to the straight lines which pass through the origin and are inclined at an angle of 45° to the straight hne x=2. 5. Find the equations to the straight lines which pass through the origin and are inclined at an angle of 60° to the straight line x + y>JS = 1. G. Find the angle between the straight lines x + y = 1, y = x + 2; also. find the co-ordinates of the point of intersec- tion. 7. Find the angle between the straight lines x+y\J^ = and x—yiJZ = 2. 8. Find the angle between a; + 3y = 1 and a; — 2^ = 1. 9. Find the equations to the straight lines passing through a given point in the axis of x, and making an angle of 45° with the axis of x. 10. Find the equation to the straight line which passes through the origin and is perpendicular to the straight line a; + y = 2. 11. Find the perpendicular distance of the point (1,-2) from the straight line a; + y — 3 = 0. 12. Find the length of the perpendicular from the point (a, J>) on the straight line - + ^ = 1. EXAMPLES. CHAPTER III. 51 13. Find the co-ordinates of the point of intersection of the straight lines - + f = 1 and t + - = 1- a a 14. Fiad the equation to the straight line which passes through the point (a, b), and through the intersection of the straight lines - + | = 1 and r + - = 1. a a 15. Shaw what loci are represented by the equations : (1) a? + y'^.Q, (,2) a?-f=0, (3) x' + xy^Q, (4) xy^Q, (5) ^+f + a^±:^0, (6) x{y-a) = 0. 16. Interpret the .equations : (1) {x-a){y-'h)=0. (2) (:x-aY + (jj-h)'=0, (3) {x-y + aY+{^x + y-ay=Q. 17. Determine what straight lines are represented by the equation y' — 4ixy + Sa;' = 0. 18. Shew that Sj/" — Sa-y — 3a;' + 30ar — 27 = represents two straight lines at right angles to one another. 19. Find the equations to the diagonals of the four-sided figure, the sides of which are represented by the equations x=4t, y = o, y=x, ,y = 2x. 20. ABGDEF is a regular hexagon; take A for the origin, AB as axis of x, and a straight line through A at right angles to AB as axis of y : find the equations to all the straight lines joining the angular points of the hexagon. 21. Given the co-ordinates of the angular points of a triangle, find the equation to the straight line which joins the middle points of two sides. 22. Find the tangent of the angle between the straight lines y — True = and my + a; = 0, when referred to oblique axes. 4—2 62 EXAMPLES. CHAPTER III. 23. Shew that whether the axes be rectangular or oblique the straight lines y + a; = and y — x = are at right angles. 24. Given the lengths of two sides of a parallelogram, and the angle between them, write down the equations to the two diagonals and find the angle between them ; taking one of the comers as origin, and the two sides which meet at that comer as axes. 25. In the figiire to Art. 75, take BA and BG as the axes of x and y ; suppose BA = a, BG = c ; and let h, k be the co-ordinates of D : then form the equations to AG, BD, AD, CD. 26. With the notation of the preceding Example, find the co-ordinates of the middle point oi AG and those of the middle point of BD, and form the equation to the straight line passing through these two points. 27. With the same notation find the co-ordinates of the middle point of EF, and thus shew that this point lies on the straight line joining the middle points oi AC and BD. 28. If --|-r= 1 and — , -j-f, =1 be the equations to two a b a b ^ straight lines, which with the co-ordinate axes (rectangular or oblique) contain equal areas, and ai', y' be the co-ordinates of the point of their intersection ; shew that y'_&-y X a —a 29. Determine what points on the axis of x are at a per- pendicular distance a from the straight line - -1- ? = 1. 30. Form the equation for determining the abscissa of a point, in the straight line of which the equation is - + f = 1, a b whose distance from a given point (a, fi) shall be equal to a. given straight line c. Shew that there are in general two such points, and in the particular case in which those points coincide c\a' + b') = {a^ + b2-ab)\ 31. Find the tangent of the angle between the two straight lines represented by the equation Aj^ + Ban) + Gx* = 0. EXAMPLES. CHAPTER III. 53 32. Find the points of intersection of the straight lines a; + 2y-5 = 0, 2x + y-7 = 0, and 2/-a;-l = 0; and shew 3 that the area of the triangle formed by them is ^ . 33. The area of the triangle formed by the straight lines y = x tan a, y — x tan fi, ^ = a; tan 7 + c, c* sin (a — /8) cos' 7 2 sin (a — 7) sin (/8 — 7) " 34. Given the equations to two parallel straight lines, find the distance between them. 35. Determine the angle between the straight lines - = 4cos^ + 3sin^, - = 3cos0 — 4sinft r r 36. Interpret F(ff) = 0; forexample, sin 3d = 0. 37. If the axes be inclined at an angle eo, the condition that the straight lines Ax + By + C = 0, A'i = or Wj = 0. Suppose we take r, = 0. CO ABRIDGED NOTATION FOR STRAIGHT LINES. Then from the second and third of equations (2) we deduce either that w, = or else that w, = and f , = 0; in the former case the three given straight lines all pass through the point (a:,, y,); in the latter case the straight lines v = and w = both pass through the two points (a;,, y^) and (x^, y^), that is, two of the given straight lines coincide so that all three will reduce either to two intersecting straight lines or to two parallel straight lines. Suppose we take «;,= in conjunction with w, = 0. Then the straight line w = passes through the given points («,, y^ and (a;,, y^. From the third of equa- tions (2) we have — = -* : and thus the straight lines m = «2 ''2 ... . . and « = either meet on the straight Ime joinmgthe points (a;,, yj and (x^, y,), or are parallel to this straight line; that is, the straight hues m = 0, w = 0, and «; = either meet at a point or are parallel. 70. Let a = 0, yS = be the equations to two straight lines expressed in terms of the perpendiculars from the origin and their inclinations to the axis of x (see Art. 50), so that a is- an abbreviation for x cos a + y sin a — p,, and ^ is an abbre- viation for iccos^ + ysinyS— ^,; we proceed to shew the meaning of the equations a — ^ = and a + ;8 = 0. Let SA be the straight line a = 6, and SB the straight line ABRIDGED NOTATION FOR STRAIGHT LINES, 61 15 = 0; let SC bisect the angle A8B, and SD bisect the sup- plement of A SB ; the angle DSG is therefore a right angle. Take any point P in SC and draw the perpendiculars PM, PJT on 8A, SB respectively. If a;, y be the co-ordinates of P, the length of PM is a by Art. 54, and the length of PN is p. Since SC bisects the angle ASB, PM = PN; therefore for any point in SG we have /3 = a ; that is, the equation to /SC7isa = /3. Similarly, the equation to SD is a = — /S. Thus a — yS = and a + ^ = represent the two straight lines which pass through the intersection of a = and /8 = and bisect the angles formed by these straight hnes. The student must distinguish between the straight lines a — ;8 = and a + /8 = 0; the following rule may be used: the two straight lines a = 0, /3 = 0, will divide the plane in which they lie into four compartments ; ascertain in which of these compartments the origin of co-ordinates is situated ; o — /8 = bisects that angle between a = and /S = in which the origin of co-ordinates lies. This is obvious from the in- vestigation in the present Article and the remarks made in Arts. 53, 54. The equation a + Xj8 = represents a straight line such that X is numerically equal to the ratio of the perpendicular from any point of it on a = to the perpendicular from the same point on /S = 0. If \ is posit(ive the straight line a-(-X/3=0 lies in the same two of the four compartments just alluded to as the straight line a + ^ = ; if X be negative the straight line a + X/8 = lies in the same two compart- ments as the straight line a — /8 = 0. From the figure we see that PM = PS sin PSM, and PN = PS sin PSN; hence X or PM sin PSM ., , . ^ . V, i- ir J.! ■ e ■^;rrr = — — TTrrrr; that IS, X expresses the ratio oi the sme of PN sm PSN' ^ the angle between a == and a + X/3 = to the sine of the angle between j8 = and a -1- X^S = 0. 71. We shall continue to express the equation to a straight line by the abbreviation a= when the equation is of the form a; cos a + y sin a — jj = ; when we do not wish to 62 EXAMPLES OF ABRIDGED NOTATION. restrict ourselves to this form, we shall use such notation as tt = 0, v = 0, m' = 0, Let M = 0, v = be the equations to two straight lines, the axes being rectangular or oblique ; then' u — Xv = Q and M 4- Xt) = represent two straight lines passing through the intersection of the first two. Suppose, asin Art. 70, that SA, SB are the first two straight lines and SC, SD the second two ; then will sin CSA ^ sin PSA sin CSB sin JDSB' For by Art. 57 it appears that if p be the perpendicular from a point (x, y) on the straight line m = 0, then p = /am, where /u. is a constant quantity ; similarly if p denote the per- pendicular from the same point on ?; == 0, then p = /i'w, where /u.' is a constant quantity. Hence the equation m — \o = 0, or 4- = shews that — , = -7- ; thus we see that numerically without regard to algebraical sign sin CSA _ Xfi sin CSB ~-/7 ■ „. ., , sin DSA \fi sin CSA sin BSA therefore sinCSB~sitLl)SB' 72. We will apply the principles of the preceding Arti- cles to some examples. Let 2 = 0, (8 = 0, 7 = be the' equations to three straight lines which meet and fdrm a triangle, and suppose the origin of co-ordinates within the triangle ; then the equations to the three straight lines bisecting the interior angles of the triangle are, by Art. 70, ;8-7 = 0...(l); 7-a = 0...(2); a-/3 = 0...(3). These three straight lines meet at a point; for it is obvious that the values of x and y which simultaneously satisfy (1) and (2) will also satisfy (3). EXAMPLES OP ABRIDGED NOTATION. 63 Again the equations to the three straight lines which pass through the angles of the triangle and bisect the angles supplemental to those of the triangle are /3 + 7 = 0...(4); 7 + a=0...(5); a+/3 = 0...(6). It is obvious that (3), (4), and (5) meet at a point ; simi- larly (5), (6), and (1) meet at a point; so likewise (4), (b"), and (2) meet at a point. In all our propositions and examples of this kind, we shall always suppose the origin of co-ordinates within the triangle, unless the contrary be stated. 73. If a = 0, /? = 0, 7 = be the equations to three straight lines which form a triangle, then any straight line may be represented by an equation of the form la + «iyS -I- W7 = ; for the exceptional cases noticed in Art. 69 cannot occur here. Let a, b, c denote the lengths of the sides of the triangle which form parts of the straight lines a = 0, /3 = 0, 7 = re- spectively. Take any point within the triangle and join it with the three angular points ; thus we obtain three triangles the areas of which are respectively — -9- , — 5- , and — -^ . Hence aa + 6/8 -1- 07 = a constant ; the constant being in fact twice the area of the triangle taken negatively. This result holds obviously for any point within the tri- angle determined by a = 0, /S =^0, 7 = 0. It will be found on examining the different' cases which arise that it is also true for any point without the triangle. Hence it is uni- versally true. Suppose we require the equation to a straight line par- allel to the straight line la + m^ -M17 = 0. This required equation may be written la +m^+ 717+ k= 0, where i is a constant. (Art. 38.) Or, since aa -I- 6/9 + 07 is a constant, the required equation may be written, Za -f- ?n/3 -J- M7 + k' {aa + 6/8-1- 07) = 0, where k' is a, constant. 64 EXAMPLES OF ABRIDGED NOTATION. 74. The straight lines represented by the equations u = 0,v=0,w = 0, will meet at a point, provided lu+ mv + ntv is identically = ; I, m, n being constants. For if lu + mv + nw = identically, we have w = always. Hence the equation w = may be written = 0, that is, the straight line w = is a straight line passing through the intersection of m = and v = Q. 75. The following example will furnish a good exercise in the subject. li A Let ABCD be a quadrilateral ; dra^^^the diagtoals AC, BD ; produce BA and CD to meet at E, aSid ^|0Vd BC to meet at F; join EF, forming what is calfed^he^ fAird diagonal of the quadrilateral. Suppose u = 0, the equation to AB, (1), t;"=0, the equation to BC, (2), w = 0, the equation to CD, (3). We propose to express the equations to the other straight lines of the figure in terms of u, v, w, and constant quan- tities. Assume for the equation to BD lu — mv = Q (4), and for the equation to OA mv — nio= (5). EXAMPLES OF ABRIDGED NOTATION. 65 These assumptions are legitimate, because (4) represents some straight line passing through B, whatever he the values of the constants I and m ; hy properly assuming these con- stants, we may therefore make (4) represent BD. Also (5) represents some straight line through C, and hy giving a suitable value to n, we may make it represent OA. We may if we please suppose one of the three constants I, m, n, equal to unity, but for the sake of symmetry we will not make this supposition. The equation to AD is lu — 7nv +nw = 0-. (6); for (6) represents a straight line passing through the intersec- tion of lu — mv =i} and lo = 0, that is, a straight hne through D; also (6) represents a straight line passing through the intersection of m = and mv — nw = 0, that is, a straight hne through A. Hence (6) represents AD. The equation to EF is lu + nw = (7); for (7) obviously represents some straight line through E, and since lu + nw = lu — mv + nw + mv, (7) represents some straight line through F. Hence (7) represents EF. Let G be the intersection oi AC and BD. The equation to EG is lu — nw = 0:... (8); for (8) represents a straight line passing through the inter- section of (1) and (3), and also through the intersection of (4) and (5). The equation to FG is lu — 2mv + nw = (9); for (9) represents a straight line^ passing through the inter- section of (4) and (5), and also through the intersection of (2) and (6). Suppose BD produced to meet EF at H, and AG and EF produced to meet at K ; then it may be shewn that the equation to AH is 2lu-mv+nw=0, that to GHis mv+nw^O, ' that to KB is lu + mv = 0, that to KD is lu—mv + 2nw = 0. We have introduced this example, not on account of any importance in the results, but as an exercise in forming the equations to straight lines. We proceed to another example. T. C. s. 5 66 EXAMPLES OF ABRIDGED NOTATION. 76. If there be two triangles such that the straight lines joining the^corresponding angles meet at a point, then the inter- sections of the corresponding sides lie on one straight line. JC-- s denote the angle between the two straight lines ; m—m' ., then, by Art. 41, tan A = z^—, ; . where ' -^ ^ 1+ mm _ \ cosoL + fi COS /3 + y cos 7 "" X. sin a + /* sin /3 + 1* sin 7 ' ,_ X' cos a + /^' cos j3 + v cos 7 and "* -~\'sina + A<.'sin/3-l-v'sin7" Hence, substituting and reducing, we find — tan is equal to a fraction of which the numerator is (/xv'-/iV) sin {7- j8) -I- (v\'- i-'X) sin (a-7) -f-(X/i.' -X» sin(jS-a), and the denominator is X\' -f- (ifi + vv + (/ui/' -t- /iV) cos (7 - /3) + (kX' + v'\) cos (a - 7) -t- (X/i' + X» cos (yS- a). Now we can express the angles is equal to the following, expression with the double sign prefixed (fiv — fi 'v) sin A + (v\'—v'\) sinB + (\/jL—\'fi) sin (7 XK'+/iijf+w'—(jiv'+fiv) cosA—{vX'+ v\) cosB—(Kfi+X'fi) cosC " Again, by Art. 41, . _ m — m' ^''^ "p ~ v(i+m')va+'«'0 ■ Proceeding in the same way we find that sin must be zero. III. Let ABC be the triangle of reference ; and suppose the straight line denoted by h + w/S -I- 717 = Q to cut the sides of the triangle at D, E, F respectively. At D we have a = 0, and therefore m^ -1- W7 = 0. Here /S denotes the length of the perpendicular from Don AC, so that /8 = CD sin C ; and 7 denotes the length of the perpendicular from D on AB, so that 7 = BD sin B. TBILINEAR CO-ORDINATES. 71 Thus mCD sin C=- nBD sin B. Similarly at E we have /3 = 0, 7 = ^^sin A, a=GE sin C; therefore nAE smA=— I CE sin C. And at F we have 7 = 0, a = -BFsmB, ^ = AFsmA; therefore IBF sin B = mAF sin A. Hence by multiplication we obtain CD.AE.BF= BD. CE.AF. See Appendix to Euclid, Arts. 56... 58. IV. Let ABC be the triangle of reference : we shall shew how the constants I, m, n in the equation to a straight line la + m^ + ny = may be expressed in terms of the sides of the triangle and the perpendiculars from its angles on the straight line. Let p, q, r denote the perpendiculars drawn from A, Bj C respectively ; any two of them will be considered to be of the same sign or of contrary signs according as they fall on the same side of the straight line or on contrary sides. Proceeding as in IIL we have mCD sin C = - nBD sin B>^ . CD r *'"* BD i' therefore mr sin C = nq sin B, therefore mrc = nqb. Similarly npa = Ire, and lQf> = mpa. 72 TRILINEAK CO-OEDINATES. „ I m n Hence — = — j = — ; pa qb re and the equation to the straight line becomes paa + qb^ +rcy = 0. V. To find the length, of the -perpendicular dravm from a given point on a given straight line. Let (a', ^8', 7') be the given point, and \a + /^/S + 1^7 = the given straight line. By Art. 49 the perpendicular distance is Xa' + AtyS' + iV where — P=Xcosa+/iCos j8+i'C0S7, — Q =\sina+/isin/8 + i'sin7. Thus P'+ Q" = X°+ /*''+ k' + 2/ii/ COS (/3 — 7) + 2i'\ cos {7 — a) + 2\/i cos (a — y3) = X,'' + yu,'+ v^—%ij.v cos A — 2i/\ cos 5— 2\/t cos C. VI. Suppose we take for the fixed point the vertex A of the triangle of reference, so that = Q and 7' = ; and use the values of I, m, n found in IV. Thus the length of the per- pendicular from A on the straight line paa. + jft/S + rc7 = is yag' tj(p'a''+ g'b''+ r^c^— 2qrbc cos A — 2rpca cosB — 2pqab cos C) ' and this perpendicular is equal to p. Moreover if A denote the area of the triangle of reference a'a = 2A. Hence finally. 4A'=pV+g^6*+r'c'— 2qrbc coaA — 2rpca cosB—2pqdb cosC. This relation then must hold between the lengths of the perpendiculars drawn from A, B, G on any straight line. Substitute for cos A, cos B, and cos G their values in terms of the sides of the triangle ; then the result may be put in the form 4A== a' [p -q){'p-r) + ¥{c[-r)[q -p) + c' (r -p) (r - q). This may be easUy verified. For we see that if it be true for one straight line it must be true for every parallel straight line, since it involves only the differences of the per- TBILINEAR CO-ORDINATES. 73 pendiculars p, q, r. It will be su£Scient then to shew that the result is true for every straight line which passes through an angular point of the triangle. Take any straight line through A, suppose it to make an angle 6 with A£, and an angle ^ with AG; so that 6 + ^ + ^ = 180°. Then p = 0, q = c sin 6, r = 6 sin ^. We have then to shew that q'b* + r'c' — 2qrbo cos A = 4 A^ The left-hand member = 6V(sin'^+sin'^ — 2 sin ^ sin ^ cos jil) = JV{sin»0 + sin"^ -|- 2 sin ^ sin )] = JVJsin'^ (1 - sin»^) + sih> (1 - sin' 6*) + 2 sin 6 sin cos cos } = iV {sin'0 cos'^ + sin^*^ cos' 5 + 2 sin ^ sin <^ cos cos ^} = JV sin*(5 -1- ^) = 6V sinM. This establishes the required relation. VII. We have seen in Art. 69 that every straight line can be represented by an equation of the form la + m,^ + rvf = Q. We shall now shew conversely that every equation of this form, with a single exception, will represent some straight line. Develope the equation as in I. ; then we see that it must represent a straight line except when I cos a+m cos P + n cos 7=0, and ^sina-t-»»siny3-f-msin7 = 0. Eliminate n; thus Z sin (7 — a) + m sin (7 — )8) = 0; therefore -: — -, ^t = -: — ; r ; sm (7 — p) sin (a — 7) and in the same way we find that each of these is equal to n sin (jS — a) ■ 74 TRILINEAU CO-ORDINATES. Thus by what is shewn in I. we have I m n sinA~ sinB sin C It follows that la + wi/3 + ny = will always denote a straight line except when I, m, and n are proportional to sin A, sin B, and sin C, that is to a, h, and c. And we have seen that aat + 6;3 + cy expresses double the area of the triangle of reference, so that it cannot be equal to zero. VIII. In the equation Ax + By + 0=0, suppose that A and B diminish indefinitely while G remains constant. The straight line represented by the equation then moves away to an indefinite distance from the origin ; for the inter- cepts on the axes are — -j and — -^ . In like manner if I, m, n are in proportions to each other which differ infinitesimally firom the proportions of a, h, c the straight fine la + m^+ny= is situated at an indefinitelygreat distance from the triangle of reference. For abbreviation it is usual to speak of the equation aa + 6/3-1- 07 = as denoting a straight line at an infinite distance, or a straight line at infinity; very often the equation is said to represent the straight line at infinity, which is open to the objection that it seems to imply that there is some definite position towards which the straight line tends as it moves away from the triangle of reference. IX. To find the eqiuition to the straight line which passes through two given points. Let (a,, ;8,, 7,) and (a,, j8j, 7,) be the two points. Then, as in Alt. 3.5, assume for the equation to the straight line la + 'm0 + ny = 0. Thus ?a, + »n/3, + wy, = 0, and Zbj + m/3, -1- ny^ = 0. Hence we deduce I _''*_''. ^i7,-^ii7i ~ 7,<», - 7,«i " « A - "A ' TBILINEAa CO-ORDINATES. 7o and the required equation is a (Av. - ^.7.) + ^ (7.a, - 7,«.) + 7 (a.^s - '^d = <>• Hence the condition -which must hold in order that the point (Oj, ^3, Yj) may be on the straight line which joins the points (a„ ^„ yj and (a„ /3j, 7,) is "3(^172 - /3j7i) + /3»(7ia2 - 7i,ai) + 78(ai^2 - <^2^i) = <>. X. Denote the condition just obtained by 2' = for abbreviation. The expression for the same condition in com- mon rectangular co-ordinates is by Art. 36 which we will denote by 0=0. We may infer that if we transform from trilinear co-ordi- nates to common rectangular co-ordinates the condition T=0 will become (7=0; so that, whether the three points are in the same straight line or not, T can only differ from C by some constant factor which does not depend on the co-ordinates of the points. But, by Art. 11, when the three points are not in the same straight line G expresses double the area of the triangle which can be formed by joining them. Hence we conclude that the area of this triangle can also be expressed by kT, where k is some constant. We may find the value of k by considering a particular case. Let the three points be the vertices of the triangle of reference ; so that we may take ^Sj = 0, 7, = 0, a, = 0, 7, = 0, a^=0, ^83 = 0. Thus T reduces to a,/8j73, which is equal to —r- ; therefore k — ^ = A; therefore k = ttxi; . abc aoc 8/\- Hence the area of the triangle formed by joining the points (a„ /3,, 7J, (a^, j8j, 7,), and (a,, jS,, 7,) is 1^, K(i8»7, - A7.) + /3s (7A - 7,«i) + %{<^A - «A) • XI. The student should carefully notice in this subject that geometrical theorems may often be obtained by inter- preting equations which naturally present themselves in our investigations. For example in Art. 72 we have shewn the meaning of the equations /3-l-7 = 0, 74-a=0, a-|-y3 = 0: 76 EXAMPLES. CHAPTER IV. ■we are naturally led to consider the meaning of the equation o + ;8 + 7 = 0. The straight Une thus denoted passes through the intersection of yS + 7 = and a = 0, and through two analogous points. Hence we have this result : the straight hues which pass through the angles of a triangle and hisect the supplemental angles meet the respectively opposite sides in three points which lie on. one straight line. Similarly we may interpret the following equations: ^ + 7-a=0, 7+a-^ = 0, a + ;8-7=0. XII. It is very easy to pass from trilinear co-ordinates to common ohlique co-ordinates. Suppose we have any equation between a, /9, and 7 ; we can express 7 in terms of a and fi, by means of the relation aa +b^ + cy== 2A, and thus trans- form the given relation into one involving only o and fi. Let ABC be the triangle of reference. Suppose CA the axis of X, and CB the axis of y. Let P be any point ; x, y its co-ordinates. Draw PM parallel to BG, meeting AG aX M. Then if a and j8 refer to the point P, we have /3 = Pilf sin G=y sm C; and similarly a = a; sin C. Thus if we substitute x sin C for a, and y sin C for ^8, we finally transform the equation into one involving the common oblique co-ordinates x and y. EXAMPLES. 1. Find the equation to the straight line passing through the origin and the point of intersection of the straight lines ? + y=l ^ + 2^ = 1 a a EXAMPLES. CHAPTER IV. 77 2. A, A' are two points on the axis of x, and B, B' two points on the axis of y, at given distances from the origia; AB and A'B intersect at P, and AB' and A'B at Q; find the equation to the straight line PQ, and shew that the axes are divided harmonically by it. 3. If a = 0, j8 = 0, 7 = be the equations to the sides of a triangle ABG opposite the angles A, B, G, prove that a sin A— fi sin B=0 is the equation to the straight line bisecting AB from C. 4. Prove by means of such equations as that given in the preceding Example the first proposition in Art. 64. 5. Shew that a cos A — /S cos .B = is the equation to the perpendicular from C on AB. 6. Hence prove. the second proposition in Art. 64. 7. If a, b, c be the lengths of the sides of a triangle opposite the angles A, B, G, respectively, prove that a COS A — /S.cos B — -x (sin B cos A — sin A cos B) = is the equation to the straight line which bisects AB and is at right angles to it. The equation may also be written / asin£sinCN . /_ 5 sin C sin J."\ _ „ (a h— • — A — ).cos.il-h8 ^-^ — s— )cos5 = 0. \ 2 sin J. / V 2 sm i> / 8. Hence prove the third proposition in Art. 64. 9. Interpret the equation aa + 6y8 = 0. 10. Shew that aa + 6;8 — <;7 = is the equation to ihe straight line which joins the middle points of AG and BG. 11. Shew that a cos -4+^8 cos £ — 7 cos (7 = is the equation to the straight line which joins the feet of the perpendiculars from A on BG, and from B on AG. 12. If straight lines be drawn bisecting the angles of a triangle and the exterior angles formed by producing the sides, these straight lines will intersect at only four points besides tJie angles of the triangle. 78 EXAMPLES. CHAPTER IV. 13. If M = 0, v = 0, w = be the equations to three straight lines, find the equation to the straight line passing through the two points u _v _w J^_''_^ Y > and Yr — ' — ~/ " 14. Find the equation to the straight line passing through the intersections of the pairs of straight lines 2au +bv + cw=0, bv— cw=0; and 2bu + av + cw = 0, av — cw = 0. 15. If a = 0, /3 = 0, 7 = be the equations to the sides ol a triangle ABC, shew that the equation to the straight line which joins the centres of the inscribed circle and the circum- scribed circle is a (cos .B — cos (7) + /8 (cos C— cos.4.) + 7(cos J. — cosjB) =0. 16. If the equations to the sides of a triangle ABC be u=0, v = 0, w = 0, and to the sides of a triangle A'B'C, u = a, v=b, w = c, then AA', BB, and CC meet at a point. 17. If the straight lines AA', BF, CC, in the last Example meet respectively the sides of the triangle ABC at D, E, F, shew that the intersections of DE and AB, of EF and BC, of FD and CA, will all lie on one straight line ; and that a similar property will hold for the intersections of the same straight lines with the sides of the triangle A'B'C. 18. In Art. 75, suppose the straight line joining F and G to meet AB at P and CD at Q ; then find the equations to CP, DP, AQ, BQ, in terms of the notation of that Article. 19. From the middle points of the sides of a triangle straight lines are drawn at right- angles (all internal or all ex- ternal) and proportional to those sides: prove_that the straight lines which join the angles with the extremities of the oppo- site perpendiculars pass through one point. 20. Let the three diagonals of a quadrilateral be produced to meet each other at three points, and let each of these points be joined with the two opposite comers of the quadri- lateral : the six straight lines so drawn 'vyill meet each other three and three at four points. RXAWPLES. CHAPTER IV. 79 21. In the figure constructed in the preceding Example the four straight lines which meet each other at any comer of the quadrilateral are so related that two of them are parallel to the sides, and two to the diagonals of some parallelogram. 22. Shew that the three points of intersection which are found in Examples 4, 6, 8, lie on the straight line a sin .4. cos J. sin (B — C) + ^ sin ^ cos Bsm.{C — A) -f 7 sin cos C sin {A — B) = 0. 23. Let any point P be taken in the plane of a triangle ABO, and from the angular points A, B, G let straight lines be drawn through P cutting the opposite sides at D, U, F re- spectively : if the equations to BG, CA, AB heu=Q,v = 0, w = respectively, shew that the equations to AP, BP, GP may bo taken to be mv — nw = 0, nw — lu = 0, lu — mv = ; and find the equations to EF, FD, BE. 24. With the notation of the preceding Example let EF and BG be produced to meet at A', let FB and GA be pro- duced to meet at B^, and JDE and AB at C": then shew that A', B', G' lie on one straight line. 25. With the notation of the preceding Example shew that BB', CG', and AB meet at a point ; also GO', AA', and BE; and AA', BF and OF. 26. Three points A', S, C in the sides BG, GA, AB of a triangle being joined form a second triangle of which any two sides make equal angles with the side of the former at which they meet. Shew that AA', BB', GO' are perpen- diculars to BO, GA, AB. 27. ABO is any triangle, the centre of the inscribed circle, 0' the centre of the escribed circle which touches BG. The straight line 00' meets BO at 2), and any straight line drawn through D meets AG aX E and AB at F. The straight ' lines OF and O'E meet at P, and the straight lines OE and O^F at Q. Shew that A, P, and Q lie on one straight line peipendiculai' to 00'. 28. Find the equations to the two straight lines which bisect the angles formed by the straight lines Za+my3 + n7=0, and fa -H m'yS 4- n'7 = 0. 80 EXAMPLES. CHAPTER IV. 29. Shew that the co-ordinates of the point of inter- section of I' a. 4- mfi -t- «'7 = 0, and I'a. + rn'fi -V w"7 = 0, are given by a _ i8 _ y m'w" - m"n' ~ nT - «T ~ ZW - i'W 2A^ ~ a (m'»" - mVj -f 6 (n7" - w'T] + c {I'm" - l"m')' 30. Find the length of ^the perpiendicular drawn from the intersection of To + «i /8 H- n'7 = 0, and Va. + m'fi + w'y = 0, on Ix -J- m^ + 717 = 0. 31. Shew that the area of the. triangle formed by the straight lines la + m0 + ny= 0, I'a + m'/3 + n'y = 0, and l"cc + ot"/3 + n"y = 0, . Aahc [I (m'n" - mfn') + m (n'l" - ru'V) 4- n {I'm"- l"m')}' '^ BD'D" where D = a {m'n" - m"n') + b {n'l" - nl') + c (I'm"-- V'm'), iy = a {m'n - mn") + b {n"l - nl") +c {V'm - Im"). D" = a {mn - m'n) + b {nl' - n'l) + c {Im' - I'm). 32. Find the condition which must hold in order that the equations :r- = — , — = -, — = r may represent three parallel \ fjU fJL V V n, straight lines. 33. When the condition in the preceding Example is satis- fied find the conditioite, which must hold in order that the straight line la + m^-^)vY=0 may be parallel to the three straight lines. 34. Find the condition which must hold in order that the equations = = J- — '- may represent a straight line. 35. Determine what is represented by the equations ia the preceding Example when the condition is not satisfied. EXAMPLES. CHAPTER IV. 81 30. Through any point P within ABC, the triangle of reference, straight lines AP, BP, CP are drawn meeting the opposite sides at I), E, F respectively : if the equations to AP, BP, CP are m^.-ivy = 0. ivf-h. = Q, lx-m^ = 0, compare the areas of AEF and JDEF with that of ABC. 37. Perpendiculars are drawn from the angles of a tri- angle on the opposite sides, and a second triangle is formed by joining the feet of these perpendiculars : shew that the two triangles are homologous, and that the equation to the axis of homology is acosA+^cosB + ycosC = 0. 38. Investigate the condition which must hold in order that the following equation may represent two straight lines: ^a' + B0'+ Cy* + 2D^y + 2Eya + 2Fafi = 0. 39. Investigate the following expressions for the square of the distance between the points (Oj, 0^, 7,) and (a„ )8„ 7,): («. - ot.)' + (yS. - /3.)' + 2 (a. - a,) (/3, - /3.) cos C sin^ a (a. -aj'sin2^ + (;8, - /8,)' sin2g+ (7, - 7.)'sin2g 2 sin A sin B sin G ' (/g.-/g.)('y -V.) sinJ.+(7 -7,)(a,-g,) sin.g+(a - aJ(/3,-;3,) sinC sin.^ sinB sin £7 40. If in Example 34 each of the fractions is equal to the distance between the points (a, 0, 7) and (a', yS', 7';, shew that the following condition must also hold : V sin2-il +/*' sm2B+v^ sin2(7= 2 sin^ sin5 sinC. T. c. s. ( 82 ) CHAPTER V. TRANSFORMATION OF CO-ORDINATES. 79. We have seen in the preceding Articles that the general equation to a straight Une is of the form y = mx + c, but that the equation takes more simple forms in particular cases. If the origin is on the straight line the equation be- comes y = mx; if the axis of x coincides with the straight line, the equation becomes y = 0. In a similar manner we shall see as we proceed that the equation to a curve often assumes a more or less simple form, according to the position of the origin and of the axes. It is consequently found convenient to introduce the propositions of the present Chapter, which enable us when we know the co-ordinates of a point with respect to any origin and axes, to express the co-ordinates of the same point with respect to any other given origin and axes. It will be seen that these propositions might have been placed at the end of the first Chapter, as they involve none of the results of the succeeding Chapters. 80. To change the origin of co-ordinates without changing the direction of the axes, the axes being oblique or rectangular. Let OX, or be the original axes; O'X', OT'the new axes ; so that O'X' is parallel to OX, and O'Y' parallel to CHAl^GE IN DIRECTION OF EECTANGULAH AXES. 83 OY. Let h, k be the co-ordinates of 0' with respect to 0. Let P be any point ; as, y its co-ordinates referred to the old axes ; x', y' its co-ordinates referred to the new axes. Let Y'(y produced cut OX at A ; draw PM parallel to OY meeting O'T at N ; then * OA=h., Aa^le; x=OM=AM + OA = 0'N + OA =x' + h, y = PM=PN+NM=FN + AO'=y' + k. Hence the old co-ordinates of P are expressed in terms of its new co-ordinates. 81. To change the direction of the axes without changing the origin, both systems being rectangular. Let OX, OF be the old axes; OX', OY' the new axes, both systems being rectangular ; let the angle XOX' = 6. Let P be any point; x, y its co-ordinates referred to the old axes ; x', y' its co-ordinates referred to the new axes. Draw PM parallel to OF, PM' parallel to OY', M'N parallel to OY, and M'R parallel to OX. Then x = 0M= ON- MN^ ON- M'R = OM' cQsXOX'-PM sinM'PR = x cos 6 — y'Bm0; y = PM= RM+ PR = M'N+ PR = x' sind + y' cos 0. Hence the old co-ordinates of P are expressed in terms of its new co-ordinates. 6—2 84 .TO CHANGE THE DIRECTION OF OBLIQUE AXES. 82. In the preceding Article 9 is measured from the positive paxt of the axis of x towards the positive part of the axis of y ; therefore if in any example to which the formulae are applied, OX' fall on the other side of OX, must be con- sidered negative. * From the formulae of the preceding Article, we see that a^ + f = x-' + y"; this of course should be the case, since the distance OP is the same whichever system of axes we use. 83. To change the direction of the oases without changing the origin, both systems being oblique. Let OX, OF be the old axes; OX', OY' the new axes. Let (XY) denote the angle between OX, OY; and- let a similar notation be used to express the other angles which are formed by the straight lines meeting at 0. Let P be any point ; X, y its co-ordinates referred to the old axes ; x', y' its co-ordinates referred to the new axes. Draw PM parallel to OY, and PM' parallel to OY'; from P and M' draw PL, M'N perpendicular to Y; from M' draw M'R perpendicular .to PL. Then x=OM, y=PM\ x' = OM', y' = PM'. TO CHAl^QE THE DIBECnON OF OBUQTTE A^ES, 85 Now TL = perpendicular from Jtf" on 0F= a; sin (XT), also PL=RL + PR = M'N+PR = OJf ' sin X'OY+ PM' sin FOF = a;'sin(Xr)+ysin(rr); therefore x sin (XF) = x' sin (ZT) + y'sin (FF) (1). Similarly by drawing from P and M' perpendiculars on OX we may shew that y sin ( FT) = x' sin (Z'X) + y' sin (FZ) (2). Equations (1) and (2) express the old co-ordinates of P in terms of its new co-ordinates; {YX) and {XY) denote the same angle, but we use both forms for greater symmetry. LetXOX = a, XOT =P, XOY=(c; then (1) and (2) become ajsino) =a;'sin (w — a) -(- 1/' sin (<» — y9) (3), ysin s) = a;'sina +y'sin^ (4). 84. Two particular cases of the general proposition in the preceding Article may be noticed. If the original axes are rectangular w = s^ , and the equa- tions (3) and (4) become x = x' cos oi + y cos j8, y=x' sin a + y' sin ^. TT If the new axes be rectangular P = -^ + a, and the equa- tions (3) and (4) become a; sin a> = x sin (o> — a)—y' cos (to — a), 2/ sin ft) = x' sin a +y' cos a, 85. Suppose we require to change both the origin and the direction of the axes ; let x, y be the co-ordinates of a point referred to the old axes ; x', y' the co-ordinates of the same point referred to the new axes. By Arts. 80 and 83 we have a; = ar, + A, y^y^ + h, where A and k are the co- ordinates of the new origin referred to the old axes, and _ x' sin (a) — g) + y sin (m — j8) _ x' sin a+y' sin ^ *" sino> ' "' sin© ' The expressions for a;, and y, will simplify when one or each of the systems is rectangular. (See Art. 84.) 86 POLAR AND EBCTANGULAR CO-ORDINATES. 86. The formulae -whicli connect the rectangular and polar co-ordinates of a point in the particular case in which the origin is the same in both systems, and the axis of x coincides with the initial line, have already been given. (See Art. 8.) The following is the general proposition. To connect the polar and rectangular co-ordinates of a point. Let OX, OF be the rectangular axes; let ;Si be the pole and 8A the initial line. Let h, k be the co-ordinates of S referred to ; draw SX' parallel to OX, and let the angle A8X' = a. Let P be any point ; x, y its co-ordinates referred to the rectangular axes; r, 6 its polar co-ordinates. Draw PM, 80 parallel to OY, the former cutting 8X' at N. and ioin 8F; then ^ x=OM, y = PM, r = 8P, ^ = the angle PSA. And x=0C+GM=0G+8N = A + rcos(5 + a) (1), y = MN+PN=8G + PN = i-hrsin(& + a) (2). If o = we have J x = h + rcos6 (3), y = k+rsai0 (4). POLAB UJD BECTA17GULAB CO-OBDINATES. 87 87. By means of the formulae of the present Chapter we shall sometimes he able to simplify the form of an equation ; for example, the axes being rectangular, suppose we have y* + a;*+6a;y=2 (1). This equation represents some locus, and by ascribing different values to a; and determining the corresponding values of y from the equation, we can find as many points of the locus as we please. The equation however will be simplified by turning the axes through an angle of 45°. In the formulae of Art. 81 put j ioi 6; thus X —y X +y ,_, ^=^' ^ = ^ (')■ Substitute these values in (1); thus (x' + y'Y + {x' -yy + Q {x" - yy = 8; therefore 2 («'* + 6a;Y' + y") + 6 {x" - y'J = 8, or «;'*+y" = l (3). Since (3) is a simpler form than (1), we shall find it easier to trace the locus by using (3) and the new axes, than by using (1) and the old axes. The student must observe that we make no change in the locus by thus changing the axes 'or the origin to which we refer it; that is, equation (1) represents precisely the same assemblage of points as (3); for instance, the point for which x =\ and y =Q\s, obviously situated on the locus (3); now this point will by (2) have for its co-ordinates referred to the old system * = "To ' y~l2' and these values satisfy (1), that is, this point is on the locus (1). We may remark that we cannot alter the degree of an equation by transforming the co-ordinates. For if in the expression Aafy^ we substitute the values of a; and 7/ in terms of x' and y' given in Aits. 80... 84, we obtain A {ax' + by' + h)" {ex -\- ey +lcY, where a, h, c, e, h, k are all constant quantities; by expanding this expression we shall obtain a series of terms of the form A'x'yy'*, where 7 + S cannot be greater than a -I- ^. Hence 88 EXAMPLES. CHAPTER V. the degree of an equation cannot be raised by transformation of co-ordinates. Neither can it be depressed; for if from a given equation we could by transformation obtain one of a lower degree, then by retracing our steps we should be able from the second equation to obtain one of a higher degree, which has been shewn to be impossible. EXAMPLES. 1. Change the equation r" = a' cos 20 into one between X and 1/. 2. Shew that the equation iusy — 3a^ = a* is changed into x' — 4y' = a', if the axes be turned through an angle whose tangent is 2. 3. Transform *Jx+ tjy = ijc so that the new axis of x , may be inclined at 45° to the original axis. 4. The equation to a curve referred to rectangular axes is y + 4aycota — 4ax=0; find its equation referred to oblique axes inclined at an angle a, retaining the same axis of ;i;. 5. Shew that the equation cfj^ = a{a? + r^) will admit of solution with respect to y' if the axes be moved through an angle of 45". 6. If X, y be co-ordinates of a point referred to one system of oblique axes, and d, if the co-ordinates of the same point referred to another system of oblique axes, and X = ma! ^r ny', y = m'x' + n'y, shew that m' + my — 1 _ mm' . n' + n' — 1 ~ nn ' ( 89 ) CHAPTER VI, THE CIRCLE. 88. We now proceed to the consideration of the loci represented hy equations of the second degree ; the simplest of these is the circle, with which we shall commence. To find tlie equation to the circle referred to any rectangular axes. N M Let C he the centre of the circle ; P any point on its cir- cumference. Let c be the radius of the circle ; a, b the co- ordinates of C; x,y the co-ordinates of P. Draw CN, PM paraUel to QY, and CQ parallel to OX. Then GQ^ + PQ^ = CP^; thatis, {x-ay + {y-bY = c' (1), or af + y'-2ax~2hy-^a^-^h^-c^ = Q (2). This is the equation required. The following varieties occur in the equation. L Suppose the origin of co-ordinates at the centre of the circle ; then a = 0, and 6 = j thus (1) and (2) become a?+f-c^ = (3). 90 EQUATION TO THE CIRCLE. II. Suppose the origin on the circumference of the circle ; then the values x = 0, y = 0, must satisfy (1) and (2); therefore which relation is also ohvious from the figure, when is ow the circumference ; hence (2) hecomes ai' + y*-2aa;-2by = (4). III. Suppose the origin is on the circumference, and that the diameter which passes through the origin is taken for the axis of X ; then 6 = 0, and a' = c'; hence (2) becomes «'+y'-2aa! = (5). Similarly if the origin be on the circumference and the axis of y coincide with the diameter through the origin, we have a = 0, and 6' = c* ; hence (2) becomes a^ + f-2by = (6). Hence we conclude from (2) and the following equations, that the equation to a circle when the axes are rectangular is always of the form a:^+f + Ax + By + E = 0, where A, B, E are constant quantities any one or more of which in particular cases may be equal to zero. 89. We shall next examine, conversely, if the equation a? + y''+Ax + By + E=Q (1) always has a circle for its locus. Equation (1) may be written ['-ih{y*^''^-' (^)- I. If j1* + £* — 4iE be negative, the locus is impossible. II. li A^ + B^ — 4iE — 0, equation (2) represents a pmnt A B the co-ordinates of which are — ^ , — -g- . This point may be considered as a circle which has an indefinitely small radius. III. If A? ■\-W — iE be positive, we see by comparing equation (2) with equation (1) of the preceding .Article that it TANGENT TO A CIRCLE. 91 represents a circle, such that the co-ordinates of its centre are - J , -|, and its radius ^{A^ + lf- 4>E)K It will be a useful exercise to construct the circles repre- sented by given equations of the form i^+y^ + Aa; + By+E=0. For example, suppose as' -I- y* + 4a; — Sy — 5 = 0, or (a; -h 2)" -(-(y- 4)' =5 4-4 + 16 = 25. Here the co-ordinates of the centre are — 2, 4, and the radius is 5. Tangent and Normal to a Circle. 90. Definition. Let two points be taken on a curve and a secant drawn through them ; let the first point remain fixed and the second point move on the curve up to the first ; the secant in its Umiting position is called the tangent to the curve at the first point. 91. To find the equation to the tangent at any point of a circle. Let the equation to the circle be a? + f=c' (1). Let x', y' be the co-ordinates of the point on the circle at which the tangent is drawn ; and as", y' the co-ordinates of an adjacent point on the circle. The equation to the secant through \x, y') and {x", y") is, by Art. 35, -i/=i-i (''-''') ■ (2)- " " X —X Now since {x', y) and (x", y") are both on the circumfer- ence of the circle, x' + y'' = c\ x"' + y"'=c'; therefore by subtraction, a;"" - x" + y"" - y" = 0, or {x"-x') {x" -J- x') + {y"-y') [y" +y') = ; «"-j/ x"±x therefore ^^r:^ y+^' ' 92 TANGENT TO A CIRCLE. Hence (2) may be written 3'-y = -<^'(— ^') (3)- Now in the limit when (x", y") coincides with {x, y'), we have «" = x', and y" = y'; hence (3) becomes Thus the equation to the tangent at the point {x\ y) is y-y — ^(a'-a^') (*)• This equation may be simplified ; by multiplying by y' and transposing we have xx' + yy' = x" + y''^ ; therefore asB'+yy' = c' (5). 92. The equation to the tangent can be conveniently ex- pressed in terms of the tangent of the angle which the straight line makes with the axis of x. For the equation to the tan- gent at (x, y) is yy + xx' = c°, or y = — >« + -,. If if t Let — -, = m ; thus the equation becomes V = mx + — . y We have then to express — in terms of m. Now x' = —my', and a;'* + y" = c*; therefore y'" (1 + m'^ = c", and y' = —— 5: . Hence the equation to the tangent may be written y = mx + c V(l + w'). Conversely every straight line whose equation is of this form is a tangent to the circle. 93. The definition in Art. 90 may appear arbitrary to the student, and he may ask why we do not adopt that given by TANGENT TO A CIBCLE. 93 Euclid (Def. 2, Book in.). To this we reply that the defini- tioa in Art 90 will be convenient for every curve, which is not the case with Euclid's definition. The student however cannot at first be a judge of the necessity or propriety of any definition ; he tnust confine himself to examining the conse- quences of the definition and the accuracy of the reasoning based upon it. We may easily shew however that the straight line re- presented by the equation xx +yy' = c* (1) touches, according to Euclid's definition, the circle a?-^f = «+-/. y y Hence the equation to a straight line through {x', y) at right angles to the tangent at that point is ■Zr SD Since this equation is satisfied by the values x=0, y = 0, the normal at any point passes through the origin of co-ordi- nates, that is, through the centre of the circle. 99. From any external point two tangents can be drawn to a circle. Let the equation to a circle be <^+y' = c' (1). and let h, k be the co-ordinates of an external point. Sup- pose x', y' the co-ordinates of a point on the circle such that 96 CHORD OF CONTACT. the tangent at this point passes through (A, k). The equation to the tangent at (x , y') is axc' + yy'^c' (2). Since this tangent passes through (A, k) h£ + hy' = c' (3). Also since {x', y) is on the circle a;" + y" = c' (4). Equations (3) and (4) determine the values of x and y', (c' — hx'\* — k — / """» therefore a" (A' + *») - 2c'Aa;' + c' (c" - F) = 0. The roots of this quadratic equation will be found to be both possible since (A, k) is an external point and therefore h' + i? greater than c\ To each value of x' corresponds one value of y' by (3) ; hence two tangents can be drawn from any external point. The straight line which passes through the points where these tangents meet the circle is called the chord of contact. 100. Tangents are drawn to a circle from a given external point; to find the equation to the chord of contact. Let A, k be the co-ordinates of the external point; a;,, y, the co-ordinates of the point where one of the tangents from (A, k) meets the circle; ar„ y^ the co-ordinates of the point where the other tangent from (A, k) meets the circle. The equation to the tangent at (a;,, y,) is asc.+yyi=c* (1). Since this tangent passes through (A, k), we have Aar, + Ay. = c' (2). Similarly, since the tangent at (x^, yj passes through (A, k), hx^ + ky^ = c' (3). Hence it follows that the equation to the chord of con- tact is xh + yk = c* (4). For (4) is obviously the equation to some straight line; CHORD OF CONTACT. 97 also this straight line passes through («,, y^, for (4) is satisfied by the values x = x^, y = y^, as we see from (2); similarly from (3) we conclude that this straight line passes through (»,, y,). Hence (4) is the required equation. Thus we may use the following process to draw tan- gents to a circle from a given external point: draw the straight line which is represented by (4) ; join the points where it meets the circle with the given external point, and the straight lines thus obtained are the required tangents. 101. Through any fixed point chords are drawn to a circle, and tangents to the circle drawn at the extremities of each chord: t}i£ locus of the intersection of the tangents is a straight line. Let h, k be the co-ordinates of the point through which the chords are drawn ; let tangents to the circle be drawn at the extremities of one of these chords, and let (a;,, y^) be the point at which they meet. The equation to the correspond- ing chord of contact is, by Art. 100, xx^ +yyi = c^ But this chord passes through (h, k) ; therefore to, -f- %, = c". Hence the point (a;,, y,) lies on the straight line xh + yk = c'; that is, the locus of the intersection of the tangents is a straight line: this straight line is at right angles to that which joins the point (A, k) with the centre. We will now demonstrate the converse of this proposition. 102. If from any point in a straight line a pair of tan- gents be drawn to a circle, the chords of contact will all pass through a fixed point. Let Ax + By+C=0 (1) be the equation to the straight line ; let (x, y) be a point in this straight line from which tangents are drawn to the circle; then the equation to the corresponding chord of contact is wx' + yy'^c' (2). Since {x, y') is on (1) we have Ax +By' -irG = Q; , Ax + G , therefore (2) may be written xx —y — „ — = c , T. C. s. 7 98 or INTERPRETATIONS OP AN EQUATION. Now, whatever be the value of x, this straight line passes through the point whose co-ordinates are found by the simul- taneous equations x — ~ = 0, ^- -(- c* = ; that is, the point Be* Ac* for which y = — j^ , x = — — : this point is on the straight line drawn from the centre perpendicular to (1). 103. The student should observe the different interpreta- tions that can be assigned to the equation xh + yk — c*= 0. I. If (A, k) be any point whatever, the equation repre- sents the locus of the intersection of tangents at the extre- mities of each chord through {h, k). (Art. 101.) II. If (A, k) be an external point, the equation represents the chord of contact. (Art. 100.) III. If (h, k) be on the circle, the equation represents the tangent at that point. (Art. 91.) In the preceding figures Q denotes the point {h, k), and BR the straight line xh + yk = &. CntCIiE EEFEREED TO OBLIQUE AXES. 99 In the first figure Q is within the circle, and the straight line RR receives only the interpretation I. In the second figure Q is withovi the circle, hence the straight line RR receires hoth interpretations I. and II. ; if therefore tangents he drawn from Q to the circle they will meet it at the points where RR intersects it. If Q he on the circle, then RR becomes the tangent at Q. Oblique Axes. 104. To Jind the eqvution to the circle referred to any oblique axes. Let (0 be the iaclination of the axes ; let be the centre of the circle ; P any point on its circumference. Let c be the radius of the circle ; a, b the co-ordinates of C ; x,y the co-ordinat«s of P. Draw CN, PM parallel to OY, and GQ parallel to OX. Then CP' = GQ" + PQ- - 2GQ . PQ cos GQP = GQ'+P(^ + 2GQ.PQ cos a:; that is, (a; - af + (y — by + 2 {x — a)(i/ — b) cos o) = c^; or, ar" + y' + 2xy cos (o — 2{a+bcos(o)x — 2{b + acos(o)y + a' + b'+2ab cos to - c" = 0. 7-2 100 POLAR EQUATION TO THE CIRCLE. Hence the equation to the circle referred to oblique axes is of the form where A, B, E are constant quantities. Polar Equation. 105. To find the polar equation to ilve circle. s X Let 8 be the pole, 8X the initial line ; C the centre of the circle, P any point on its circumference. liet 80=1, C8X=a, so that I, a are the polar co-ordi- nates of C; let c be the radius of the circle ; and let r, 6 be the polar co-ordinates of P. Then Cr = PS' +G8'-2P8 .08 . cos PS C; that is, (^ = 7* + ^- 2lr cos {6 -a) (1), or r' — 2rl (cos a cos ^ + sin o sin ^) -h f — c" = (2). Hence the polar equation to the circle is of the form r' + Ar cos e + BrsinO + E = (3), where A, B, E are constant quantities. The polar equation may also be deduced from the equa- tion referred to rectangular axes in Art. 88, by putting r cos 9 and r sin for ar and y respectively. If the initial line be a diameter we have a = 0, hence (1) becomes r*-2J>-cos0-f-J'-c* = O (4). PEEPENDICULAE ON THE TANGENT. 101 If, in addition, the origin be on the circumference P = c", therefore r= 2Zcos^ (5). 106. To express the perpendicular from the origin on the tangent at any point in terms of the radius vector of that point. Let 8Q be the perpendicular from the origin on the tan- gent at P, and suppose 8Q=p; then SC' =SP' + PCr'-2SP.PC cos SPO = SP» + P(7» - 2SP . PC sin SPQ; that is, f = r'.+ c* — 2cp. In the figure 8 and C are on the same side of the tangent at P. If we take P so that the tangent at P falls between 8 and C, we shall find f = r* + c° + 2cp. 107. These equations aie sometimes useful in the solu- tion of problems, or demonstration of properties of the circle. For example, take the equation (4) in Art. 105, r'-2rlcose + P-c^=0; by the theory of quadratic equations we see that th.e product of the two values of r corresponding to any value of d is P — c', which is independent of 0. This agrees with Euclid ni. 35, 36. Also the sum of the two values of r is 2? cos 0; hence if a straight line be drawn through the pole at an inclination to the initial line, the polar co-ordinates of the middle point of the chord which the circle cuts off from this straight line are — - — , and 0; that is, I cos 0, and 0. Hence the polar equation of the locus of the middle point of the chord is r = Z cos 0; this by (5) in Art. 105, is a circle, of which the diameter is I. 102 EXAMPLES. CHAPTEK VI. EXAMPLES. 1. Determine the position and magnitude of the circles (1) a;»+3/'' + 4y-4a;-l = 0, (2) a!' + f + 6x-3y-l=0. 2. Find the points of intersection of the straight lines y + x = — l, y + x = — 5, and 3y + 4a; = — 25, with the circle x*-^y' = 25. 3. A circle passes through the origin and intercepts lengths h and k respectively from the positive parts of the axes of X and y : determine the equation to the circle. 4. A circle passes through the points (h, k) and {h', k') : shew that its centre must lie on the straight line iJ.-K'){x-'^)^ik.-k'){y-t^)=0. 5. On the straight line joining {x', y') and {x", y") as a diameter a circle is described : find its equation. 6. A and B are two fixed points, and P a point such that AP= mBP, where w is a constant : shew that the locus of P is a circle, except when «i = 1. 7. The locus of the point from which two given unequal circles subtend equal angles is a circle. 8. Find the equation which determines the points of intersection of the straight line t + t— 1 = 0, and the circle »* + ^ — 2ax — 2by = 0. Deduce the relation that must hold in order that the straight line may totich the circle. 9. Find the equation to the tangent at the origin to the circle x^ + y^ — 2y — 3x = 0. 10. Shew that the length of the common chord of the circles whose equations are (a;-a)' + (y-6)*=c', {x-hY + {y-ay=d'. is V(4c'-2(a-6)'}. EXAMPLES, CHAPTER VI. 103 11. A point moves so that the sum of the squares of its distances from the four sides of a square is constant : shew that the locus of the point is a circle. 12. A point moves so that the sum of the squares of its distances from the sides of an equilateral triangle is constant : shew that the locus of the point is a circle. 13. A point moves so that the sum of the squares of its distances from any given numher of fixed points is constant : shew that the locus is a circle. 14. Shew what the equation to the circle becomes when the origin is a point on the perimeter, and the axes are in- clined at an angle of 120°, and the parts of them intercepted by the circle are h and k respectively. 15. Find the inclination of the axes in order that the equation ai' + y' — xy — hx — hy = may represent a circle. Determine the position and the magnitude of the circle. 16. Find the inclination of the axes in order that the equation a? + y* + a;y — hx —hy = may represent a circle. Determine the position and the magnitude of the circle. 17. Determine the equation to the circle which has its centre at the origin, and its radius = 3, the axes being in- clined at an angle of 45°. 18. Determine the equation to the circle wluch has each 1 . . 2 of the co-ordinates of its centre = — 5 and its radius = -75 , 3 yo the axes being inclined at an angle of 60°. 19. The axes being inclined at an angle a, find the radius of the circle a;" + y' + 2xy cos co—hx — ky = 0. 20. Shew that the equation to a circle of radius c referred to two tangents inclined at an angle a as axes is x' + ^+ 2xy cosay — 2 {a; + y)c cot o + c' cot" „ = 0. 21. Shew that the equation in the preceding Example may also be written x + y — 2 */(ay) sin ^ = c cot ^ . 104 EXAMPLES. CHAPTER VI. 22. Find the value of c in order that the circles {x - af +{y- by = c', and (x - 6)' + (y- a)' = c\ may touch each other. 23. ABC is an equilateral triangle ; take A as origin, and AB as axis of x: find the rectangular equation to the circle which passes through A, B, C. Deduce the polar equa- tion to this circle. 24. If the centre of a circle he the pole, shew that the polar equation to the chord of the circle which subtends an angle 20 at the centre is r = c cos ^ sec {6 — a), where a is the angle between the initial line and the straight line from the centre which bisects the chord. Deduce the polar equa- tion to a straight line touching the circle at a given point. 25. Find the polar equation to the circle, the origin being on the circumference and the initial line a tangent. Shew that with this origin and initial line, the polar equation to the tangent at the point 6' is r sin (26' — 0)=2o sin'^'. 26. Shew that if the origin be on the circumference and the diameter through that point make an angle a with the initial line, the equation to the circle is r = 2c cos {& — a). 27. Determine the locus of the equation r = Acos(0-a) +Bco3(0- 0)+ Ccos{e -y) + 28. AB is a given straight line; through A two inde- finite straight lines are drawn equally inclined to AB, and any circle passing through A and JB meets those straight lines a.t L, M: shew that the svm, of AL and AM is constant when L and M are on opposite sides of AB, and that the difference of AL and AM is constant when L and M are on the same side of AB. 29. ABG is an equilateral triangle : find the locus of P yfhen PA = FB + PC. 30. There are n given straight lines m aking with another fixed straight line angles a, /8, 7, ; eT'^JIw; P is taken such that the sum of the squares on the perpendiculars from EXAMPLKS. CHAPTER VI. 105 it on these n straight lines is constant : find the conditions that the locus of F may be a circle. 31. A point moves so that the sum of the squares of its distances from the sides of a regular polygon is constant : shew that the locus of the point is a circle. 32. A straight line moves so that the sum of the perpen- diculars AP, BQ, from the fixed points A and B is constant : find the locus of the middle point of FQ. 33. is a fixed point and AB a. fixed straight line ; a straight line is drawn from meeting AB at P ; in OP a point Q is taken so that OP. OQ = ¥ : find the locus of Q. 34. A straight line is drawn from a fixed point 0, meet- ing a fixed circle at P; ia OP a point Q is taken so that OP.OQ = k': find the locus of Q. 35. Shew that (hy - kx)* - c' {(x - A)' +iy- W] repre- sents the. two tangents to the circle, se' + if = c^, which pass through the point {h, k). 36. Determine what is represented by the equation r»-ra cos 26 sec d - 2a' = 0. 37. The polar equation to a circle being r = 2c cos 0, shew that the equation 2c cos /8 cos a = r cos (/8 + a — 6) represents a chord such that the radii drawn to its extremities from the pole, make angles a, /3 with the initial line. 38. Tangents to a circle at the points P and Q intersect at r ; if the straight lines joining these points with the ex- tremity of a diameter cut a second diameter perpendicular to the former at the points p, q, t, respectively, shew that pt = qt. 39. Find the equation to the circle which passes through three points whose co-ordinates are given. 40. 'Shew that the co-ordinates of the centre and the radius of the circle in the preceding Example are always finite except when the three given points are on a straight line. ( 106 ) CHAPTER VII. RAOICAl AXIS. POLE AND POLAE. Radical Axis. 108. We have sheum in Art. 88 that the equation to a circle is (a; — a)" + (y — 6)' — c* = 0. We shall write this for abbreviation S=0. If the point {x, y) be not on the circum- ference of the circle, S is not = ; we may in that case give a simple geometrical meaning to 8. I. Let {x, y) be without the circle ; draw a tangent from {x, y) to the circle ; join the point of contact with the centre of the circle (a, 6) ; also join (x, y) with (a, h). Let G re- present the point {a, h), Q the point (ar, y), and T the point of contact of the tangent. Thus we have a right-angled triangle formed, and since {x — af + {y — by =$(?', it foUows that S = QT^ ; that is, 8 expresses the square of the tangent from {x, y) to the circle. By Euclid III. 36, the square of the tangent is equal to the rectangle of the segments made by the circle on any straight line drawn from {x, y), and thus 8 will also express the value of this rectangle. II. Let {x, y) be within the circle ; then 8 is negative. Let G and Q have the same meaning as before, and produce GQ to meet the circle at T and T'; then _ 8= GT' - GQ' = {GT - GQ) {GT + GQ) = TQ . T'Q. Hence by Euclid ill. 3-5, if any straight line PQP' be drawn meeting the circle at P and P', the value of the rectangle PQ.FQ is -8. 109, Let 8 denote (a; - a)' H- (y - bf - c", and fif denote (x - a')*-h (y - b')' - c'* ; sothat 8 = (1), and S' = (2), RADICAL AXIS. 107 are the equations to two circles : we proceed to interpret the equation S-S' = (3). S—8' contains only the first powers of w and y; therefore 8—S' = is the equation to some straight line. Also if values of x and y can be found to satisfy simultaneously (1) and (2), these values will satisfy (3). Hence when the circles represented by (1) and (2) intersect, (3) is the equa- tion to the straight line which joins their points of inter- section. Also suppose that from any point in (3), external to both circles, we draw tangents to (1) and (2) ; then, by Art. 108, these tangents are equal in length. Hence whether (1) and (2) intersect or not, the straight line (3) has the following property: if from any point of it straight lines he drawn to touch both circles, the lengths of these straight lines are equal. 110. An equation of the form A{a^ + y')+Bx + I>y + U = will represent a circle ; for after division by A we obtain the ordinary form of the equation to a circle. We shall say that the equation to a circle is in its simplest form when the co- eflBcient of ar* and y' is unity. Definition. If 8 = 0, S' = 0, be the equations to two circles in their simplest forms, the straight line 8 — S'=0 is called the radical axis of the circles. The axes of co-ordinates may here be rectangular oir oblique. Or we may give a geometrical definition thus. A straight line can always be found such that if from any point of it tangents be drawn to two given circles, these tangents are equal ; this straight line is called the radical axis of the circles. 111. The three radical axes belonging to three given circles meet at a point. Let the equations to the three circles be 8, = (1). 5,= (2). S,= (3). 108 RADICAL AXIS. The equations to the radical axes are /Sf,-/Sf, = 0, belonging to (1) and (2), 'S,--Sf3 = 0, (2) and (3), 8,-S,=^0 (3) and(l). These three straight lines meet at a point ; since it is ob- vious that the values of x and y which simultaneously satisfy two of the equations, will also satisfy the third. 112. A large number of inferences may be drawn from the preceding Articles by examining the special cases which fall under the general propositions. (See Pliicker Analytisch- Geometrische Entwickelungen, Vol. i. pp. 49... 69.) We notice a few of these respecting the radical axis of two circles. 113. The radical axis is perpendicular to the straight line joining the centres of the two circles. Let the equations to the circles be {w-ay+{y-bY-c''=0, {x -a'f + (y-h'y-c'* = 0; then the equation to the radical axis is («; - a)' - (a; - aT+ (2/ - 6)' - (y - 6')' - c' + c" = ; that is, X {a - a)+y{b'-b} +i {a'-a"+ 6' - 6" - c» + c") = 0...C1). And the equation to the straight line joining the centres of the circles is (Art. 35) 2'-^ = ^^*-") (2); (1) and (2) are at right angles by Art. 42. 114. When two circles touch, their radical axis is the common tangent at the point of contact. For the radical axis passes through the common point and is perpendicular to the straight line joining the centres of the circles. 115. Suppose the radius of one of the circles to become indefinitely small, that is, the circle to become a point ; the radical axis then has the following property: if from any CENTRES OF SIMILITUDE. 109 point of the radical axis we draw a straight line to the given point, and a tangent to the given circle, the straight line and the tangent will be equal in length. 116. The radical axis of a point and a circle falls without the circle, whether the point he without or within the circle. For if the radical axis met the circle, the co-ordinates of the points of intersection would satisfy the equation to the point as well as the equation to the circle. But the equation to the point can be satisfied by no co-ordinates except the co-ordi- nates of that point; therefore the radical axis cannot meet the circle. If the point be on the circle, the radical axis is the tangent to the circle at this point. 117. Suppose both circles to become points. Then the straight lines drawn from any point in the radical axis to the two fixed points are equal in length. Hence the radical axis belonging to two given points is the straight line which bisects at right angles the distance between the two given points. 118. Suppose in Art. Ill that each circle becomes a point; the theorem proved is then the following : the straight lines drawn from the middle points of the sides of a triangle at right angles to the sides meet at a point. 119. It is a well-known geometrical problem to draw a straight line which shall touch two given circles : see Appendix to Euclid, Art. 4. If the circles do not intersect, four com- mon tangents can be drawn; two of them will be equally inclined to the straight line joining the centres, and will intersect on that straight line between the circles ; the other two will also be equally inclined to the straight line joining the centres, and wiU intersect on that straight line beyond the smaller circle. These two points of intersection are called centres of similitude. We will briefly explain some of the properties of centres of similitude. I. Let a centre of similitude of two circles be taken as the pole, and the straight line passing through the centres of the circles as the initial line. By Art. 105 the equations to the two circles will be of the forms r'-'2.rlcose + F-(? = 0, r'-2rr cos 0+ r-c'' = ..,(1). 110 CENTRES OF SIMILITUDE. From the first equation r-=«cosfl±V(c'-f sin'6') (2). When the two values of r are equal the radius vector becomes a tangent : this takes place when F sin' 6 = c*. Since the circles have common tangents passing through the pole c' c" d c . 1 j^ = jii, and therefore -p = + y. If the lower sign is taken b i lb the centre of similitude is between the centres of the two circles ; if the upper sign is taken the centre of similitude is on the production of the straight line which joins the centres : we may call the former the inner centre of similitude, and the latter the outer centre of similitude. Since j^ = jt the second of equations (1) may be written r = j{Zcos^ + V(c'-Z»sin»^)} (3). From (2) and (3) we have the following result : Let A be the centre of one circle, and B the centre of another, and let T be a centre of similitude ; let any straight line through T cut the former circle at K and L, and the latter at M and If, so that T^is less than TL, and TM less than TJUT: then TKTL _TA TM~ TN~ TB' II. When two circles intersect only one pair of common tangents can be drawn ; and when one circle is entirely within the other no common tangent can be drawn. Nevertheless two points always exist such as the point T just considered ; so that we may take the following as the most general defi- nition of the centre of similitude of two circles : A centre of similitude is a point on the straight line joining the centres or on this straight line produced such that its distances from the centres are proportional to the radii of the corresponding circles. The essential property of a centre of similitude may be considered to be that expressed by the final result in I. CExntES OP smiLinrDE. Ill III. Let y be a centre of similitude of two circles ; draw from T two straight lines, one cutting the circles at K, L, M, N; and the other at k, I, m, n. Now we have just shewn that TK^Tk TM Tm' therefore the triangles TKk and TMm are similai-, and Mm is parallel to Kk. Hence the angle Kkl = the angle Mmn; and therefore the angles MNn and KM are supplemental, by Euclid iii. 22, so that a circle would pass round NKkn : let Nn and Kk be produced to meet at R, then RK.Rk = RN.Rn, by Euclid III. 36. Cor. Hence the tangents from R to the two circles are equal, by Euclid III. 36 ; and therefore R is on the radical axis of the two circles. Similarly Nn is parallel to LI ; and Mm and LI if pro- duced meet on the radical axis. IV. Suppose there are three circles ; since each pair has two centres of similitude there will be six centres of simili- 112 POLE AND POLAR. tilde on the -whole : we shall shew that four straight lines can be drawn each containing three centres of similitude. Let A,B,C be the centres of three circles; htp, g', r be their radii. Let F he a. centre of similitude of the circles which have their centres at A and B; draw a straight line through F meeting CA and CB at E and D respectively. By page 71 we have AE.GD.BF=GE.BD. AF. ^^* BF~q' CD _p GE ^'^"^ BD~qAE' Now suppose that £ is a centre of similitude of the circles which have their centres at A and G ; then GE r ^, . GD r TTi = -\ therefore DTi = -- AE p BD q Hence D is a centre of similitude of the circles which have their centres at B and G. In this way we obtain results which can be enunciated definitely thus : the outer centre of similitude of two circles, and the two inner centres of simili- tude of these two circles and any third circle lie on a straight line ; also the three outer centres of similitude lie on a straight line. Pole and Polar. 120. Definitiok. If the equation to a given circle be a? + }f = + cos' ^) — 2xy tan ^ + y' sin* = 0, make angles a, /3 with the axis of ai, shew that tan a - tan /3 = 2. 3. One side of a square a comer of which is at the origin makes an angle a with the axis of x : find the equations to the four sides and the two diagonals. 4. Find the equations to the diagonals of the parallelogram formed by the straight lines ah ah ha ha and shew that the diagonals are at right angles. 5. The distance of a point (aj^ , y^ from each of two straight lines which pass through the origin of co-ordinates is S: shew that the two straight Unes are represented by the equation (^.y-zcy.)' = («;' + 2^)8'. 6. Find the condition that one of the straight lines re- presented by Ay^ + Bxy + Gx' = may coincide with one of those represented by ay* + bxy + cjs" = 0. 7. If a= 0, /3 = 0, 7 = be the equations to the three sides of a triangle, and a, 6, c be the perpendicular distances EXAMPLES. CHAPTER VH. 115 between these sides and those of another triangle parallel to them respectively, the straight line joining the centres of the inscribed circles wiU be represented by any of the equations a—p ^— 7 7— a a — b b — c c — a 8. Shew that the equation to the straight line passing through the middle point of the side £G of a triangle ABG and parallel to the external bisector of the angle A is /8 + 7-|(sin5 + sin(7) = 0, 9. The equation to the straight line drawn parallel to BG through the centre of the escribed circle which touches BG is (a + /9) sin 5 + (a + 7) sin Cf = 0. 10. Find the equations to the straight lines which pass through the intersection of the straight lines la+mfi + nr/=0, l'a + m'^ + n'y = 0, and divide the angles between them into parts having their sines in a given ratio. 11. Find the equations to the two straight lines which bisect the angles between the straight lines represented by Af + Bxt/+Gar' = 0. 12. Find the condition in order that the straight lines Ay" + Bxy + Cx" = and ay" + hxy + ca^ = may have their angles bisected by the same pair of straight lines. 13. If M = 0, « = 0, be the equations to two circles, shew that by giving a suitable value to the constant \, the equation M + Xw = will represent any circle passing through the points of intersection of the given circles. 14. A fixed circle is cut by a series of circles, all of which pass through two given poiuts : shew that the straight lines which join the points of intersection of the fixed circle with each circle of the series all meet at a point. 8—2 ( "6 .) •CHAPTER VIII. THE PARABOLA. 123. There are three curves which we now proceed to define ; we shall then deduce their equations from the defini- tions, and investigate some of their properties from their equations. Definition. A conic section is the locus of a point which moves so that its distance from a fixed point bears a constant ratio to its distance from a £xed straight line. If this ratio be unity, the curve is called a parabola, if less than unity, an ellipse, if greater than unity, an hyperbola. The fixed point is called the focus, and the fixed straight line the directrix. 124. It will be shewn hereafter that if a cone be cut by a plane, the curve of intersection will be one of the following; a parabola, an ellipse, an hyperbola, a circle, two straight lines, one straight line, or a point. Hence the term conic section is applied to the parabola, ellipse, and hyperbola, and may be extended to include the circle, two straight lines, one straight line and point. We shall also shew that every curve of the second degree must be a conic section in this larger sense of the term. At present we confine ourselves to tracing the consequences of the definitions in Art. 123. 125. To find the equation to the Parabola. A parabola is the locus of a point which moves so that its distance from a fixed point is eqvud to its distance from a fixed straight line. Let S be the fixed point, YY' the fixed straight line. Draw SO perpendicular to YY'; take as the origin, 08 EQ^JATION TO THE PABABOLA. 117 as the direction of the axis of a;, OF as that of the axis of y. Suppose 0/S=2a. Let P be any point on the locus ; join BP ; draw TM parallel to OF and PJV parallel to OX; let OM.=x, By definition /SfP = PiV; therefore £fP' = PJT* ; therefore FM* + SM^ = FN'', that is, y" + (a: - 2a)' = <^ ; therefore y' = 4a {x — a) (1). This is the equation to the parabola with the assumed origin and axes. The curve cuts the axis of a; at a point A ■which bisects OS; for when y = in (1), we have x = a. The equation will be simplified if we put the origin at A ; let X = AM, then x' = x—a, and (1) becomes y' = ^ax'. We may suppress the accent, if we remember that the origin is now at A ; thus we have for the equation to the parabola f = ^ax (2). 126. To trace the parabola from its equation y' = 4ax. From this equation we see that for every positive value of X there are two values of y, equal in magnitude, but of 118 FOBM OF THE FARASOLA. opposite sign. Hence for every point P on one side of the axis of X, there is a point P' on the other side, such that P'M=PM. Thus the curve is symmetrical with respect to the axis of x. Negative values of x do not give possible values of y ; hence no part of the curve lies to the left of the origin. Asa; may have any positive value, the curve extends without limit on the right of the origin. A is called the vertex of the curve, and AX the axis of the curve. 127. We have drawn the curve concave towards the axis of X ; the following proposition will justify the figure. The ordinate of any point of the curve which lies between the vertex and a fixed point of the curve is greater than the corresponding ordinate of the straight line joining the vertex and the fixed point. Let P be the fixed point ; x', y' its co-ordinates ; then the equation to AP isy=-,x = »/(— r ) . m, since y" = 4iax'. Let X denote any abscissa less than x', then since the ordi- nate of the curve is \J{4iax), and that of the straight line is K/\~) • ^ °^ \/\') ^ VC^o*). it is obvious that the ordi- nate of the cun'e is greater than that of the straight line. All points may be said to be outside the curve for which FOCAL DISTANCE OF IlSY POINT. 119 j^ — 4aa; is positive ; that is, points for which x is negative, or for which x is positive and less than ^ . And all points may he said to he inside the curve for which y' — 4cm; is nega- tive. Since the square of the distance of any point from the focus is y + (a; — a)', that is (a: + a)' + y* — ^nw;, it follows that the distance of any point, not on the curve, from the focus is greater or less than its distance from the directrix according as the point is outside or inside the curve. 128. Definition. The double ordinate through the focus of a conic section is called the Latus Rectum. Thus in the figure in Art. 126, LSL' is the Latus Eectum. Let x = a, then from the equation y' = 4aa;, y = + 2a. Hence L8 = L'S= 2a; and LL' = 4a. 129. To express the focal distance of any point of the parabola in terms of the abscissa of the point. The distance of any point on the curve from the focus is equal to the distance of the same point from the directrix. Hence (see figure to Art. 125), SP=AM+AS, =x + a. Tangent and normal to a Parabola. 130. To find the equation to the tangent at any point of a parabola. (See Definition, Art 90.) Let x, y' be the co-ordinates of the point, x", y" the co- ordinates of an adjacent point on the curve. The equation to the secant through these points is 2'-y' = fcl'('"-^') ^^^' since {x', j/) and («", y") axe on the parabola y'* = iax', y"' = 4!ax"; therefore y" — y" = 4a (»" — x') ; y" - y _ 4a therefore x"-x'-y"+y' hence (1) may be written y - y' = „ > {x - x). if '^ if -120 TANGENT TO A PARABOLA. Now in the limit y" = y' ; hence the equation to the taflr gent at the point {«', y) is y-y' = ~{x-x) (2). This equation may be simplified ; multiply by y', thus yy' = 2a (a; — x) + y' = lax — lax + 4aa;' = 2a (« + «') (3). 131. The equation to the tangent can be conveniently expressed in terms of the tangent of the angle which the straight line makes with the axis of the parabola. For the equation to the tangent at (a;', y) is yy' = 2a{x-i-x'), 2a 2ax' 2a iax' or y = -rX-1 r- ='~r x + -„ ,- " y y y ^y -Y'*i m- Let —r="m; therefore k = — ; thus (1) may be written y ' 2m' \ I J y='>^ + ^ ^^^' this is the required equation. Conversely, every straight line whose equation is of this form is a tangent to the parabola. 132. It may be shewn as in Art. 93, that a tangent to the parabola meets it at oiJy one point. Also, if a straight line meets a parabola at only one point, it will in general be the tangent at that point. For suppose the equation to a parabola to be f = ^ax (1), and the equation to a straight line to be y =mx->r c (2). To determine the abscissae of the points of intersection, we have the equation (wia? + c)' = ^ax, or mV + (2mc-4a)a; + c''=0 (3); NORMAL TO A PARABOLA. 121 tMs quadratic equation will have two roots, except when (mc — 2aY = m^c*. that is. when c = — . Hence if the straight Une (2) meets the parabola, it will meet it at two points, unless c = — , and then the straight line m is a tangent to the parabola by Art. 131. If, however, the equation (2) be of the form y = c, so that ■the straight line is parallel to the axis of x, then instead of (3) we have the equation c' = iax, which has but one root ; hence a straight line parallel to the axis of the parabola meets it at only one point, but is not a tangent. 133. The axis of y is a tangent to the curve at the vertex. For the equation to the tangent at {x, y') is ■y\/ = 2a{x+x'); and when a;' = and y = 0, this becomes x = 0. 134. To find the equation to the normal at any point of a parabola. (See Definition, Art. 97.) Let x', y be the co-ordinates of the point ; the equation to the tangent at that point is y = -7 (« + «')• (!)• The equation to a straight line through {x', y') at right angles to (1) is y-y'=-|-(^-*') (2). This is the equation to the normal at {x, y). 135. The equation to the normal may also be expressed in terms of the tangent of the angle which the straight line makes with the axis of the curve. For the equation to the normal isy = — ■^a; + y'+^. y=-Ta''+y'^& w- 122 PBOFEBTIES OF THE PARABOLA. Let y — ^ = m; therefore y' = — 2am ; thus (1) may be written y = mx — 2am — an^ (2). 136. We shall now deduce some properties of the para- bola from the preceding Articles. Let x, y' be the co-ordinates of P; let PT be the tangent at P and PG the normal at P. The equation to the tangent at P is yy =2a{x-\- x). T ^^ Z^ \ T A I S I •I C X Let y = 0, then x = ~x'; hence AT= AM. AlsoST = AT+AS, =AM + A8, =5fP (Art. 129). Hence the triangle 8TP is isosceles, and the angle STP is equal to the angle 8PT. Thus if PJf be parallel to the axis of the curve, P2^ and PS are equally inclined to the tangent at P, so that the tangent bisects the angle between PS and NP produced. Since the angle PTS is half the angle PSX, it follows that the angle between two tangents to a parabola is half the angle between the focal distances of the points of contact. . PBOPEBTIES OF THE PASABOLA. 123 137. The equation to the nonual at P is At the point G, where the normal cuts the azis^ y —0; hence from the above equation x—x' = 2a; thus MG = 2a = half the latus rectum. Also 8G = SP = ST. 138. To find the lociis of the intersection of the tangetvt at any point with the perpendicular on it from the focus. Let x', if be the co-ordinates of any point P on the curve ; the equation to the tangent at P is y^^ix^x") (1). if The equation to the straight line through the focus per- pendicular to (1) is y = -£(^-a) (2X We have now to eliminate id and y by means of (1), (2), and y"=4cw;' (3). From (3) we find x in terms of y, and thus (1) may be written 3'=7'«+f W- Thus the problem is reduced to the elimination of y from (2) and (4); from (2) 3,'=--^ (5); " x—a ^ " ,.•..• /^x ii (x—a)x ay substitute in (4); then y = —- ; therefore y'{x — a) + (x — a)'x+ay''=0, or {y'+{x-ay}x = (6). If the factor j^+{x— a)* be equated to 7,ero, we have y=0, x = a (7). 124 LOCUS OBTAINED BY ELIMINATION. The point thus determined is the focus ; this however is not the locus of the intersection of (1) and (2), for the values in (7), although they satisfy (2), do not satisfy (1). We conclude therefore that the required locus is given by the equation a; = 0, which we obtain by considering the other factor in (6). This result can be easily verified ; for if we put a; = in (1) we obtain y = — ;- = ^; and if we put a; = in (2), we also obtain y=%-; thus (1) and (2) intersect on the straight line a; = 0. Or we may equate the right-hand members of (1) and (2), and by reduction obtain {4a' + y'') a; = ay'* — 4aV, which is zero : therefore a; = 0. Thus, if in the figure in Art. 136, Z be the intersection of the tangent at P with the axis of y, SZ is perpendicular to the tangent. 139. The process of the preceding Article is of frequent use and of great importance. We have in (1) and (2) the equations to two straight lines ; if we obtain the values of x and y from these simultaneous equations, we thus determine the point of intersection of the straight lines ; the values of x and y will depend upon those of x and y, thus giving dif- ferent points of intersection corresponding to the different straight lines represented by (1) and (2). If from (1), (2), and (3) we eliminate x and y' we obtain an equation which holds for the co-ordinates of every point of intersection of (1) and (2). This is, by our definition of a locus, the equation corresponding to the locus of the intersection of (1) and (2). Sometimes the elimination produces, as in the preceding Article, an equation which does not represent the required locus. The student has probably noticed in solving alge- braical questions that he often arrives at other results besides that which he is especially seeking. We can frequently interpret these additional results; thus in the preceding Article, since, whatever x' and y may be, the values x = a, y = 0, satisfy one of the equations which we use in eflfectiog the elimination, we might anticipate that our result would involve a corresponding factor. PEEPENDICULAK ON THE TAKGENT. 125 140. If the straight line from the focus, instead of being perpendicular to the tangent, meet it at any constant angle, the locus of their intersection will still be a straight line. We ■will indicate the steps of the investigation. Suppose ^ the angle between the tangent and the straight line from the focus; equation (1) remains as in Art. 138; instead of (2) we have, by Art 45, — r+tanfl », y ' \ 2a + y'tany8, y Instead of (5) in Art. 138, we shall find , _ 2g (g; — g) + 2ay tan j8 ^ y — {x — a) tan yS The result of the elimination is y[y— (a;— o)tan/3} {a; — a + ytanyS} — x{y — {x — a) tan 0\^ — a{x—a + y tan /8)' = 0. Now, guided by the result of Art. 138, we may anticipate that y* + (x — a)' will prove a factor of the left-hand member of the equation; and we shall find by reduction that the equa- liion may be written {y* + (a; — af} {y tan /S — a- tan'/3 — a) = 0. Hence the required locus is determined by y = x tan /3 + a cot ^. 141. To find the length of the perpendicular from the focus on the tangent at any point of the parabola. The equation to the tangent at the point {x', y) is 2a, The perpendicular on this from the point (a, 0), by Art. 47, = 2a^±£l=^^±^=V{a(a+a^')]. /^{y* + 4ia) v{*a(a-(-a;)J ^ ^ ' ■ Call the focal distance of the point of contact r, and the 126 TWO TANGENTS FROM AN EXTERNAL POINT. perpendicular p ; then, by Art. 129, r = o + «'; therefore p = n/iar). Also PG = twice 8Z= 2^{ar) : see Ait. 136. 142. From any external point two tangents can be dravm to a parabola. Let the equation to the parabola be ^ = 4aa; ; and let h, k be the co-ordinates of an external point. Suppose x', y' the co-ordinates of a point on the parabola such that the tangent at this point passes through (A, i). The equation to the tangent at (a;', y) is yy' = 2a(a; + a;')- Since this tangent passes through (A, A) %' = 2a (A -»-«') (1). Also since («', y"^ is on the parabola ^" = 403;' (2). Equations (1) and (2) determine the values of x and y'. Substitute from (2) in (1), thus ky' = 2ah + ^ , therefore y'* — 2ki/ + iah = 0. The roots of this quadratic •will be found to be both possible, since (h, k) is an external point and therefore k^ greater than 4aA. To each value of y' cor- responds one value of x' by (1); hence two tangents can be' drawn from any external point. The straight line which passes through the points where these tangents meet the parabola is called the chord of contact. 143. Tangents are drawn to a parabola from a given external point : to find the equation to the chord of contact. Let h, k be the co-ordinates of the external point ; a;,, y, the co-ordinates of the point where one of the tangents from {h, k) meets the parabola; a;,, _y, the co-ordinates of the point where the other tangent from {h, k) meets the parabola. The equation to the tangent at {x^, y^ is yy^=2a{xJfx^ (1). Since this tangent passes through (A, k) we have ky^ = 1a{h + x^ (2). CHORD OP CONTACT. 127 Similarly, since the tangent at (ar,, y^ passes through (h, k) ky^ = 1a{h + x^ (3). Hence it follows that the equation to the chord of con- tact is %=2a(a; + A) (4). For (4) is obviously the equation to some straight hne; also this straight line passes through {x^, y,), for (4) is satis- fied by the values a; = aij, y = y,, as we see from (2) ; similarly from (3) we conclude that this straight line passes through (*s> yJ- Hence (4) is the required equation. Thus we may use the following process to draw tangents to a parabola from a given external point. Draw the straight line which is represented by (4), join the points where it meets the parabola with the given external point, and the straight lines thus obtained are the required tangents. 144. Through any fixed point chords are drawn to a parabola, and tangents to the parabola drawn at the extremi- ties of each chord : the locus of the intersection of the tangents is a straight line. Let h, k be the co-ordinates of the point through which the chords are drawn ; let tangents to the parabola be drawn at the extremities of one of these chords, and let (a;,, y,) be the point at which they meet. The equation to the corre- sponding chord of contact by Art. 143 is yy^ = 2a{x + a;,). But this chord passes through Qi, k) ; therefore ky^ = 2a (A + a;,). Hence the point (a;,, y^ lies on the straight line ky = 2a{x + h); that is, the locus of the intersection of the tangents is a straight line. We will now prove the converse of this proposition. 145. If from any point in a straight line a pair of tangents be drawn to a parabola, the chords of contact will all pass through a fixed point. Let Ax + Sy+G=0 (1) be the equation to the straight line ; let {x', y) be a point in this straight line from which tangents are drawn to the parabola; then the equation to the corresponding chord of contact is yy' = ia{x + x') (2). 128- TANGENTS FROM AN VXTERSAL POINT. Since {x', y) is on (1) we have Ax +Bt/' + C = ; there- fore (2) may be written y {Ax +G)+ 2aB (x + x) = 0, or (Ay + 2aB)x'+ Cy + laBx = 0. Now whatever be the value of x', this straight line passes through the point whose co-ordinates are found by the simul- taneous equations Ay + 2aB = 0, Gy-\- 2aBx = j that is the ^ • * <• 1-1, 2a5 G pomt tor which y = ^ , a; =-j . The student should observe the different interpretations that can be assigned to the equation ky = 2a(x + h). The statements in Art. 103 with respect to the circle may all be appHed to the parabola. 146. Some interesting geometrical investigations relat- ing to tangents to a parabola from an external point may be noticed. To draw the two tangents to a parabola from any external point. Let denote the external point and S the focus. On OS as diameter describe a circle, and let it cut the tangent at the vertex at Z and z. Join OZ and Oz: these straight lines, produced if necessary, are the tangents from by Art. 138 and Euclid iii. 31. Or we may proceed thus. Join 08. With centre and radius 08 describe a circle, and let it cut the directrix at Q and q. Through these points draw parallels to the axis meet- TANGENTS FBOU AN EXTEKNAL POINT. 129 ing the parabola at P and p. Then OP and Op axe the re- quired tangents. For join OQ and SP. Then in the triangles OPS and OPQ we have 0S= OQhj construction, PS = PQ by the nature of the parabola, and OP common. Therefore the angle OPS = the angle OPQ; and OP is the tangent at P by Art 136. Similarly Op is the tangent at p. The two tangents to a parabola from an external point svbtend equal angles at the focus. Since the triangles OPS and OPQ are equal in all re- spects, the angle 0/SP = the angle OQP; and similarly the angle 0/^ = the angle Oqp: and the angles OQP and Oqp are equal, for they are the complements of the equal angles OQq and OqQ. The angle between a tangent atid a straight line parallel to the axis is equal to the angle between the other tangent and the straight line from the external point to the focus. Draw OH parallel to the axis. The angle QOH = the angle qOM; that is twice the angle P08— the angle SOH = twice the angle pOS + the angle SOH ; therefore the angle POS = the angle pOH, and therefore also the angle P02f= the angle pOS. The student should observe the extension thus given to the result in Art. 136 : at any point of the curve the straight hne which bisects the angle between the focal distance of the point and the parallel to the axis is at right angles to the tangent, and at any external point the straight hne which bisects the angle between the focal distance and the parallel to the axis is equally inclined to the two tangents. The circle which passes through the intersections of three tangents to a parabola will pass through the focus. T. c. s. ^ 9 130 DUHKTEBS. Let P, Q.Rhe the points of contact, and pqr the triangle formed by iJie tangents. Since Fr and Qr subtend equal angles at 8 the angle FSr is half the angle PSQ. Similarly the angle PSq is half the angle PSR. Hence the angle qSr is half the angle Q8R ; that is by Art. 136 the angle qSr is equal to the angle qpr: therefore 8 is on the circumference of the circle which passes round pqr. Diameters. 147. To find the length of a straight line drawn from any point in a given direction to meet a parabola. Let x', y be the co-ordinates of the point from which the straight line is drawn ; x, y the co-ordinates of the point to which the straight line is drawn ; 6 the inclination of the straight line to the axis oi x; r the length of the straight hne; then (Art. 27) x = x' + rcos6, y = y' + rsm0. If {x, y) be on the parabola, these values may be substituted in the equation if = iax ; thus (i/' + rsin6y=ia {x' + r cos 0); or r*sm^0+ 2r (y' sin 5 - 2a cos ^ -h y" - 4aa;' = 0. From this quadratic two values of r can be found, which are the lengths of the straight lines that can be drawn from {x', ^ PR = A!!, -1- PR = A + y* sin ft TO A DIAMETER AND TANGENT AS AXES. 133 Substitute these values in the equation y'= 4aic; thus {k+y'smef = 4>a(h + x+y'cose), or y" sin*5 + 2y' {k sin 0-2a cos e)+k^-iah = 4ax. But, k = 2a cot 6, and i' = 4aA; thus we have y''sin''^ = 4aa;', which is the reqiiired equation. We may shew that ^^ = SA'; for SA'=a + h (Art 129); A' and A = -J— = a cot*^; therefore a + h = Hence the equation may be written y^ = 4aV, where a' = /&4'; or suppressing the accents on the variables y' = ia'x, 152. The equation to the tangent to the parabola wiU be of the same form whether the axes be rectangular, or the obliqiie system formed by a diameter and the tangent at its extremity; for the investigation of Art. 130 wiU apply with- out any change to the equation y' = 4ia'x which represents a parabola referred to such an oblique system. 153. Tangents at the extremities of any chord of a para- bola meet on the diameter which bisects thai chord. Refer the parabola to the diameter bisecting the chord, and the corresponding tangent, as axes ; let the equation to the parabola be _y'' = 4a'ar; let af, y' be the co-ordinates of one extremity of the chord; then the equation to the tangent at this point is yy'^^a'^x + x") (1). The co-ordinates of the other extremity of the chord are x', —y\ and the equation to the tangent there is -yy'=2(i{x+x^ (2). The straight lines represented by (1) and (2) meet at the point for which y= 0, x = — x\ this demonstrates the theorem. 134 POLAK EQUATION. Polar Equation. 154. To find the Polar Equation to the parabola, ^ focus being the pole. Let SP = r, ASP =6, (see figure to Art. 123); then SP=PN, by definition; that is, SP=08+ SM; or r = 2a + r cos [ir — d); therefore r (1 + cos 6) = 2a, 2a and " 1 + cos ^ ' If we denote the angle XSP by ff, then we have as before 2a SP = 0S+8M; thus»- = 2a + roos^, andr=T ^. ' 1 — cos 155. The polar equation to the parabola when the vertex is the pole may be conveniently deduced from the equation y" = 4aar by putting r cos and r sin 6 for x and y respec- tively ; we thus obtain r = — . „„ . ■' ' SID. We add a few miscellaneous propositions on the parabola. Definitiok. a chord passing through the focus of a conic section is called a focal chord. 156. Iftamgents he drawn at the extremities of any focal chord of a parabola, (1) the tamgents will intersect on the directrix, (2) the tangents will meet at right angles, (3) the straight line drawn from, the point of intersection of the tan- gentsto the focus will he perpendicidar to the focal chord. (1) If the tangents to a parabola meet at the point (h, k) the equation to the chord of contact is, ky = 2a{x + h) by Art. 143. Suppose the chord passes through the focus ; then the values x = a, y=0, must satisfy this equation; therefore = 2a{a + h); therefore A = — a; that is, the point of intersection of the tangents is on the directrix. FOCAL CHORDS. 135 (2) The equation to the tangent to a parabola may he written (Art. 131) y = mx -\ — . Suppose (h, k) a point on the tangent; therefore Am' — km + a = 0. This quadratic will determine the inclinations to the axis of the parabola of the two straight lines that may be drawn through the point {h, k) to touch the parabola. Suppose tjIj, m, the tangents of these in- clinations, then by the theory of quadratic equations Tra.mt. = ■=- . /Ir If h = — a, mjn^ = — 1 ; that is, the two tangents are at right angles. (3) The equation to the straight line passing through the k focus and (A, k) isy= , _ (x — a). l£h = — a, this becomes k y = — -^{x — a); the straight line is therefore perpendicular to the focal chord of which the equation is yk =2a{x — a). 157. If through any point within or vrithowt a parabola, two straight lines be drawn parallel to two given straight lines to meet the curve, the rectangles of the segments will be to one another in an invariable ratio. Let {x', y') be the given point, and suppose a and /8 respectively the inclinations of the given straight lines to the axis of the parabola. By Art. 147, if a straight line be drawn through (x', y') to meet the curve and be inclined at an angle a to the axis, the lengths of its segments are given by the equation r* sin'o + 2r (y' sin a — 2a cos o) + y'^ — ^ax' = 0. Therefore by the theory of quadratic equations the rect- ai^le of the segments = - — :-j . Similarly the rectangle of the segments of the straight line i/* — 4iax' drawn through {x', y') at an angle ;8 = ■ ^a • sm pi Hence the ratio of the rectangles =-=-5—, and this ratio " sm'a is constant whatever x' and y' may be. 136 EECTANGLE OF THE SEGMENTS OF A STRAIGHT LINE. Let be the point through which the straight lines OPp, OQq, axe drawn inclined to the axis of the parabola at angles a, ^, respectively ; then we have shewn that OP. Qp ^ sin'/g OQ.Oq~ sin'a * Let tangents to the parabola be drawn parallel to Pp, Qq, meeting the parabola at E and D respectively; let 8 be the focus ; then by Art. 151, 8E si^, therefore ^^-QP ^^ 52) sin'a' tiieretore^^.^^_-g^. Suppose to coincide with T; then OP . Op becomes TE' xadOQ.Oq becomes TD'; therefore TE' 8E SB' EXAMPLES. CHAPTEE Tin. 137 EXAMPLES. 1. Find the equation to the straight line joining ^ and L. (See figure to Art. 126.) 2. Find the equation to the circle which passes through A, L, L'. (See figure to Art. 126.) 3. A point moTes so that its shortest distance &om a given circle is equal to its distance from a given fixed dia- meter of that circle : find the locus of the point. 4. Trace the curves y* = 4oa!, and a? + iay = Q; and determine their points of intersection. 5. Determine the equation to the tangent at L. (See figure to Art. 126.) 6. Find the angle between the straight lines in Exam- ples 1 and 5. 7. Determine the equation to the normal at L. 8. Find the point where the normal at L meets the curve again, and the length of the intercepted chord. 9. Find the point in a parabola where the tangent is inclined at an angle of 30° to the axis of x. 10. The length of the perpendicular from the inter- section of the directrix and axis on the tangent at (»', y") is a {of — a) V{a (»' + «)}• 11. Find the points of contact of tangents the perpen- diculars on which from the intersection of the directrix and axis are equal to one-fourth of the latus rectum. 12. A circle has its centre at the vertex A oi a, parabola whose focus is S, and the diameter of the circle is ZAS: shew that the common chord bisects ^;S. 138 EXAMPLES. CHAPTER VHI. 13. Trace the curve y=x — a?, and determine whether the straight line x+y = l is a tangent to it. 14. The tangent at any point of a parabola will meet the directrix and latus rectum produced at two points equally distant from the focus. 15. PM is an ordinate of a point P on a parabola; a straight line is drawn parallel to the axis bisecting PM and cutting the curve at Q ; MQ cuts the tangent at the vertex AaAT: shew that AT = f PJIf. 16. If from any point P of a circle PC be drawn to the centre G, and a chord PQ be drawn parallel to the diameter AGB and bisected at R, shew that the locus of the inter- section of GP and AR is a parabola. 17. Find the ordinates of the points where the straight line y = mx + c meets the parabola; hence determine the ordinate of the middle point of the chord which the para- bola intercepts on this straight line. 18. A is the "origin, 5 is a point on the axis of y, BQ is a straight line parallel to the axis of « ; in AQ, produced if necessary, P is taken such that its ordinate is equal to BQ : shew that the locus of P is a parabola. 19. From any point Q in the straight line BQ which is perpendicular to the axis GAB of a parabola whose vertex is A, PQ is drawn parallel to the axis to meet the curve at P : shew that if GA be taken equal to AB, the straight lines AQ and GP will intersect on the parabola. 20. At the point (af, y') a normal is drawn: find the co-ordinates of the point where the normal meets the curve again, and the length of the intercepted chord. 21. If the normal at any point P meet the curve again at Q, and SP = r, and p be the perpendicular from S on the tangent at P, then PQ = -^— , 22. P is any point on a parabola, A the vertex ; through A is drawn a straight line perpendicular to the tangent at P, EZiKPLIB. CHAFTEB Vm. 139 and through P is drawn a straight line parallel to the axis ; the straight lines thus drawn meet at a point Q : shew that the locus of Q is a straight line. Find also the equation to the locus of Q' the intersection of the perpendicuku: from A and the ordinate at P. 23. PQ is a chord of a parabola, PT the tangent at P. A straight line parallel to the axis of the parabola cuts the tangent at T, the arc PQ at E, and the chord PQ at F. Shew that TE : EF :: PF : FQ. 24. In a parabola whose equation is ^ = iax, pairs of tangents are drawn at points whose abscissas are in iJas ratio of 1 : /I ; shew that the equation to the locus of their inter- section vdll be ^ = (/I* + /t"*)' oas when the points are on the same side of the axis, and ^ = —(ji*~ //■'*)* ax when they are on different sides. 25. Two straight lines are drawn from the vertex of a parabola at right angles to each other ; the points wheire these straight lines meet the curve are joined, thtis forming a right-angled triangle : find the least area of this triangle. 26. Let r and r' be the lengths of two radii vectores drawn at right angles to each other from the vertex of a parabola : then {rr')* = IGa' (r* + r% 27. Find the polar equation^ to the parabola referred to the intersection of the directrix and axis as origin and the axis as initial line. 28. If a straight line be drawn from the intersection of the directrix and axis cutting the parabola, the rectangle of the intercepts made by the curve is equal to the rectangle of the parts into which the parallel focal chord is divided by the focus. 29. Find the polar equation to the parabola when the intersection of the directrix and the axis is the origin and the initial line the directrix. 30. A system of parallel chords is drawn in a parabola: find the locus of the point which divides each chord into segments whose product is constant. 140 EXAMPLES. CHAPTER VHI. s 31. In a triangle ABC- if" tan A tan g = 2, and AB be fixed, the locus of G will be a parabola whose vertex is A and focus B. 32. Find the equation to the parabola referred to tan- gents at the extremities of the latus rectum as axes. 33. Find the equation to the parabola referred to the normal and tangent at Zr as axes. 34. P is a point on a parabola; x, v^ are its co-ordinates : find the equation to the circle described on SP as diameter. 35. Shew that the circle described on 8P as diameter toitches the tangent at the vertex. 36. If the straight line y = m (a; — o) meets the parabola at (x', y') and {x", y"), shew that «'+a," = 2a + ^; afx" = a'; l/ + y" = ^; vf 4a\ 37. A circle is described on a focal chord of a parabola as diameter ; if m be the tangent of the inclination of this chord to the axis of x, the equation to the circle is a;» - 2aa; fl + 4) +»*-—- 3 (2aa; - a'). . The equation to J.'P is 2/ = ^ ar, ox y=- \ — -x\x, since (x', y") is on the ellipse. " Let X denote any abscissa less than x', then since the 10—2 148 FOCAL DISTANCES OF AlfT POINT. ordinate of the curve is -i^{2ax — x^ or -./( ^j^, and that of the straight line is ~ a/(~;"~1 ) a;, it is obvious that the ordinate of the curve is greater than that of the straight line. 165. AA' and BS are called axes of the ellipse. The axis AA' which contains the two foci is called the major axis and sometimes the transverse axis ; BB' is called the minor axis and sometimes the conjugate axis. The ratio which the distance of any point in the ellipse from the focus bears to the distance of the same point from the corresponding directrix is called the excentricity of the ellipse. We have denoted it by the symbol e To find the lotus rectum (see Art. 128) we put x=CH, that is = oe, in equation (1) of Art. 162 ; thus ,_ 6V(l-e') y ^~ a" 'a'' V 2b* therefore LH= — , and the latus rectum = — . a a Since V = a'— oV ; therefore V + oV = a' ; that is, CB'+CH' = a'; therefore BH=a; similarly BS = a. 166. To express the focal distances of any point of the ellipse in terms of the abscissa of the point. Let 8 be one focus, E'K' the corresponding directrix ; H the other focus, EK the corresponding directrix. Let P be a point on the ellipse ; x, y its co-ordinates, the centre being the origin. Join SP, HP, and draw IfPN parallel to the major axis, and PM perpendicular to it. Then 8P= ePN' = e {E'C + CM) =e(^ + s^=a + ex. Also HP = ePN =e{CE-CM) = e{^-ii^=a-ex. FOCAL DISTANCES OP AITT POINT. 149 Hence 8P + HP = 2a ; that is, the sum of the focal dis- tances of any point on the ellipse is equal to the major axis. K' JST' r \7» ^^'^^'"^ ( __- -"^l \ £!• 1 s c M3I jr. X It is obvious from Euclid, i. 21, that the sum of the focal distances of any point outside the ellipse is greater than the major axis, and the sum of the focal distances of any point inside the ellipse is less than the major axis. of f It is easily seen that — j + ^ — 1 is positive for any point outside the ellipse, and negative for any point inside the ellipse. Let the co-ordinates of any point Q be a; and y ; then SQ' =y'+{x-aey= {ex - a)' + / -f- (1 - e') (a;* - a") Thus H^ is greater or less than e* ( a; ) according as Q is outside or inside the ellipse ; therefore the focal distance of any point not on the curve bears to the distance of the point from the corresponding directrix a ratio which is greater or less than e according as the point is outside or inside the ellipse. 150 EXCENTBIC ANGLE. 167. The equation jf = -^ (a* — a?) may be written a y'= -J (a — a;) (a + a;). Hence (see the figure to Art. 162) PM' A'M.MA BG" 'AC 168. Let a circle be described on the major axis of the el- lipse as a diameter; its equation referred to the centre as origin will be y' = a' — as*. Hence if any ordinate MP of the ellipse be produced to meet the circle at P we have PM^ = —^P'M'; therefore Join P" with C the centre of the ellipse; let P'CM= if>, and let w, y be the co-ordinates of P; then jB = CP' cos ^ = a cos ^, y = -P'Jlf =-osin^ = 6sin^. , These values of x and y are sometimes useful in the solu- tion of problems. The angle P'CM is called the excentric angle of the point P. CONNEXION OF THE ELLIPSE AND PAHABOLA. 151 The circle described on the major axis of an ellipse as diameter is sometimes called the auxiliary circle. 169. From Art. 160 we see that the equation to the ellipse when the viertex is the origin is ^ = 2pex — (1 — e') a;'. If we suppose e = 1, this becomes y' = 2px, which is the equation to a parabola whose latus rectum is 2p. Also in the ellipse a = -- ^ . ,h=a V(l - e") = .,f^_ ,.. , ^jffor a{l-e) = ^. ^ ' 1 + e If we now make e = l, we have a and 6 infinite, and a (1 — c) =^ . Thus if we suppose the distance between the vertex and the nearer focus of an ellipse to remain constant while the excentricity approaches continually nearer to unity, the major and minor axes of the ellipse increase indefinitely and the ellipse about the vertex approximates to the form of a parabola. Thus if any property is established for an ellipse we may seek for a corresponding property in the parabola by referring the ellipse to the vertex as origin and examining what the result becomes when e is made to approach continually to unity, while the distance between the vertex and the nearer focus remains constant. Tangent and Normal to an Ellipse. 170. To find the equation to the tangent at any point of an ellipse. (See Definition, Art. 90.) Let x', y' be the co-ordinates of the point, x", y" the co- ordinates of an adjacent point on the curve. The equation to the secant through these points is 2'-2'' = fci'(^-'^^ (1)' since {x, y') and {x", y") are points on the ellipse, aY + hV = a^h\ aV" -I- 6 V = a'6' ; therefore a' (j"' - y") + V («'" - x") = ; 152 TANGENT TO AN ELLIPSE. therefore •-;; — —, = — » . -^t-, — . • X —X a y +y Hence. (1) may be written « — « = 5 . -ij-, > [X — X). Now in the limit x" = x', and y" = y; hence the equation to the tangent at the point {x, y') is y-y'^-^i^-^l (2). This equation may be simplified ; multiply by o^, thus a^yy + b'xx' = aV' + 6 V = d'b'. 171. The equation to the tangent can be conveniently expressed in terms of the tangent of the angle which the straight line makes with the major axis of the ellipse. For the equation to the tangent at (x, y") is oryy + 1" i'^^ = tt I or 1/ = j-; a; + — . ^y y Vx' y Let — 5— , = m; thus the equation becomes y — mx-'r—; ay ^^ y we have then to express -; in terms of m. y Now 6V = - aYm, and ay + i V» = a'6» ; therefore aV » + ^-^ = o'6' ; therefore y" (aW + 6') = 6*, therefore -, = V(aW + 6»). Hence the equation to the tangent may be written y = mx + >J(a'm' + V). Conversely every straight line whose equation is of this form is a tangent to the ellipse. It may be shewn as in Arts. 93, 94, that the tangent at any point of an ellipse meets it at only one point, and that NORMAL TO AX ELLIPSK 153 a straight line ■which meets an ellipse at only one point is the tangent at that point. 172. The tangents at the extremities of either axis are parallel to the other axis. For the co-ordinates of A are a, 0. (See the figure to Art. 162.) Hence, putting x' = a, and y' = 0, the equation d^yy' + Vxx = aV becomes x= a, which is the equation to a straight line through A parallel to GY. Similarly the tan- gent at A' is parallel to GY, and the tangents at B and B' are parallel to GX. 173. To find the equation to the normal at any point of an ellipse. (See Definition, Art. 97.) Let x', y' be the co-ordinates of the point ; the equation to the tangent at that point is ^^"^''V' ^^' The equation to the straight line through (x, y') at right angles to (1) is y-y=^{^-«'') (2). This is the equation to the normal at {x', y). 174. The equation to the normal may also be expressed in terms of the tangent of the angle which the straight line makes with the major axis of the ellipse. The equation to the normal at {x', y) is y = j^x—(rt — ljy'. Let TT^, = m ; thus the equation becomes ox y=mx p— y (1); a' — V we have then to express — vy— y in terms of m. Now, 6V = ^ , and aY + h'x" = a'V; therefore aV°+Trn, = o''6'; 154 PROPERTIES OF THE ELLIPSE. therefore tf'*{bW + a") = hW. Hence (1) becomes y = mx — AjibW + a") .(2). 175. We shall now deduce some properties of the ellipse from the preceding Articles. f Let x, y be the co-ordinates of P; let FT be the tangent at P, and Pff the normal at P; PM, PAT" perpendiculars on the axes. The equation to the tangent at P is ai'yy' + Ifxx' = a'V. Let y=0, then x = — , hence CT= -^ ; therefore GM.CT=GA\ Y Similarly, if the tangent at P meet (7 Fat T'. ON. or = Off. 176. The equation to the normal at P is At the point O where the normal cuts the major axis. PROPERTIES OF THE ELLIPSE. 155 y = 0, hence from the ahove equation x — x' = j- ; thercr Or CM. fore x = x' {\-^ = eV. Thus GQ = e' At the point G' where the normal cuts the minor axis, aV oV a; = 0, hence from the above equation y = y' — ^ = — rj- y. ThMS GG'=^PM. Suppose the focal distance PH produced to meet the ellipse again at p. Let Q denote the middle point of i^, and through Q draw a straight line parallel to the major axis meeting the normal PG at K. Then, by similar triangles, QK EG ae- e^x QP HP a-ex = e: thus QK=e.QP = ^.Pp. If K had denoted the point of intersection of the straight line through Q and the normal at p, we should have obtained the same value of QK; hence we have the following result : the straight line parallel to the major axis which passes through the intersection of normals at the ends of a focal chord bisects that chord, 177. The lengths of PG and PG' may be conveniently expressed in terms of the focal distances of P. PG^ = PJf » + GM' = y" + {x - e'xy Let SP=r', HP = r; then r' = a + ex', r = a-ex'; thus -P^* =—;«-• O TV Similarly, it may be shewn that PC* = -tj-. 156 PROPERTIES OF THE ELLIPSE. 178. The normal at any point bisects the angle between the focal distances of that point. Let x', y be the co-ordinates of P ; the co-ordinates of S are —ae, Oj hence the equation to /SPis (Art. 35) y = -, — Ix + ae). The equation to the normal at P is y — y' = -rr-r ^ — <>>)• Hence the tangent of the angle GPS ay if i V a;' + oe . . (a' - &') x'}/ + a'e/ ~ ■ ay ~ oy ' -I- fcV + 6Vae "•"ftVCa'-l-ae) _ a^e'x'y' -t- g'e/ _ ea/ ~ a't'-l-ftVae 6' " ihe equation to HP is y = -J^ — (x — ae); hence it may be shewn that the tangent of the angle GPH also = -—-; therefore the angle SPG = the angle HPG. Hence the angle fifPr' = the angle HPT; that is, the tangent at any point is equally inclined to the focal distances of that point. 179. The preceding proposition may also be established thus: GG = e'x', (Art. 176); therefore SG = ae + ^x', and HO — ae — ^ai, Also SP = a + ex', HP =a — ex'; hence SG _ SP HG~HP' therefore by Euclid, vi. 3,PG bisects the angle 8PH. 180. To find the locus of the intersection of the tangent at any point with the perpendicular on it from the focus. PEBPENDICULAB ON THE TANGENT. 157 Let y = mx+t/(b*+m''a*) (1) be the equation to a tangent to the ellipse (Art. 171) ; then the equation to the perpendicular on it from the focus H is (see the figure to Art. 175) y = ~-i^-ae) (2). K we suppose x and y to have respectively the same values in (1) and (2), and eliminate m between the two equa- tions, we shall obtain the required locus. From (1) y — nia; = V(6' + niV); from (2) my + x = ae; square and add, then (y + a*) (1 + m') = 6' + «iV + aV = a'(l + m'); thus y* + a' = a* is the equation to the required locus, which is therefore a circle described on the major axis of the ellipse as diameter. We have supposed the perpendicular drawn from H; we shall arrive at the same result if it be drawn from S; hence if HZ, SZ' be these perpendiculars, CZ and CZ" each = a. 181. To find the length of the perpendicular from iJie focus on the tangent at any point. The equation to the tangent at the point (a;', y") is y= — i-,x + —. The co-t)rdinates of the focus H are ae, 0. But if p de- note the length of the perpendicular from a point (a;,, y^ on the straight line y = mx + c, then by Art. 47 ^~ 1-hwi' • In the present case x^-=ae, y^^O, m = — riy c = — 5 i/ if (Vx'ae V\ ' _ g'y (g - ex'Y _ a*b* (a - ex" -6V' f b'x'ae 6'Y , V"tfV" v) _ a*b*{a-ex')' _ a%\a-exj thus p T"^ aV'-h iV a\a'b'-Vx'^) + b' ay loS TWO TANGENTS FKOM AN EXTERNAL POINT. a'bXa—ex^y h'Ca — ex') b^r ... _„„ = -vA a-k = — ^ 1 = —r • C-A-rt. 17n. a'{a—ex^ a + ex r ^ 6V Since »•' = 2a — r we have p' = s . Similarly if p' be the perpendicular from S on the tan- gent at (a;', y) we shall find ^" = — ■ Therefore pp' = 6'. From the point G in the figure of Art. 175 suppose a perpendicular drawn on PH meeting it at i2; then by similar triangles -p^ = -™, ; therefore PR = PG x ^frp • Substitute the value of HZ just obtained, and the value of PQ from 12 Art. 177, and we have PR = - . Thus the length PR is constant and equal to half the latus rectum. 182. Frcym -any external point two tangents can he drawn to an ellipse. Let the equation to the ellipse be a'y' + JV = o*&', and let h, k be the co-ordinates of an external point. Suppose x', y' the co-ordinates of a point on the ellipse, such that the tangent at this point passes through (Ji, k). The equation to the tangent at (»', y') is a'yy + i'xx = a'fe*. Since this tan- gent passes through (h, k) a^ky' + VU =.a%^ (1). Also since [x', y) is on the elUpse ay* + Vx"' = a%^ (2). Equations (1) and (2) determine the values of x' and y'. Substitute from (1) in (2), thus (^^JZ^fl^j + JV" = o'i!.', or a!\a^F+yh?)-2a^b^hx' ^- a* [V-¥) = Q. The roots of this quadratic will be found to be both possible since {h, k) is an external point and therefore d^W + hVi? greater than aW, The straight line which passes through the points where these tangents meet the ellipse is called the chord of contact. 183. Tangents are drawn to an ellipse from a given ex- ternal point: to find the equation to the chord of contact. CHORD OF CONTACT, , 159 Let h,khe the co-ordinates of the external point; a;,, y, the co-ordinates of the point where one of the tangents from (h, k) meets the ellipse ; a;,, y, the co-ordinates of the point where the other tangent from (h, k) meets the ellipse. The equation to the tangent at (a;,, y,) is a^yy^ + b^xx^ = a'b*; since this tangent passes through {h, k) we have a%, + 6'Aa;, = a»J' (1). Similai:ly, since the tangent at (x^, y,) passes through (A, k) ' a'h/^ + b'hx^^a'b' (2). Hence it follows that the equation to the chord of contact is a^ky + Vhx=a?b* (3). For (3) is ohviously the equation to some straight line; also this straight line passes through {x^, y^ for (3) is satisfied by the values a; = a;,, y = y, as we see from (1) ; similarly from (2) we conclude that this straight line passes through (a;,, y^. Hence (3) is the required equation. Thus we may Ose the following process to draw tangents to an ellipse from a given external point : draw the straight line which is represented by (3); join the points where it meets the ellipse with the given external point, and the straight lines thus obtained are the required tangents. 184. Through any fixed point chords are drawn to an ellipse, and tangents to the ellipse are drawn at the extremities of each chord: the locus of the intersection of the tangents is a straight line. Let h, k be the co-ordinates of the point through which the chords are drawn; let tangents to the ellipse be drawn at the extremities of one of these chords, and let {x^, y,) be the point at which they meet. The equation to the corresponding chord of contact is, a'yy^ + h^xx^ = aV, by Art. 183. But this chord passes through {h, k); therefore a%, + S'AaTj = o'J'. Hence the point (x^,y^) lies on the straight line d^ky+Vhx=a^b^; that is, the locus oi the intersection of the tangents is a straight line. We will now prove the converse of this proposition. 160 TANGENTS FROM AN EXTERNAL POINT. 185. If from any point in a straight line a pair of tan- gents he dragon to an ellipse the chords of contact will all pass through a fixed point. Let Ax + By+ C=0 (1) be the equation to the straight line ; let {a/, y) be a point in this straight line from which tangents are drawn to the ellipse} then the equation to the corresponding chord of contact is a'yy' + b'xx' = a'b' (2), Since (x, y) is on (1), we have Ax' + By + C = ; therefore (2) may be written Vxx' —^^— a^y = aV, or. (6.^_^)a,'-^-a'6* = 0. Now, whatever be the value of x', this straight line passes through the point whose co-ordinates are found by the simul- taneous equations ox ^ = 0, ^ + o o = 0, that is, the point for which y = ^ , a; = — ^r- • The student should observe the different interpretations that can be assigned to the equation a^ky + Vhx = a^b*. The statements in Art. 103 with respect to the circle may all be applied to the ellipse. 186. Some interesting geometrical investigations relat- ing to tangents to an ellipse from an external point may be noticed. To draw the two tangents to an ellipse from any external Let denote the external point, and S either focus. On OS as diameter describe a circle and let it cut the circle described on the major axis as diameter at Z and z. Join OZ and Oz. Then these straight lines, produced if necessary, are the tangents from by Art. 180 and Euclid, iii. 31. Or we may proceed thus. Let denote the external point, *Si the more remote focus. With S as centre and radius equal to the major axis of the ellipse describe a circle. Let H be the other focus. With as centre and radius equal to OM describe another circle cutting the former at Q and q. Join TANGENTS FBOH AN EXTERNAL POINT. 161 ■SQ and Sq cutting the ellipse at P and p; then OP and Op are the required tangents. For join OS, Off, OF, and OQ. Then in the triangles OPQ and OPff we have OQ = Off by construction, PQ = 2a — SP = Pff, and OP common. Therefore the angle OPQ = the angle OPff; and OP is the tangent at P by Art. 178. Similarly Op is the tangent at p. The two tangents to an ellipse from an external point sub- tend equal alleles at each focus. Join .Sp and Oq. The triangles OiSfQ and OSq are equal in all respects ; thus the tangents OP and Op subtend equal angles at S. Also the angle OffP = the angle OQP, and the angle OHp = the angle Oqp : thus the tangents OP and Op subtend equal angles at ff. The angle between a tangent and a focal distance of the external point is eqwal to the angle between the other tangent and the other focal distance. The angle SOQ = the angle SOq ; that is, twice the angle SOP+ the angle SOH = twice the angle HOp + the angle SOH; therefore the angle SOP = the angle ffOp, and also the angle ffOP = the angle iSOp. T f a W X62 EXAMPLES. CHAPTER IX. The student should notice the extension which is thus obtained of the result in Art. 178. At any point of the curve the straight line which bisects the angle between the focal distances is at right angles to the tangent ; at any external point the straight line which bisects the angle between the focal distances bisects the angle between the two tangents. EXAMPLES. 1. Find the excentricity of the ellipse 2a!'' + St/* = C*. 2. Find the equation to the tangent at the end of the latus rectum L. (See the figure to Art. 162.) Also find the lengths of the intercepts of this tangent on the axes. 3. Write down the equation to the normal at L. 4. If the normal at L passes through the extremity of the minor axis B', find the excentricity of the ellipse. 5. Find the equation to A'B and to CL. (See the figure to Art. 162.) Find the excentricity of the ellipse if these straight lines are parallel. 6. Find the equation to B'H, and determine the abscissa of the point where this straight line cuts the ellipse again. 7. Find the equation to AL, and determine the angle between this straight line and the tangent at L. 8. If from the point P whose abscissa is x', a straight line be drawn through H, determine the abscissa of the point where it meets the ellipse again. 9. Find a point in the ellipse such that the tangent there is equally inclined to the axes. 10. Find a point in the ellipse such that the intercepts made by the tangent on the co-ordinate axes are proportional to the corresponding axes of the ellipse. 11. P is a point on an ellipse, y its ordinate : shew that tan^P^' = --^. EXAMPLES. CHAPTER IX. 163 12. P- is a point on an ellipse, y its ordinate : shew that the tangent of the angle between the focal distance and the 6' tangent at P is — . 13. If denote the angle mentioned in the preceding Example, shew that PG—>J[a^— V cot'^). 14. From P a point on an ellipse straight lines are drawn to A, A', the extremities of the major axis, and from A, A' straight lines are drawn at right angles to AP, A'P: shew that the locus of their intersection will be another ellipse, and find its axes. 15. If any ordinate MP be produced to meet the tangent at L at Q, prove that QM= PH. (See the figure to Art. 162.) 16. If a series of ellipses be described having the same major axes the tangents at the ends of their latera recta will pass through one or other of two fixed points. 17. If the focus of an ellipse be the common focus of two parabolas whose vertices are at the ends of the major axis, these parabolas will intersect at right angles, at points whose distance &om each other is equal to twice the minor axis. 18. Shew that the length of the longer normal drawn from a point in the minor axis of an ellipse at a distance c from the centre and intercepted between that point and the curve is {^•^ff 19. If any parallel straight lines be drawn from the focus H and the extremity A of the major axis of an ellipse, and if M and iV be the points where they meet the minor axis, or the minor axis produced, then the circle whose centre is M and radius NA will either touch the ellipse, or fall entirely outside of it. 20. A and A' are the extremities of the major axis of an ellipse, T is the point where the tangent at the point P of the curve meets AA' produced ; through T a straight line is drawn at right angles to AA' and meeting AP and AP' produced at Q and It respectively : shew that Q T= RT. 11—2 164 EXAMPLES. CHAPTER IX.. 21. If , ' be the excentric angles of two points, the equation to the chord joining the points is ? cos^' + f sin *±-^' = cosi^^. a 2 b 2 2 22. Express the equation to the tangent at any point in terms of the excentric angle of that point, 23. Shew that the equation to the normal at the point whose excentric angle is ^ is ax sec (f> — by cosec ^ = a^ — b\ 24. The locus of the middle point of PG (see Art. 176) is an ellipse of which the excentricity e is connected with that of the given ellipse by the equation 1 — e' = (1 + e')*(l — e**). 25. Determine the point of intersection of the tangent at L with the straight line HB ; find the value of the excentri- city of the ellipse when these straight lines are parallel. 26. A tangent at any point P of an ellipse meets the directrix EK at T and E'K' at 2" : shew that TE varies as the cotangent of PffS, and T'E' varies as the cotangent of PSH. (See the figure to Art. 162.) 27. If the straight line y = mtc + c intersect the ellipse a'y + iPa^ = a'6', shew that the length of the chord will be 2ah V{(1 + to') (mV + &' - c')} niW + b' Hence find the relation between the constants that this straight line may be a tangent to the ellipse. 28. Find the equation to the circle described on HP as diameter, supposing x', y' the co-ordinates of P. 29. Shew that any circle described on HP as diameter, touches the circle described on the major axis as diameter. 30. From a point (k, k) two tangents are drawn to an ellipse : find the sum of the perpendiculars from the foci on the chord of contact. 31. Any ordinate PM of an ellipse is produced to meet the circle on the major axis at Q, and normals to the ellipse and circle at P and Q respectively meet at B: find the locus of R. EXAMPLES. CHAPTER IX. 165 32. Two ellipses have a common centre and their axes coincide in direction ; also the sum of the squares of the axes is the same in the two ellipses : find the equation to a common tangent. 33. If 0, & he the inclinations to the major axis of the ellipse of the two tangents that can he drawn from the point (A, A;), shew that tan0+tan0'= = — r^i tan0tanfl' = -s — ^,. ar—hr a — ft 34. Find the locus of a point such that the two tangents from it to an ellipse are at right angles. 3.5. Shew that the two tangents which can he drawn to an ellipse through the point (h, k) are represented hy (a'-h')(if-ky+2{y-k){x-h)Ak+{¥-k'){a!-hy=0, or hy {a'k^ + ¥h^ - a'i') (oy + 6V - aW) = (a'Ay + ¥hx - a'by. 36. Tangents are drawn to an ellipse from the point (A, k) : shew that the straight lines drawn from the origin to the points of contact are represented by-j + ^=f— j+-j|j . 37. Pairs of radii vectores are drawn at right angles to each other from the centre of an ellipse : shew that the tan- gents at their extremities intersect on the ellipse o* "*" 6* ~ «*''"&" ■ 38. From an external point T whose co-ordinates are h and k a straight line is drawn to the centre G meeting the ellipse at H: shew that GE' a'b' • 39. From an external point {h, k) tangents are drawn : if X , x^ he the abscissae of the points of contact, shew that 2haV _ a*(y-fc ') "''^'"^'aV + b-'h!" "''"'' a'k'+bV 40. From an external point (h, k) tangents are drawn meeting the ellipse at P and Q : find the value of HP . HQ, H being a focus. 166 EJUMPLES. CHAPTEB IX. 41. From an external point Tthe straight lines TP, TQ are drawn to touch the ellipse at P and Q. GT cats the ellipse at B, and JBiV is drawn parallel to HT to meet the major axis at N: shew that ffP.HQ = EN\ 42. Two ellipses of equal excentricity and whose major axes are parallel can only have two points in common: prove this, and shew that if three such ellipses intersect, two and two, at the points P and P', Q and Q', E and JR', respectively, the straight lines PP', QQ', RR', meet at a point. 43. Two concentric ellipses which have their axes in the same direction intersect, and four common tangents are drawn so as to form a rhomhus, and the points of intersection of the ellipses are joined so as to form a rectangle : prove that the product of the areas of the rhombus and rectangle is equal to half the continued product of the four axes. 44. The ordinate at any point P of an ellipse is produced to meet the circle described on the major axis as diameter at Q : prove that the perpendicular from the focus S on the tan- gent at Q is equal to SP. 45. Find the equation to the ellipse referred to axes passing through the extremities of the minor axis, and meet- ing at one extremity of the major axis. 46. If from points of the curve t + -5 = (a*— b^y, tangents be drawn to the ellipse -j + p = 1, the chords of contact will be normal to the ellipse. 47. Pi-ove the proposition in Art. 180 in a manner similar to that used in Art. 138. Also prove the proposition in Art. 138 in a manner similar to that used in Art. 180. 48. Find the equation to the ellipse the origin being the point (h, k) on the ellipse and the axes parallel to the axes of the ellipse. 49. From a point P on an ellipse two chords PQ, PQ are drawn meeting the ellipse at Q, Q; if A, k be the co-ordi- nates of P referred to the centre, and mx -|- ny = 1 the equation to QQ' referred to P as origin, and axes parallel to the axes of the ellipse, shew that with P as origin the straight lines EXAMPLES. CHAPTER IX. 167 -PQi -PQ' are represented by «• , y* , f2xh . 2yk\ , a« + & + (-^ + -^j^'"^ + "^) = 0- 50. Let P be any point on an ellipse ; draw PP parallel to the major axis and cutting the curve at F ; through P draw two chords PQ, PQ, making equal angles with the major axis; join QQ : shew that QQ' is parallel to the tangent at P'. 51. From the equation y = mx-\- ^[rr?a^ + 6') deduce the equation to the tangent to the parabola. 52. In the figure of Art. 175 suppose GP produced to a point Q such that GQ = n. GP, and fcid the locus of Q. 53. ? If PN be any ordinate of a circle, and from the ex- tremity A of the corresponding diameter AB, AQ be drawn meeting PN at Q, so that AQ = PJT, find the locus of Q and the position of its focus. 54. Express the tangent of the angle between CP and the normal at P in terms of the co-ordinates of P. 55. Find the greatest value of the tangent of the angle between CP and the normal at P. 56. The major axis of an ellipse is equal to twice the minor axis ; a straight line of length equal to half the major axis is placed across the major axis with one end on the curve and the other on the minor axis : shew that the middle point of the straight line is on the major axis. 57. A circle is inscribed in the triangle formed by two focal distances and the major axis of an ellipse : find the locus of the centre. 58. If SZ', HZ be perpendiculars on the tangent at the point P of an ellipse, SZ and HZ' will intersect on the normal at P. 59. Shew that the equation to the two straight lines which join the point (A, k) with a focus of the ellipse i» (% - hxY - aV(y - kf = 0. 60. Shew that the straight lines in Examples 35 and 59 have the same bisectors of their angles. ( 168 ) CHAPTER X. THE ELLIPSE CONTINUED. Diameters. 187. To find the length of a straight line dravm from any point in a given direction to meet an ellipse. Let of, y' be the co-ordinates of the point from which the straight line is drawn ; x, y the co-ordinates of the point to which the straight line is drawn; 6 the inclination of the straight line to the axis of x; r the length of the straight line; then (Art. 27) a; = a;'-|-rcos^, y = y* -1- r sin 0. If (x, y) be on the ellipse these values may be substituted in the equation a^jf + Va? = a'6'; thus a" (y' + r sin 6)' + V(af + r cos Of = a'ft'; therefore r»(a* sin'^ + b' cos*^ + 2r(a'y' sin ^ -h 6V cos 6) From this quadratic two values of r can be found which are the lengths of the two straight lines that can be drawn from (x', y') in the given direction to the ellipse. 188. To find the diameter of a given system of parallel chords in an ellipse. (See Definition, Art. 148.) Let 6 be the inclination of the chords to the major axis of the ellipse ; let x', y be the co-ordinates of the middle point of any one of the chords ; the equation which determines the lengths of the straight lines drawn from (x', y) to the curve is (Art. 187) r'(o» sin'e -j- V cos*^) -|- 2r (ay sin 6 + 6V cos 0) -faY'-}-6V'-a'6' = (1). DIAMETERS OF THE ELLIPSE. 169 Since {x, y') is the middle point of the chord, the values of r furnished by this quadratic must be equal in magnitude and oj^osite in sign ; hence the coeflScient of r must vanish ; thus ay sin 5 + 6Vcos ^ = 0, or y* = - -5 cot ^ . x' (2). Considering x and y' as variable, this is the equation to a straight line passing through the origin, that is, through the centre of the ellipse. Hence every diameter passes through the centre. Also every straight line passing through the centre is a diameter, that is, bisects some system of parallel chords ; for by giving to a suitable value the equation (2) may be made to represent any straight line passing through the centre. If be the inclination to the axis of x of the diameter which bisects all the chords inclined at an angle 6 we have from (2) tan ff — — ■, cot ^ ; therefore tan 6 tan v = ; . a 189. If one diameter bisect all chords parallel to a second diameter, the second diameter will bisect all chords parallel to the first. Let 6^ and 5, be the respective inclinations of the two diameters to the major axis of the ellipse. Since the first bisects all the chords parallel to the second, we have tan P, tan 0^= — rj . And this is also the only condition that must hold in order that the second may bisect the chords parallel to the first. 190. The tangent at either extremity of any diameter is parallel to the chords which that diameter bisects. Let h, k be the co-ordinates of either extremity of a diameter ; 6 the inclination to the major axis of the ellipse of the chords which the diameter bisects. Then the values x = h, y = k must satisfy the equation a'y sin d + b'x cos 0= 0; Jfh therefore tan = — jr . But, by Art. 170, the equation to the Of iC 170 CONJUGATE DIAMETERS OF THE ELLIPSE. 121 tangent at (h,k) is i/ — k= ^ {x — h). Hence the tangent is parallel to the bisected chords. 191. Definition. Two diameters are called conjugate when each bisects the chords paiuUel to the other. From Art. 190 it follows that each of the conjugate dia^ meters is parallel to the tangent at either extremity of the other. 192. Given the co-ordinates of one extremity of a diameter to find those of either extremity of the conjugate diameter. Let AGA', BCE be the axes of an ellipse ; POF, BCD' a pair of conjugate diameters. Let x', y be the given co-ordinates of P; then the equa- tion to CP is y=lx. .(1). Since the conjugate diameter DiX is parallel to the tangent at P, the equation to DD' is 2^ = -^^ (2)- CONJUGATE DIAMETERS OF THE ELLIPSE. 171 We must combine (2) with the equation to the ellipse to find the co-ordinates of D and If. Substitute the value of y from (2) in aY + 6V = a'6'; then a' -^ ^' o? + 6V = a'6' ; therefore (6V* + a^y"^ a? = aV ; therefore a;' = ^' = ^'; therefore x=±-r-; therefore from (2) y=+ — . In the figure the abscissa of D is negative and that of D' positive ; hence the upper sign applies to D' and the lower sign to 1). The properties of the ellipse connected with conjugate diameters are numerous and important j we shall now give a few of them. 193. The sum of the squares of two conjtigate semi-dia- meters is constant. Let x, y be the co-ordinates of P; then by the preceding Article h' "*■ a' = a' + 6'. Thus the sum of the squares of two conjugate semi-diame- ters is equal to the sum of the squares of the semi-axes. Moreover CD' = a' + V-x'-y'^ = a' + V-x'^ - ^] (a' - a;") = a' - (l - -]) x^ = a* - eV = SP.HP by Art. 166. 194. The area of the 'parallelogram formed by tangents at the ends of conjugate diameters is constant. Let POP', BCD' be the conjugate diameters (see the figure to Art. 192). The area of the parallelogram described so as to touch the ellipse at P, D, F, D', is ^GP.GDsmPGD, or 172 PEEPENDICULAB FROM THE CENTKE ON THE TANGENT. 4p . CD, where j> denotes the perpendicular from C on the tangent at P. Let x', y be the co-ordinates of P; then the equation to the tangent at P is y = j-, !e-\ — > . Hence (Art. 47) ^ == _^_£_ = -^^-,^^^^ . therefore 4p . GH = 4o6. Thus the area of any parallelogram which touches the ellipse at the ends of conjugate diameters is equal to the area of the rectangle which touches the ellipse at the ends of the axes. 195. Let a', b' denote the lengths of two conjugate semi- diameters ; a the angle between them ; by the preceding Article a'b' sin a = a6 ; therefore ., _ a'b" 4a'y 4a'y ^""^ a"b" (a" 4- by - (a" - b")' ~ (a' + by - (a'^ - by ' Hence sin' o has its least value when a' = V, and then 2ai Bma= ' 196. From Art. 194 we have a'6» aV ^ CD* ~ a* + b'- OF* ^^^' ^^^^- This gives a relation between p the perpendicular from the centre on the tangent at any point P and the distance GP of that point from the centre. In Arts. 177 and 193 it is shewn that PC" = -] GI^. Hence p.PQ = b\ Similarly j».PG' = a'. We may also express p in terms of the angle its direction makes with the major axis ; for let yjr denote the angle, then EQUATION BEFEBSED TO CONJUGATE DIAUETEBS. 17S the equation to the tangent at {x', y') is a'yy + Vxa^ = a'b', and tms may also be put in the form (Art 20) X cos -^ + y sin ■\^=p. Hence -r^ = - , - ^ , = — ; sm^^ y cos'^ x ^, . , «J*sini/r J , a'b cos-sir therefore ay = , ox = — : and therefore a'6* = — j- (6' sin*'^ + a' cos' ■^) ; therefore p* = b* sin''^ + a' cos*'^ = a" (1 — e' sin*'^). 197. Let if> and ^' be the excentric angles corresponding to P and D respectively (Art 168). Then a;' = ocos^ (1), y = &sin^ ..(2). Y-acosf (3), bx , . ,, — = &sm0 ..(4). From (2) and (3) cos ' = — sin ^, from (1) and (4) sin ^' = cos ; therefore ^' = ^ + ^. 198. To find the equation to the ellipse referred to a pair of conjugate diameters as axes. Let CP, CD be two conjugate semi-diameters (see the figure to Art. 192), take CP as the new axis of x, CD as that of y ; let PGA = a, DC A = /8. Let x, y be the co-ordinates of any point of the ellipse referred to the original axes ; x, y' the co-ordinates of the same point referred to the new axes ; then (Art. 84) a; = a;' cos a -I- y* cos /8, y = a/ sin a + y' sin /3. Substitute these values in the equation ay-l-6V = a'6'; then a^(x' sin a -|- y' sin /3)' + &•(«' cos a + y cos /8)' = a*V. or a;'»(a»Bin*a + W cos* a) H- y"(a' sin»/3 + V cos'^) + 2a;'y' (a* sin a sin /3 + 6' costt cos jS) = a'6'. 174 EQUATION REFERRED TO CONJUGATE DIAMETERS. But, since CP and CD are conjugate semi-diameters, tan a tan /8 = — ^ ; hence the coefficient of xy vanishes, and the equation becomes x'* (a' sin" a + 6' cos" a) + y" (a* sin» y3 + 5' cos' /3) = a%\ In this equation, suppose x = 0, then 'I _ o'i' ^ o*sin''/S + t'cos'';8" This is the value of CD', which we shall denote by J**; similarly we shall denote GP^ by o", so that a sin a + cos a Hence the equation to the elUpse referred to conjugate diameters is a" ^6" ^' or, suppressing the accents on the variables, 199. A particular case of the preceding is when a = b'; then o' sin' fi + b^ cos* /3 = a* sin' a + 6' cos' a ; therefore a' (sin* /3 — sin* a) = 6* (cos* a — cos' /8) = 6' (sin* /3- sin' a); therefore (a* — 5*) (sin* /9 — sin' a) = ; therefore sin' ^ = sin' a; therefore /3 = tt — a. And since a" = J" each of them = ° ^ , (Art. 193). Hence from the value of a'* in the preceding Article, we have g' + y ^ a'y . 2 o' sin' a + 6' cos' a' TAIJGENTS AT THE EX.TBEMITIES OF A CHORD. 175 therefore (o* + J») j(a' - b') sia» o + 6'} = 2aV; therefore sm'a=. ..,.., — j^, = » , ., . (o" + 6") (o* — b) a* + b' This shews that the eqiuil conjugate diameters are parallel to the straight lines BA and BA'. 200. The equation to the tangent to the ellipse will be of the same form whether the axes be rectangular or the oblique system formed by a pair of conjugate diameters ; for the in- vestigation of Art. 170 will apply without any change to the equation a'*y' + i'V = o"6'' which represents an ellipse re- ferred to such an oblique system. 201. Tangents at the extremities of any chord of an ellipse meet on the diameter which bisects that chord. Eefer the ellipse to the diameter bisecting the chord as the axis of X, and the diameter parallel to the chord as the axis of y; let the equation to the ellipse be a"y^ + b'*x'^ = a'%'\ Let «', y' be the co-ordinates of one extremity of the chord ; then the equation to the tangent at this point is a'^yy' + b'^xx=a%'* (1). The co-ordinates of the other extremity of the chord are x', — y, and the equation to the tangent there is -a"yj/ + b'^xx' = a"b'* (2). The straight lines represented by (1) and (2) meet at the '2 point for which y = 0, x = —r: this proves the theorem. Supplemental chords. 202. Definition. Two straight lines drawn from a point of the ellipse to the extremities of any diameter are called supplemental chords. They are called principal sup- plement^ chords if that diameter be the major axis. 203. If a chord and diameter of an ellipse are parallel, tlie suj^lemental chord is parallel to the conjugate diameter. Let FF be a diameter of the eUipse; QP, QP' two sup- 176 SUPPLEMENTAL CHORDS. plemental chords. Let x', y' be the co-ordinates of P, and therefore — x, — y the co-ordinates of P'. Let the equation to PQ be (Art. 32) y-'!/ = m,{x-x') (1), and the equation \x> FQ y + y' = mXx + x') (2). The co-ordinates of the point Q satisfy (1) and (2) ; if then we suppose x, y to denote those co-ordmates, we have from (1) and (2) by multiplication 2/*-y" = mm'(a;*-a!'^ (3). But since {x, y) and {sd, y') are points on the elUpse ay 4- 6 V = a^b\ a 'y" + h^x'* = o'6' ; therefore a' {y^ - y") +V{^- x") = ; therefore •t^-^=^-\ (af-uP) (4). From (3) and (4) we have wm^ = ; . But we have shewn in Art. 188 that if this relation be satisfied, the two straight lines represented by w = mx and y = m'x are conjugate' diameters ; this proves the theorem. POLAK EQUATION TO THE ELLIPSK 177 Polar Equation. 204. To find the polar equation to the ellipse, the focus being the pole. Let SP = r, A'SP = d, (see the figure to Art. 158) ; then SP = ePJSr. by definition ; that is, SP=e{OS+ SM) ; or r = a (1 - e") + er cos (tt - 0), (Art. 161) ; therefore r (1 + e cos 0) = a (1 — e*), and 1 + e cos ff ' If we denote the angle ASP by 0, then we have as before SP = e(OS+SM) ; thus r = a(l- e") + er cos d. . a{l-e") and r = ^j— ^^ '-j, . 1— e cos ff 205. We shall make use of the preceding Article in finding the polar equation to a, chord, from which we shall deduce the polar equation to the tangent. Let P and P' be two points on the ellipse ; suppose that A'SP=a-^, and A'SP' = a + ^, so that PSP'=2^; and let I be the semi-latus rectum of the ellipse, so that Z = a (1 — e') : it is required to find the polar equation to the straight line PP'. Assume for the equation (see Art. 29) Arcos0 + Brsm0 + C = O (1). T.C.S. 12 178 POLAR EQUATION TO A CHORD. Since the straight line passes through P, equation (1) must he satisfied by the co-ordinates of F; now A'8P = a — /8, and therefore SP= =— -, 5s > *^^s from (1) 1 + e cos (a — (8) Z{ilcos(a-/S)+J5sin(a-/S)}+C{l+ecos(a-/3)}=0 (2). Similarly, since the straight line passes through P', Z{^cos(a+/3)+5sin(a+/3)} +C{l+ecos(a+/3)} = (3). From (2) and (3), by subtraction, I (A sin a sin /3 — .B cos a sinyS) + (7e sin a sin /S = ; therefore { (A sina — £cosa) + Cesina = (4). From (2) and (3), by addition, Z(^cosacos/3 + £sinacos^) + (7(1 -t-ecosacos/S) =0; therefore Z(J.cosa+Psina) + 0(sec/8+ecosa) =0 (5). From (4) and (5) we find lA + C(sec)3 cosa + e) = 0, IB -\- (7 sec ^ sin a = 0. Substitute the values of A and B in (1) and divide by C; thus r \ (sec /8 cos a + e) cos 5 + sec /3 sin a sin ^ [• — Z = ; therefore r = - -. -^ ; rr. (6). ecost^-l-secy8cos(a — P) ^ ' If SQ bisect the angle P/SF, we have PSQ=(3, &ndLA'SQ = a. Now suppose ^ to diminish indefinitely ; then the chord PP' becomes the tangent at Q, and we obtain its polar equation by putting /3 = in the preceding result ; thus we have I ecos0 + cos(a— ^) " The investigations of this Article will apply to the para- bola by supposing e — 1. The investigation of (6) has been put in the following MISCELLANEOUS PROPOSITIONS. 179 brief form by Mr F. G. Landon. The equation to 8P is 6 = a—^, and that to SP" is6 = a + p; therefore the equation d—a=±fi represents the two straight lines SP and SP" : this may be written cos (^ — o) = cos ^, or cos (0— a) secy3 = 1. The equation to the ellipse is — e cos = 1. Combining these equations we get — e cos = cos {0 — a) sec /8, which must be satisfied at the points P and P' ; and as this is the equation to a straight line it is the equation to the straight line PP'. 206. The polar equation to the ellipse referred to the centre is sometimes useful; it may be deduced from the equation oy + fcV = o'i', by putting rcos^, rsin ^, for x and y respectively : we thus obtain r^Xa*sm^0+b'cos*0) = a'6'. We add a few miscellaneous propositions on the ellipse. 207. If tangents he drarvn at the extremities of any focal chard of an ellipse, (1) the tangents wiU intersect on the corre- sponding directrix, (2) the straight line drawn from the point ^intersection of the tangents to the focus will be perpendicular to the focal chord. (1) If two tangents to an ellipse meet at the point {h, k) the equation to the chord of contact is, by Art 183, a% + b^hx = a*b\ Suppose the chord passes through the focus whose co-ordi- nates are x = — ae, y = ; then — fe%ae = o'6', therefore A = — ; e that is, the point of intersection of the tangents is on the directrix corresponding to this focus. (2) The equation to the straight line through (h, k) and Jc a the focus is y = T {x+'ae)j If A = --, this becomes y ~ KZr^ (* + ffl«) = iTi (a: + a«)j ^^^ *^s straight line is therefore perpendicular to the focal chord of which the ^. . Vhx ^ 6" equation is y = ir + t • 12—2 180 BECTANGLE OF THE SEGMENTS OF A STRAIGHT LINE. 208. Iftkrmigh any point within or without an ellipse, two straight lines he drawn parallel to two given straight lines to meet the curve, the rectangles of, the segments will he to one an- other in an invariable ratio. Let (a;, y') be the given point and suppose a and /3 respec- tively the inclinations of the given straight lines to the major axis of the ellipse. By Art. 187 if a straight line be drawn from {x, y') to meet the curve, and be inclined at an angle a to the major axis, the lengths -of its segments are given by the equation r' (a' sin'a + ¥ cos' a) + 2r (aV sin a + 6V cos a) + aY' + 6'ai'-a,'i»=0; therefore the rectangle of the segments = ^ . , — ' , ~ „ . Similarly the- rectangle of the se^ents of -the straight line drawn from {x,.y) at an angle ^ = a'^in' ^ + V oos' ^ • TT XI, *• e■^^. * 1 a'sin'^+ft'oos'yS , Hence the ratio of the rectangles = ■ . ■ . ,. — i- : and ^' asm"a+6cosa this ratio is eonstant whatever-a;' and y'- may b& Let be the point through which the straight lines OPp, OQq, are drawn inclined to the major axis of the ellipse at angles a, /3, respectively ; then OP.Op ^ o'sin'^-l- fe'cos'^ Oq.Oq a'sin*a + &'cos''a ■ RECTANGLE OF THE SEGMENTS OF A STRAIGHT LINE. 181 Draw 'the semi^iiameters CD, GE, parallel to Pp, Qq, respectively, then by Art. 206, C£»~a'sin»a + 6'cos»a ' ^, . OP. Op CD' ''^'''^'''' QQ-Oq = OF'- Let TM, TN be tangents parallel to Pp, Qq, respectively ; then if coincides with T, the rectangle- OP . Op •becomes TM' and the rectangle OQ . Oq becomes TJV*; therefore TM^_GP^ TM _GD .TN'~CE'' ^^TN~GE- The preceding investigations are very important : we will point out some inferences which may be drawn from them. Suppose that an ellipse and a circle intersect at four points : denote these points by P, p, Q, q. Then we. have seen that OP.Of _ Clf Uii. Oq~ CE'' But since tlte four points are on the arcle we have OP . Op = OQ . Oq by EucUd, ni. 35 and 36, Cor. Therefore CD' = GE'. And since CD and GE are equal they make equal angles with the major axis of the ellipse. Thus if an ellipse and a circle intersect at four ^points the common chords make equal angles with the m,ajor axis of the ellipse. Suppose that Q and ,q coincide so that OQq becomes a common tangent to the ellipse and circle ; thus we obtain the following result : if an ellipse and a circle have a common tangent and a comvion chord, the tangent and the chord make equal angles with the major acais of the ellipse. We may conceive that the three points P, Q, and q move up to coincidence. The circle in this case is called the circle of curvature of the ellipse at the point of coincidence. We do not discuss the properties of the circle of curvature in the present work ; but we may remark that we have obtained the following result : the tangent at any point of an ellipse and the chord drawn from the pmni to tiie other intersection of the 182 EXAMPLES. CHAPTER X. ellipse and ^e circle of curvature at the point make equal angles with the major aids of the ellipse. Similar remarks may be made in comiexion with Art. 157. EXAMPLES. 1. CPand CD are conjugate semi-diameters: given the co-ordinates of P {x, y'), find the equation to PD. 2. If straight lines drawn through any point of an ellipse to the extremities of anv diameter meet the conjugate CD at the points M, N, prove that CM .CN = CD". 3. CP, CD are two conjugate semi-diameters ; OF", CD' are two other conjugate semi-diameters : shew that the area of the triangle PCP" is equal to the area of the triangle DGU. 4. Normals at P and D, the extremities of semi-conjugate diameters, meet at K : find the equation to KC, and shew that KG is perpendicular to PD. 5. In an ellipse the rectangle contained by the perpen- dicular from the centre upon the tangent, and the part of the corresponding normal intercepted between the axes, is equal to the difference of the squares of the semi-axes. 6. Shew that the locus of the intersection of the perpen- dicular from the centre on a tangent to the ellipse is the curve which has for its equation r* = a'cos'^-|-6'sin*0, the centre being the origin. 7. From A the vertex of an ellipse draw a straight line ARQ to Q the middle point of HP meeting SP at JJ : shew that the locus of E is an ellipse, and also the locus of Q, 8. Find the polar equation to the ellipse, the vertex being the origin and the major axis the initial line. 9. If any chord AQ meet the minor axis produced at B, and GP be a semi-diameter parallel to AQ, then AQ.AR = 2CP''. 10. A circle is described on AA' the major axis of an ellipse as diameter; P is any point in the circle; AP, A'P EXAMPLES. CHAPTEB X. 183 are joined cutting the ellipse at points Q and Q' respectively: shew that AP A'P _a^ + V' AQ'^A'Q'~ ¥ ' 11. If circles be described on two semi-conjugate diame- ters of an ellipse as diameters, the locus of their intersection is the curve defined by the equation 2 («* + y')' = aV + b'y\ 12. CP, CD are conjugate semi-diameters; GQ is per- pendicular to PB: find the locus of Q. 13. Find the points where the ellipse a(l — e')=r+ record cuts the straight line a (1 — e"^ = r sin ^ + r (1 + e) cos 6. 14. Write down the polar equations to the four tangents at the ends of the latera recta ; also the equations to the tan- gents at the ends of the minor axis : the focus being the pole. 15. Determine the locus of the intersection of tangents drawn at two points P, Q, which are taken so that the sum of the angles ASP, ASQ, is constant. 16. If PSp be a focal chord of an ellipse, and along the straight line SP there be set off 8Q a mean proportional be- tween SP and Sp, the locus of Q will be an ellipse having the same excentricity as the original ellipse. 17. Two ellipses have a common focus and their major axes are equal in length and situated in the same straight line: find the polar co-ordinates of the points of inter- section. 18. From an external point two tangents are drawn to an ellipse : find between what limits the ratio of the length of one tangent to the length of the other lies. 19. TP, TQ are two tangents to an ellipse, and GP, GQ are the radii from the centre respectively parallel to these tangents : prove that P'Q' is parallel to PQ. 20. From a point whose co-ordinates are A, A a straight line is drawn meeting the ellipse at P and p ; and CD is the parallel semi-conjugate diameter: shew that OP.Op _h\F 184 EXAMPLES. CHAPTEK X. 21. When the angle between the radius vector from the focus and the tangent is least, the radius vector = a. 22. When the angle between the radius vector from the centre and the tangent is least, the radius vector = ( — = — j • 23. FT,ptaxe tangentsattheextremitiesof any diameter Pp of an ellipse ; any other diameter meets P,T at.T, and its conjugate meets pta,tt; .also any tangent.meets FT at T' and pt at i : shew that PT : P2" :: pt' : pt. 24. Erom the ends P, B, of conjugate diameters in an ellipse, draw straight lines parallel to any tangent line ; and from the centre C draw any straight line cutting these straight lines and the tangent at points-^, d, t, respectively : then will (7/+ C(F= ce. 25. If tangents be drawn from different points of an ellipse of lengths equal to n times the ^emi-conjugate diameter at each point, then the locus of their extremities will be a con- centric ellipse with semi-axes equal to oV(n''+l)andiv'(n' + l). 26. Apply the equation to the tangent in Art. 171 to find the locus of the intersecticm. of- tangents at the extremities of conjugate diameters. 27. If 'from a point (of, y") of an ellipse a chord be drawn parallel to a fixed straight line, shew that the length of this chord varies as — ^ , ^ , where A is the iaclination of C03.J^ the tangent at {x', y') to the axis, and a the inclination of the fixed straight line to the axis. 28. If through any point Pof an ellipse two chords PQ, PR be drawn parallel to two fixed straight lines and making angles a and /8 respectively with the tangent at P, shew that the ratio of PQ cosec a to PR cosec ^ is constant. 29. A parabola is touched at the extremities of the latus rectum by an ellipse of given magnitude : find the latus rectum of the parabola. EXAMPLES. CHAPTEB X. 183 30. Tlie perpendicular from the centre on a straight line joining the ends of perpendicular diameters of an eUipse is of constant length. 31. Chords are drawn thsough the end of an axis of an ellipse : find the locus of .their middle points. 32. Chords of an ellipse are drawn through any fixed point: find, the locus of their middle points. 33. Two focal chords are .drawn in an ellipse at right angles to each other : find their position when the rectangle contained by them has respectively its gi-eatest and least value. 34. In an ellipse if 'PP' and QQ'be focal chords at right angles to each other l-e'_ 1— e' _ -1 ,1 3' + en .C/T~ iirra"'"" SP.SP' "^^Q.S^' AC^BC^' 35. PSp, QSq are focal chords; suppose T the point where the. straight lines PQ, pg meet : shew that TS is equally inclined to the focal chords, and that ,2* is on the directrix corresponding to S. 36. If r, be the polar. co»-ordinates of a point P, shew ^, , , rrD<7 J> J l+ecos^ that tan. MPZ= -r^rr 3 — jji ^^'^ = = — a — • V(2or — r* - 67 e sm 37. Perpendiculars are drawn from P and D the ex- tremities of any pair of conjugate diameters on the diameter y = x tan a : shew that the sum of the squares of the perpen- diculars is a' sin' a + ¥ cos' a. 38. The excentric angles of two points P and Q axe and <^' respectively : shew that the area of the parallelogram formed by the tangents at the extremities of the diameters through P and Q is . , ., — T^ 5 shew also that the area ° sm (^ — 9) is least when P and Q are the extremities of conjugate diameters. 186 EXAHPLES. CHAPTER X. 39. Shew that the equation to the locus of the middle points of all chords of the same length (2c) in an ellipse is '^ " + 6V + a' + 6' ^ "• ay- 40. Chords of an ellipse are drawn at right angles to one another through a point whose co-ordinates are h, k : if GP, CQ be the radii drawn from the centre parallel to the chords, and E, F the middle points of the chords, shew that 41. Given the co-ordinates of P, find those of the inter- section of the tangents at P and D. (See the figure to Art. 192.) 42. Shew that the equation «! , ^' _ , _ ( xibx-af) y{ay' + hx) _ ]' represents the tangents at P and B, supposing x', y' the co- ordinates of P. (See the figure to Art. 192.) 43. If CP, CD be any conjugate semi-diameters of an ellipse APBDA', and BP, BD be joined and also AB, A'P, these latter intersecting at 0, shew that BBOP is a parallel- ogram. 44. Shew that the area of the parallelogram in the preceding Example = ai/ + hx' — ah, where x', y' are the co- ordinates of P ; and find the greatest value of this area. 45. If a straight line be drawn from the focus of an ellipse to make a given angle a with the tangent, shew that the locus of its intersection with the tangent will be a circle which touches or falls entirely without the ellipse according as cos a is less or greater than the excentricity of the ellipse. 46. In an ellipse 8Q, HQ, drawn perpendicular to a pair of conjugate diameters, intersect at Q : prove that the locus of § is a concentric ellipse. EXAMPLES. CHAPTEK X. 187 47. Two ellipses have their foci coincident ; a tangent to one of them intersects at right angles a tangent to the other : shew that the locus of the point of intersection is a circle having the same centre as the ellipses. 48. Find what is represented by the equation x' + ^' = c' when the axes are oblique. 49. Shew that when the ellipse is referred to any pair of conjugate diameters as axes, the condition that y = mx and J" y = rthx may represent conjugate diameters is mm'= ^ . 50. The ellipse being referred to equal conjugate dia- meters, find the equation to the normal at any point. 51. From any point P perpendiculars PM, FN are drawn on the equal conjugate diameters: shew that the normal at P bisects MN. 52. An ellipse intersects the side FQ of a triangle at r and r', the side QR at p and p, and the side RP at q and q: shew that Pr.Pr'.Qp.Qp'.Rq.Rq' = Pq.Pq'.Qr.Qr'.Rp.Rp. Shew also that a similar result is true for a polygon ; and shew what it becomes when the ellipse touches the sides. ( 188 ) CHAPTER XI. THK HYPERBOLA. 209. To find the equation,io the hyperbcda. The hyperbola is the locus of a point which moves so that its distance from a fixed point bears a constant ratio to its distance from a fixed straight line, lite ratio being greater than unity. Let H be the fixed point, TY' the fixed straight line. Draw HO perpendicular to YY' ; take as the origin, OR as the direction of the axis of a;, Y" as that of the axis of y. Let P be a point on the locus ; join MP, draw PM parallel to Y and PN parallel to OX. Let OE=p, and let e be the ratio of HP to PN. Let x, y be the co-ordinates of P. By definition HP = ePN; therefore HP^ = e^PN^- there- fore PM' + HM' = iPN\ that is, y^ + (x -p)* = eV. EQUATION TO THE HYPERBOLA. 189 This is the equation to the hyperbola with the assumed origin and axes. 210. To find' where the hypeirbda meets the axis of X we put y = in the equation to the hyperbola ; thus (x — pf'~ €W : therefore x~p= +.ex;. therefore x — —r- . \ s-i sr- - > 1 + e Since e is greater than unity, 1 — e is a negative quantity. Let OA' = —-^ , OA = , " , the former being measured e — 1 1 + e- " to the. left of 0, then A' and A are points oft the hyperbola. A and A' are called - the. i;er&'ce& of the hyperbola, and G the point midway between A and A' is > called the centre of the hyperbola.' 211. We shall obtain a simpler' form of the equation to the hyperbola by transferring the< origin to Aot: C. I. Suppose the origin at A. Since OA = :r-^— , • we put x = x' + :—-— and substitute l+« ^ 1+e this value in the equation y^ + {x—pf = eV; thus j^+(«/+3^-py=e'(x'+j^y, therefore 2,« + ^^ - 1^ = e' fo.'^+l?^) ; " 1+e \ l+ej' therefore 7/= 2pex' + (e' - 1) x" The distance A' A = — ^ + -r^— = -,- , ; we will denote e-1 1+e e' — 1 this by 2a; hence the equation becomes ^'= (e"— 1) {2ax'+x"). 190 EQUATION TO THE HYPERBOLA. We may suppress the accent, if we remember that the origin is at the vertex A, and thus write the equation f^(e'-l)(2ax + x') (1). II. Suppose the origin at C. Since CA = a, we put x=x —a and substitute this value in (1); thus 2/» = (e' - 1) {2a [x' - a) + {x - af} = (e' - 1) (a;'= - a'). "We may suppress the accent, if we remember that the origin is now at the centre C, and thus write the equation 3^=(e'-l)(^-a') (2). In (2) suppose a;= 0, then ^= — (e°— l)o'; this gives an impossible value to y, and thus the curve does not cut the axis of y. We shall however denote (e' — 1) a' by i', and measure off the ordinates GB and GB' each equal to h, as we shall find these ordinates useful hereafter. Thus (1) may be written f='^,{2a^^a?) (3), and (2) may be written y' = l,{a?-a^) (4), or, more symmetrically, -s-| = l. -Of, ay-h'x'' = -a''h' (5). 212. Since AS= iOA and OA = zr^— , we have 1+e ATT- gp _ (e-l)ep . ,v 0A = 7r^- = a, 1 + e e CH = OA + AH=a + {e-l)a = ea, rOEM OF THE HTPEEBOLA. CO=CA-OA=a-^-^a = -, e e and Qg = _p = °(^°-1) . 191 213. We may now ascertain the form of the hyperbola- Take the equation referred to the centre as origin, 2/' = |(x'-a'). .(1). For every value of x less than a,y is impossible. When x = a, y = 0. For every value of x greater than a there are two values of y equal in magnitude but of opposite sign. Hence if P be a point in the curve on one side of the axis of X, there is a point P' on the other side of the axis, such that P'M=PM. Thus the curve is symmetrical with re- spect to the axis of x, and it extends indefinitely to the right oiA. If we ascribe to x amj negative value we obtain for y the same pair of values as when we ascribed to x the cor- responding positive value. Hence the portion of the curve to the left of the axis of y is similar to the portion to the right of it. 192 FORM OF THE HYPERBOLA. As the equation (1) may be put in the form ^=^(2/' + h') (2). •we see that the axis of y also divides the curve symmetrically. Thus the curve consists- of two similar branches each extend- ing indefihiteiy. The straight line EK is the directrix, His the correspond- ing focus. Since the curve is- symmetrical with respect to the straight line BOB', it follows that if we take GS=CH and GE' = GE, and draw E'K' at right angles to GE', the point S and the straight line E'K' will 'form respectively a second focus and directrir, by> means of which the curve might have been generated. 214. The point G is called the centre of the hyperbola, because every chord of the hyperbola which passes through C is bisected at G. This is shewn in the same manner as the corresponding proposition in the ellipse. (See Art. 163.) 215. We have drawn the curve concave towards the axis of x ; the following proposition will justify the figure. The ordinate of any point of the curve which lies between a vertex and a fixed point of the curve on the same branch as the vertex is greater than tWe corresponding ordinate of the straight line joining that vertex and the fixed point. Let A be the vertex and take it for the origin ; let P be the fixed point ; x', y' its co-ordinates. Then the equation to the hyperbola is (Ait. 2-11) y' = -, (2ax+-a^). The equation to .4P ie yhr=Kx, or y = - * /(-t-+ Ijaj, since {x', y') is on the hyperbola. Let X denote any abscissa less than x', then since the ordinate of the curve is - ii/{2ax + a?) or - ^ ( 1- 1 ) a?, and that of the straight line is - ^ / (-j -|- 1 j ar, it is obvious that FORM OF THE HTFEBBOLA. 193 the ordinate of the curve is greater than, that of the straight line. AJl points may be said to be outside the curve for which n — i + 1 is positive ; and all points may be said to be inside the curve for which |j — , + 1 is negative. It is easy to see that according to this definition a point is outside the curve when no straight Une can be drawn from the point to a focus without cutting the curve. A very instructive mode of obtain- ing this result is that exempUfied in Art. 54 : the expression ^ j + 1 is negative when the point (x, y) is a focus, vanishes when {x, y) is on the curve, does not vanish in any other case, and is positive when a; = for all values of y. Hence we infer that the expression is negative for every point which can be joined to a focus by a straight line that does not cut the curve, and positive in every other case. Similar remarks might be made in connexion with Art. 127. 216. AA' and BB' are called axes of the hyperbola. The axis AA' which if produced passes through the foci, is called the transverse axis, and B^ the conjugate axis. We do not, as in the case of the ellipse, use the terms major and minor axis, because since b = a V(e* — 1) (Art. 211), and e is greater than unity, b may be greater or less than a. The ratio which the distance of any point on the hyper- bola from- the focus bears to the distance of the same point from the corresponding directrix is called the excentricity of the hyperbola. We have denoted it by the symbol e. To find the latus rectum (see Art. 128) we put x = CE, that is = ae, in equation (1) of Art. 213 ; thus , iV(e'-l) h* y- a* "a" therefore LH = - , and the latus rectum = — . a a Since V = a' (e" — 1) ; therefore 6' + o' = oV ; that is CE'+CA^=-GH*; therefore AB = CH. T. c. s. 13 194 FOCAL DISTANCES OF ANT POINT. 217. The equation to the hyperbola may be derived from the equation t.o the ellipse by writing — V for 6". We shaU find that the hyperbola has many properties similar to those which have been proved for the ellipse ; and as the demon- strations are similar to those which have been given, we shall in some cases not repeat them for the hyperbola, but refer to the corresponding Articles in the Chapters on the ellipse. 218. To express ike focal distances of any point of the hyperbola in terms of the abscissa of the point. Let 8 be one focus, E'K" the corresponding directrix ; H the other focus, UK the corresponding directrix. Let P be a point on the hyperbola; x, y its co-ordinates, the centre being the origin. Join 8P, HP, and draw PNN' parallel to the transverse axis, and PM perpendicular to it. Then SP=ePN' = e{CM+CE')^e{x + ^ = ex-\-a. ■ HP = ePN =e{CM-CE)=e[x-^ = ex-a. Hence SP — HP=2a; that is, the difference of the focal distances of any point on the hyperbola is equal to the trans- verse axis. TANGENT TO AN HYPERBOLA. 195 Let X, y be the co-ordinates of any point Q. Then S(^= {x + aeY + f={ex+aY+f-{^-l) {a? -a*) = e'(x + f)Vy-|(«:'-a'). Therefore the focal distance of any point not on the curve bears to the distance of the point from the corresponding di- rectrix a ratio which is greater or less than e according as the point is outside or inside the curve. Suppose Q a point outside the curve; join Q with the nearer focus, which we wiU denote by H; and let Qjff cut the curve at P. Let 8 be the other focus. Join SQ, SP. Then SQ is less than SP + PQ by Euclid, L 20; therefore SQ - EQ is less than SP+PQ- HQ, that is less than SP- HP. Thus the difference of the focal distances of any point outside an hyper- bola is less than the transverse aivis. Similarly we may shew that the difference of the focal distances of any point inside an hyperbola is greater than the transverse axis. 6' 219. The equation y* = — {«* — a') may be written 3'' = -.(a'-a) {x+a). Hence (see the figure to Art. 213), . „ A'M ~~AC^' Tangent and Normal to an Hyperbola. 220. To find the equation to the tangent at any point of an hyperbola. By a process similar to that in Art. 170, it will be found that the equation to the tangent at the point (x', y') is or d^yy' — b*xx' = — a'6'. These equations may be derived from the corresponding equations with respect to the ellipse by writing — 6' for V. 13—2 196 NORMAL TO AN HYPERBOLA. 221. The equation to the tangent to the hyperbola may also be written in the form y = tax + t/(mV — V); (see Art. 171). Conversely every straight Une whose equation is of this form, is a tangent to the hyperbola. 222. It may be shewn as in the case of the circle that a tangent to an hyperbola meets it at only one point. Also if a straight Une meet an hyperbola at only one point, it is in general the tangent to the hyperbola at that point. For sup- pose the equation to an hyperbola to be a'y' — 6V = — a'6', and the equation to a straight Hne y = mx + c. Then to de- termine the abscissae of the points of intersection, we have the equation a* (mx + c)' — b'a^ = — a*V, or (aW -b*)a^ + 2a*m,cx + a'((^+b') = 0. This equation has always two roots, except (1) when a*»iV = (aW-JV*(c* + 5'), or d'=mW-b\ and consequently the straight line is a tangent ; (2) when aW — J' = ; the equation then reduces to one of the first degree, and therefore has but one root. Thus a straight line which meets the hyperbola at only one point is the tangent at that point unless the inclination of the straight line to the transverse axis be + tan"'- . a 223. The tangents at the vertices A and A' are parallel to the axis of y. (See Art. 172.) 224. To find the equation to the normal at any point of an hyperbola. (See Art. 173.) It will be found that the equation to the noi-mal at (of, y) isy-y'=-^,(x- of). This may also be written in the form * {a*+V)m ,„ ._^ ,^,, y = '^" V(o'-ym') - (See Art. m.) peoperTies op the htpebbola. 197 225. We shall now deduce some properties of the hyper- bola from the preceding Articles. Let x', "if be the co-ordinates of P ; let TT be the tangent at P, Pff the normal at P ; TM, FN perpendiculars on the axes. The equation to the tangent at P is a*r/i/— b'xx = — a*6'. Let y = 0, then x=-,, hence CT = ^^ ; therefore CM.GT=GA\, Similarly GN.GT' = GB^. 226. As in Art. 176, we may shew that GG = e' CM, and GG' = ^ PM . 227. As in Art. 177, we may shew that where SP=r', EP=r. 198 PEOPERTIES OF THE HYPERBOLA. Also the result established in Art. 176 respecting normals at the ends of a focal chord holds for the hyperbola, changing major axis into transverse axis. 228. The tangent at any point bisects the angle between the focal distances of that point. For in the manner given in Art. 178, we may shew that the angle SPG' = the angle HPO ; and therefore since PT is perpendicular to GG', the angle TPS = the angle TPH. Or we may prove the result thus : GG=^oil (Art. 226) ; therefore BG = eV + ae, HG = eV — ae. Also SP = eai' + a, HP = ex' — a; hence SG__SP^ HG HP' therefore by Euclid, VI. A, PG bisects the angle between HP and SP produced, that is, the angle SPG' = the angle HPG. 229. To find the locus of the intersection of the tangent at any point vdth the perpendicular on it from the focus. It may be proved as in Art. 180, that the required locus is the circle described on the transverse axis as diameter. 230. Let p denote the perpendicular from H on the tangent at P, and p' the perpendicular from S ; then, as in Art. 181, it may be shewn that p' = ^-r , and p'^= — ; there- 6V fore pp'=V. Since r'=2a + r, we haye»' = ^r- — • ^^ ^ 2a+r The result established in the latter part of Art. 181 holds also for the hyperbola. 231. From any external point two tangents can be drawn to an hyperbola. Let h, k be the oo-ordinates of the external point, then as in Art. 182, we shall obtain the following equation for deter- mining the abscissae of the points of contact of the tangents and hyperbola, x' (aV - 6%») + Za'b'hx - a* (6» + &») = 0. The roots of this quadratic will be possible if aVA' + a* (6' + iP) {a'li* - &%») is positive ; that is, if ^d' — b'h' + a'b' is positive. TWO TANGENTS FROM AN EXTERNAL POINT. 199 But if {h,k) he an external point the last expression is positive, and therefore two tangents can he drawn to the hyperbola from an external point. The product of the two values of x' given by the ahove quadratic is — jp — iSTij ttis product is therefore positive or negative according as a^l^—Vh' is negative or positive; that is, the two tangents meet the same branch or different branches according as aV — bV is negative or positive. The case in which a'k^ — b'h^ = requires to be noticed. Here one root of the quadratic equation becomes infinite, and the other is — „,,, ' ; see Algebra, Chapter XXii. In this case the point (h, k) falls on a certain straight line called an asymptote, which we shall consider hereafter ; see Art. 255. The asymptote itself may then count as one of the two tangents &om the point (A, k). If A = and A = the point {h, k) is the origin ; in this case the two asymptotes may count as the two tangents from the point (A, k). 232. TangemAs are drawn to an hyperbola from a given external point: to find the equation to the chord of contact. Let h, k be the co-ordinates of the external point ; then the equation to the chord of contact is a'ky -b'hx=- aV. (See Art. 183.) 233. Through any, fixed point chords are dravm to an hyperbola, and tangents to the hyperbola are drawn at the extremities of each chord : the locus of the intersection of the tangents is a straight lin£. Let h, k be the co-ordinates of the point through which the chords are drawn, then the equation to the locus of the intersection of the tangents is a'ky -b'hx = - aV. (See Art. 184.) 23 4<. If from any point in a straight line a pair of tan- gents be drawn to an hyperbola, the chords of contact will all pass through a fixed point. (See Art. 185.) 200 INTERPRETATIONS OF AX EQUATION. The student should observe the different interpretations that can be assigned to the equation aVcy — Vhx = — a'6'. The statements in Art. 103 with respect to the circle may all be applied to the hyperbola. 235. Some interesting geometrical investigations relating to tangents to an hyperbola from an external point may be noticed. To draw two tangents to an hyperbola from an external point. The first method given in Art. 186 may be applied with- out any change. In applying the second method we shall have to distin- guish between the two cases which present themselves in Art. 231 ; the distinction between the two cases will be more fully appreciated by the student after he has read the next Chapter. If the external point be between a branch of the curve and the adjacent portions of the ' asymptotes, the two tangents both touch that branch of the curve : if the external point be so situated that we cannot pass from the point to the curve without crossing an asymptote, the two tangents touch different branches of the curve. The student can easily construct the figures required in the process we shall now give. I. Suppose the external point to be between a branch of the curve, and the adjacent portions of the asymptotes. Let denote the external point, H the nearer focus, S the farther focus. With centre S and radius equal to 2a describe a circle; with centre and radius OH describe another circle cutting the former at Q and q. Join SQ and 8q, and produce these straight lines to meet the curve at Pand p. Join OP and Op; these are the required tangents from 0. The demonstration is like that inArt. 186; and we can shew that OP and Op subtend equal angles at H, and also at S. II. Suppose the external point so situated that we can- not pass from the point to the curve without crossing an asymptote. Let denote the external point, H the nearer TANGENTS FROM AN EXTERNAL POINT. 201 focus, 8 the farther focus. With centre 8 and radius equal to 2a describe a circle ; with centre and radius OH describe another circle cutting the former at Q and q, the angle H8Q being less than the angle H8q. Join 8Q and produce it to meet the curve at P ; also join qS and produce it to meet the curve at p. Join OP and Op ; these are the required tan- gents from 0. The demonstration is like that in Art. 186. From the triangles 08Q and OSq we have the angle OSq equal to the angle OSQ. Thus the angle OSp is the supple- ment of the angle OSQ; so that the angles subtended at 8 by the tangent Op and the tangent OP are supplementary. Also the angle OHp = the angle OqS = the angle OQS = the supplement of the angle OQP =the supplement of the angle OHP ; so that the angles subtended at H by the tan- gent Op and the tangent OP are supplementary. 2%e straight line which bisects the angle between the focal distances of an external point is equally inclined to the two tangents from that point. In I. we have angle SOQ + twice angle QOP+ angle SOq + twice angle pOH =B60'; therefore angle 80Q + angle QOP + angle pOH = 180°, that is, angl« 80P + angle pOH = 180°; thus the angle pOH is the supplement of the angle 80P, that is, equal to the angle between SO and PO produced. In II. we have angle pOq = angle pOH = angle pOQ + twice angle POH, angle SOq = angle SOQ = angle SOp + angle pOQ ; therefore by subtraction angle SOp = twice angle POfl"— angle SOp, therefore angle SOp = angle POH. The student should observe the extension thus given to the result in Art. 228. 202 EXAMPLES. CHAPTER XI. EXAMPLES. 1. Find the equation to an hyperbola of given transverse axis whose vertex bisects the distance between the centre and the focus. 2. If the ordinate MP of an hyperbola be produced to Q so that MQ = 8P, find the locus of Q. 3. Any chord AP through the vertex of an hyperbola is divided at Q so that AQ : QP :: AG' : BG\ and QM is drawn to the foot of the ordinate MP; from Q a straight line is drawn at right angles to QM meeting the transverse axis at 0: shew that AO : A'O :: AG* : BG\ 4. PQ is a chord of an ellipse at right angles to the major axis AA'; PA, QA' are produced to meet at R: shew that the locus of R is an hyperbola having the same axes as the ellipse. 5. If a circle be described passing through any point P of a given hyperbola and the extremities of the transverse axis, and the ordinate MP be produced to meet the circle at Q, shew that the locus of Q is an hyperbola whose conjugate axis is a third proportional to the conjugate and transverse axes of the original hyperbola. 6. Find the locus of a point sucltthat if from it a pair of tangents be drawn to an ellipse the product of the perpen- diculars drawn from the foci on the chord of contact will be constant. 7. If an ellipse and an hyperbola have the same foci their tangents at the points of intersection are at right angles. 8. Shew that the equation a? + y^^k\Ax + By+Gf represents an ellipse or an hyperbola according as T^{A}-\-ff) is less or greater than unity. ( 203 ) CHAPTER XII. THE HYPERBOLA CONTINtrED. Diameters. 236. To find the length of a straight line drawn from any point in a given direction to meet an hyperbola. Let a;', y' be the co-ordinates of the point from which the straight line is drawn ; x, y the co-ordinates of the point to which the straight line is drawn ; the inclination of the straight line to the axis of x; r the length of the straight line; then (Art. 27) a; = x'-f-rcos^, y=y' + raind. If (x, y) be on the hyperbola these values may be substi- tuted in the equation o'y — h*a? = — a^b"; thus a*(yJrr sin df - b' {x' + r cos 0)' = - a*V ; therefore r' (a' sin' O-b* cos' 6) + 2r (ay sin - Vx cos 6) + aY'-6V'-l-a'6' = 0. From this quadratic two values of r can be found which are the lengths of the two straight lines that can be drawn from (x, y') in the given direction to the hyperbola. 237. To find the diameter of a given system of parallel chords in an hyperbola. (See Definition in Art. 148.) Let d be the inclination of the chords to the transveree axis of the hyperbola ; let «', y' be the co-ordinates of the middle point of any one of the chords ; the equation which deter- mines the lengths of the straight lines drawn from (x', y) to the curve is (Art. 236) r»(a' sin'^ - V cos"^) + 2r (ay sin - 6V cos 6) + aY-b'as"+a'b'=0 (1). Since (»', y') is the middle point of the chord, the values of r furnished by this equation must be equal in magnitude and opposite in sign; hence the co-efficient of r must vanishj thus b* a*y' sin 6 — bW cos = 0, or y' = -j cot . a;' (2). 204 CONJUGATE DIAMETERS OF THE HYPERBOLA. Considering »' and y' as variable this is the equation to a straight line passing through the origin, that is, through the centre of the hyperbola. Hence every diameter passes through the centre. Also every straight line passing through the centre is a diameter, that is, bisects some system of parallel chords. For by giving to ^ a suitable value the equation (2) may be made to represent any straight line passing through the centre. If & be the inclination to the axis of x of the diameter which bisects all the chords inclined at an angle Q, we have from (2) tan ^ = -5 cot ^; therefore tan Q tan ^ = -= . a" ' a" 238. If one diameter bisect all chords parallel to a second diameter, the second diameter will bisect all chords parallel to the first. Let ^, and 6^ be the respective inclinations of the two diameters to the transverse axis of the hyperbola. Since the first bisects all the chords parallel to the second, we have b' tan 0^ tan ^, = -j . And this is also the only condition that must hold in order that the second may bisect the chords parallel to the first. The definition in Art. 191 holds for the hyperbola. 239. Every straight line passing through the centre of an ellipse meets that ellipse ; this is evident from the figure, or it may be proved analytically. But in the case of an hyper- bola this proposition is not true, as we proceed to shew. 240. To find the points of intersection of an hyperbola with a straight line passing through its centre. Let the equation to the straight line he y = mx. Substitute this value of y in the equation to the hyperbola ay — Va? = — a'b'; then we have for determining the abscissae of the points of intersection the equation (a^m' — 6*) a;' = — a'b'; therefore a;' =-75 =— =. Hence the values of x are impossible o — am -"^ if aW is greater than S*. Thus a straight line drawn through CONJUGATE HTPEBBOLi. 205 the centre of an hyperbola will not meet the curve if it makes ■with the transverse axis on either side of it an angle greater than tan"' - . 241. It is convenient for the sake of enunciating many properties of the hyperbola to introduce the following im- portant definition. Definition. The conjugate hyperbola is an hyperbola having for its transverse and conjugate axes the conjugate and transverse axes of the original hyperbola respectively. 242. To find the equation to the hyperbola conjugate to a given hyperbola. Let AA', BB" be the transverse and conjugate axes re- spectively of the given hyperbola ; then BB' is the transverse axis of the conjugate hyperbola, and A A' is its conjugate axis. Let P be a point in the given hyperbola, Q a point in the conjugate hyperbola. Draw FM, QN perpendicular to 206 EQUATION TO THE CONJUGATE HTFEBBOLA. GX, GY respectively. The equation to the given hyperbola Ls y»= ^ (a;' - o») ; therefore PM^ = ^ {CM* - CA*). Hence CM.' QN* = -rrm (GN' — GR), since Q is a point on an hyperbola having GB, GA for its semi-transverse and semi-conjugate axes respectively. Thus if «, y denote the co-ordinates of Q, S V7e have ^ = t5 {y' — &')• This, therefore, is the equation to the conjugate hyperbola; we observe that it may be deduced from the equation to the given hyperbola by vrriting — o' for o' and — V for i*. The foci of the conjugate hyperbola will be on the straight line BGB' at a distance from G = AB (Art. 216) ; that is, at the same distance from G as S and M. 243. Every straight line drawn through the centre of an hyperbola meets the hyperbola or the conjugate hyperbola, ex- cept the two straight lines inclined to the transverse axis of the hyperbola at an angle = tarT* a' Let the equation to the straight line be y = mx (1). To find the abscissae of the points of intersection of (1) •with the given hyperbola, we have, as in Art. 240, the equation "^"j'-aW ^^)- Similarly to find the points of intersection of (1) with the conjugate hyperbola, we have the equation "^"aW-b* <^^- If m* be less than -3 , (2) gives possible values, and (3) b* impossible values for a: ; if m* be greater than -; , (2) gives CONJUGATE DIAMETERS. 20T impossible values, and (3) possible values fora;; ii m' = -i, (2) and (3) make x infinite. Thus the two straight lines that can be drawn at an inclination tan"' - to the transverse axis a of the given hyperbola meet neither curve ; and every other straight line meets one of the curves. 244. Of two conjugate diameters one meets the original hyperbola, and the other the conjugate hyperbola. Let the equations to the two diameters be y = TMx, y = m'x ; 6" b* then, by Art. 238, mm' = —^ ; therefore rrcm* = -j . Qi Q, b' 6" Hence if m' is less than -^ , m"* is greater than -5 ; thus the first diameter meets the original hyperbola, and the second meets the conjugate hyperbola. K m' is greater than -2, then m'* is less than -5; thus the first diameter meets d a the conjugate hyperbola, and the second meets the original hyperbola. 245. We proceed now to some properties connected with conjugate diameters. When we speak of the extremities of a diameter we mean the points where that diameter intersects the original hyperbola or the conjugate hyperbola. We may remark that the original hyperbola bears the same relation to the conjugate hyperbola as the conjugate hyperbola bears to the original hyperbola. Thus the definition may be given as follows : two hyperbolas are called conjugate when each has for its transverse axis the conjugate axis of the other. Also if a straight line bisect all parallel chords termin9,ted by one of the hyperbolas it bisects all the chords of the same system which are terminated by the other hyperbola. For the equation (Art. 237) tan 6 tan 0"= -g remains unchanged when 208 CONJUGATE HYPEEBOLAS. we write —a' for a* and — i" for 6", that is, when we pass from the original hyperbola to the conjugate (Art. 242). Both curves are comprised in the equation 246. Tlie tangent at either extremity of any diameter is parallel to the chords which that diameter bisects. See Art. 190. 247. Given the co-ordinates of one esctremity of a diameter, to find those of each extremity of the conjugate diameter. Let ACA', BCE be the axes of an hyperbola; POP', DCiy a pair of conjugate diameters. Let x, y* be the given co-ordinates of P; then the equation to GP is y-x''"- .(1). Since the conjugate diameter DU is parallel to the tan- gent at P, the equation to DD' is ay .(2). CONJUGATE DIAMETEBS OF THE HYPERBOLA. 209 We must combine (2) with the equation to the conjugate hyperbola to find the co-ordinates of D and D'. Substitute from (2) in aY - 6V = a%^; then a" ^ ^! a;' - 6V = d?V; ay therefore (6V - a'y") a? = aV; therefore a? = -^.^ = -J- ; therefore x = + -f- ; ab ~ b therefore from (2), y = + — . In the figure the abscissa of D is positive, and that of B' negative ; hence the upper sign applies to B, and the lower sign to B'. 248. The difference of the squares on two conjugate semi- diameters is constant. Let x', y be the co-ordinates of P; then, by the preceding Article, CP' - GB-" = x'^ + y'^ -%- -"-^ " b' a' -— p + a^ **-"• Hence the difference of the squares on two conjugate semi- diameters is equal to the difference of the squares on the semi- axes. Moreover CB' = x" + y" - a* + 6' = x" + -] {x" - a') - a' + J' ■^ a = x"' (l -h 1°) - a' = eV -a' = SP.HP by Art. 218. 249. The area of the parallelogram formed by tangents at the ends of conjugate diameters is constant. Let PGP', BCB' be the conjugate diameters (see the figure to Art. 247). The area of the parallelogram formed by tangents at P, B, F , D', is 4 OP. CB sin PGB, or 4p. CB, where p denotes the perpendicular from C on the tangent at P. T. c. s. 14 210 PEKPENDICULAB FROM THE CENTRE ON THE TANGENT. Let X , y be the co-ordinates of P; then the equation to the tangent at P is ^ = -,-, x — > . Hence ' (Art. 47) P = V V a'V ^/(- VW + 6V>" . li^yi. , ''■^"] _V(ay' + 6V') therefore 4p . CD = 4a6. Hence the area of any parallelogram formed by tangents at the ends of conjugate diameters is equal to the area of the rectangle formed by tangents at the ends of the axes. 250. Let a, h' denote the lengths of two conjugate semi- diameters ; a the angle between them ; by the preceding Article, a'h' sin a = ah. By making P move along the hy- perbola from A we can make CP or a as great as we please. Also since d* — 6'* is constant, h' increases with a'. Thus sin a can be made as small as we please, that is, GP and CD can be brought as near to coincidence as we please. The limiting position towards which they tend is easily found ; for from Art. 237, mm' = -^ ; thus the limit to which m and m' approach as CP and CD approach to coincidence is + - . 2.51. From Art. 249 we have • _a'y^ «y rArt948^ ^ GD'~ CP'-a'+b'- ^■^^■'^^^■) This gives a relation between p the perpendicular from the centre on the tangent at any point P, and the distance CP of that point from the centre. As in Art. 196 p.PG= V, and p.PG' = a". CONJUGATE DIAMETERS OF THE HYPERBOLA. 211 Also if ^ denote the angle which the perpendicular makes with the transverse axis, we may shew as in Art. 196 that p' = o''(l-e'sin''^). 252. To find the equation to the hyperbola refeired to a pair of conjugate diameters as axes. Let CP, CD be two conjugate semidiameters (see the figure to Art. 247), take CP as the new axis of a;, CD as that of y; let PC A = a, DC A =/3. Let x, y be the co-ordinates of any point of the hyperbola referred to the original axes ; w', y' the co-ordinates of the same point referred to the new axes ; then (Art. 84) x = x' cos oi+y cos /8, y = x' sin a + y' sin /S. Substitute these values in the equation a^ — Vo^ = — oHf; then a" [x sin a + y sin /S)" — W {x cos a + y cos ^f = - aV, or x^ (c^ sin" a-¥ cos'' a) + 2/" (a" sin= ^ - 6" cos" /9) + Ix'y (a" sin a sin /S — 6" cos a cos /3) = — a^i". But since CP and CD are conjugate semidiameters, tan a tan /S = A, ; hence the coefficient of x'y vanishes, and the equation becomes a.-" (a" sin" a-V cos" a) + y'" (a" sin" /3 - 6" cos" /S) = - a"6". In this equation suppose y = 0, then - a"6" a"6" 2 ^ __ ~ tt" sin" a — 6" cos" a 5" cos" a — a" siii" a ' This is the value of CP", which we shall denote by a". If we put a;' = in the above equation, we obtain .,_ -^y^ y "a"sin"/3-6"cos";8" Now since we have supposed that the new axis of x meets the curve, we know that the new axis of y will not meet the curve (Art. 244), so that -^-^^^ZTj/^^^ ^^ ^°* * positive 14—2 212 ASYMPTOTES. quantity ; we shall denote it by — 6". Hence the equa- tion to the hyperbola referred to conjugate diameters is -7i — j7i = l, or, suppressing the accents on the variables, 2 2 a'" V Also the equation to the conjugate hyperbola referred to the same axes is -^j — f^s = — 1. Qi The equation to the tangent to the hyperbola will be of the same form whether the axes be rectangular or the oblique system formed by a pair of conjugate diameters. (See Art. 200.) 253. Tangents at the extremities of any chord of an hyper- bola meet on the diameter which bisects that chord. (See Art. 201.) 254. If a chord and diameter of an hyperbola are parallel, the supplemental chord is parallel to the conjugate diameter. (See Arts. 202, 203.) Asymptotes. 255. The properties of the hyperbola hitherto given have been similar to those of the ellipse ; we have now to consider some properties peculiar to the hyperbola. Let the equation to the hyperbola be "f = -^ («' — a°), and let GL be the straight line which has for its equation y = — . CL Let MPQ be an ordinate meeting the hyperbola at P and the straight line CL at Q ; then if CM be denoted by x, PM=-^J{a?-a\ QM = -; a ^ a thus PQ = - {a;-v'(^-a'')] = - . -, ^ = ^ ^ a^ ^'^ ^> a x+^j{!^-d') x+^J{a?-ay If then the straight line MPQ be supposed to move parallel ASYMPTOTES. 213 to itself from A, the distance PQ contiaually diminishes, and by taking CM large enough we may make PQ as small as we please. The straight line GL is called an asymptote of the curve. Similarly the straight line CL', which has for its equa- bx . tion y = , IS an asymptote. Thus the equation — , — i5 = includes both asymptotes. We may take the following definition. Definition. An asymptote is a straight line the dis- tance of which from a point of a curve diminishes without limit as the point on the curve moves to an infinite distance from the origin. The distance of P from CL is PQsiaPQC; and as we have seen that PQ diminishes without limit as P moves away from the origin, CL is an asymptote according to the definition hero given. 256. In the same manner we may shew that CL is an 214« CONNEXION OF TANGENT AND ASYMPTOTE. asymptote to the conjugate hyperbola. For let MP be pro- duced to meet the conjugate hyperbola at P', then (Art. 242) P'Jlf = -V(a!' + a'); therefore P'Q = ^{V(^ + a') -^l = ^(,.^%^, • Hence as CM is increased indefinitely P'Q is diminished indefinitely ; therefore CL is an asymptote to the conjugate hyperbola. 257. The equation to the tangent to the hyperbola at the point {x, y) is d'yy — Vxx'= — aV, therefore y= ¥x'x f ■ y _b a.' x'x y" a") 6' ~y' „ / (v . a^\ If x' and y' are increased indefinitely the limiting form to ■which the above equation approaches is. y = — . Thus the tangent to the hyperbola, approaches continually to coincidence with an asymptote when, the point of contact moves away in- definitely irom the origin. 258. It appears from Art; 243 that every straight line drawn through the centr.e of an hyperbola must meet the hyperbola or its conjugate, unless its direction coincides with that of one of the asymptotes. And from Art. 250 it appears that as conjugate diametersancrease indefinitely they approach to coincidence with one off"the asymptotes. 259. The straight line joining the ends of conjugate dia- meters is parallel to one asymptote and bisected by the other. Let x', y be the co-ordinates of any point P on the hjrper- bola (see the figure to Art. 247) ; then the co-ordinates of J), PROPERTIES OF THE ASYMPTOTES. 215 the extremity of the conjugate diameter, are (Art. 247) -^ and — . Hence the equation to DP is , hx' y ""IT thatis, y-3'' = --(a;-a;'); and therefore DP is parallel to the asymptote y = — hx a Also the co-ordinates of the middle point of DP axe (Ai-t. 10) >(...f)„dl(,..!^, , . ay +hx , ay +bx that IS, " aj — and -^ . 26 2a These co-ordinates satisfy the equation y = — ; therefore the asymptote i/ = — bisects PD. Since the diagonals of a parallelogram bisect each other, and PD is one diagonal of the parallelogram of which CP and CD are adjacent sides, the other diagonal coincides with the asymptote, that is, the tangents at P and D meet on the asymptote. 260. The equation to the hyperbola referred to conjugate diameters as axes is •^-^ = 1 (1) Hence the equations to the asymptotes referred to these axes are h'x h'x ,„. y=-^' y=—a: (2). 216 EQUATION TO THE HYPERBOLA For we may shew as in Art. 243 that the straight lines denoted by (2) are the only straight lines through the centre which meet neither (1) nor its conjugate. Hence these straight lines are the asymptotes by Art. 258. Or the same conclusion maybe obtained thus : the original equation to the hyperbola is "^ — ^ = 1, and that to the two asymptotes — a ~ f» = 0. If by substituting for x and y their values in terms of the new co-ordinates x and y', and sup- pressing accents on the variables, the former equation is reduced to -;, — t>s = 1, the latter must beconie, by the same «* v' substitution, — r, — rTs = 0. Cb O 261. To find tlis equation to the hyperbola refeiTed to the asymptotes as axes. Let CX, GY be the original axes ; CX', C7' the new axes, so that CX' and CY' are inclined to CX on opposite sides of it at an angle a such that tan a = - . Let x, y he the co-ordinates of a point P referred to the old axes ; x', y the co-ordinates of the same point referred to the new axes. Draw PM' parallel to CY' , and PM and M'N each parallel to GY. Then a; = CJ/= GN^ NM= {x + y') cos a. So y = PM= (y' — a;') sin a. Also cos a — . , „■ , and sin a = .. , „ ; substitute V(,a + o ) V (c^ + ) these values in the equation a'^ — Vt^— — a'6' ; then a'J' (y' - xj - a»6* (y' ^-xj^- oVCo" + 6*), ,, a' -1-6' or a!y=-^— , REFERRED TO THE ASYMPTOTES AS AXES. 217 or, suppressing the accents, xy = — ^ — . The equation to the conjugate hyperbola referred to the • IK i. a*a\ d' + b* same axes is (Art. 242) a!y = r — - 262. To find the equation to the tangent at any point of an hyperbola when the curve is referred to its asymptotes as axes. Let x', y be the co-ordinates of the point; x", y" the co- ordinates of an adjacent point on the curve. The equation to the secant through these points is -y' =!"-!'(—')• X — X .(1). Since (x, y') and {x", if') are points on the hyperbola x'y=\{a'^h\ xY^W^V); therefore x''y'' = x'y'. xy -y Hence (1) may be written y — y = , , {x — x), 218 EQUATION TO THE TANGENT. y-y =-^,{oi!-x). or Now in the limit of = x; hence the equation to the tan- gent at the point («', y') is y-y' = -y-,{x-x') (2). This equation may be simplified ; multiply by x, thus / , , a • ' <^ + ^^ yx +a;y =233^^ =— g— - 263. To find where the tangent at {x, y') meets the axis of X put y = in the equation yx + xy' = — ^ — ; thus X= a . = f-=2x. 2y y Similarly to find where the tangent cuts the axis of y put a; = m the equation ; thus y = -5-; — = — r- = zy . Thus the product of the intercepts = 4ix'y' = a* + 6*. The area of the triangle contained between the tangent at any point and the asymptotes is equal to the product of the intercepts on the axes into half the sine of the included angle = ^ (a' + 6'') sin 2a = (a* + 6") sin a cos a = ah, and is therefore constant. Since the tangent at (a;', y") cuts ofif intercepts 2a;', iy, from the axes of x and y respectively, the portion of the tangent at any point intercepted between the asymptotes is bisected at the point of contact. Polar Equation. 264. To find the polar equation to the hyperbola, the focus being the pole. Let HP = r, AHP= 6; (see the figure to Art. 209); then HP = ePN, by definition ; that is, HP=:e(OH+ HM) ; or r = a (e* - 1) + er cos (tt - 6), (Art. 212) ; POLAR EQUATION TO THE HTPEEBOLA. 219 therefore r (1+ e cos 0) = a (e' — 1), J a(e' — 1) „ , and r = -^ '-. (1). 1 + e cos ^ ^ If we denote the angle XHP by 6, then we have as before HP = e{OH + HM); thus r = a (e* — 1) + er cos 6, a (e' - 1) . and r = ,— ^ ^ (2). 1 — e cos ^ ^ ' We may also proceed thus : in the figure to Art. 218 suppose fifP= r and PSH = 6 : then SP = ePiV', that is, SP = e{SM-SE'); or r = cr cos 0— 0(6" — 1); therefore r (ecos^— l) = a(e* — 1), a,d -=-^^^ (3)- e cos p — 1 ^ 265. As in Art. 205 it may be shewn that if the equation to the hyperbola be (1) of Art. 264 then the polar equation to a chord subtending at the focus an angle 2y3 is I e cos 6 + sec ^ cos {a — 6) ' a — /3 and a + j8 being respectively the vectorial angles of the straight lines which join the focus to the ends of the chord, and I the semi-latus rectum. Hence the polar equation to the tangent is I r = e cos + cos {a — 6)' 266. The polar equation to the hyperbola, the centre being the pole, is (Art. 206) r" (a' sin* d-h^ cos' 6) = - aV. Arts. 207, 208 are applicable to the Hyperbola. 220 TRACING THE HYPERBOLA. 267. It will be a good exercise to trace the form of the hyperbola from any of the polar equations of Art. 264. Take for example the equation (1); suppose = 0, then r = a(e — 1); we must therefore measure off the length a(e — l) on the initial line from the pole H, and we thus obtain the point A as one of the points of the curve. As 8 increases from to 5- we see from (1) that r increases ; cos 6 is negative when 6 is greater than =■ and r continues to increase. Let a be such an angle that 1 + e cos a = 0, that is, cos a = — , then the nearer 6 approaches to a the greater r becomes, and by taking 6 near enough to a, we may make r as great as we please. Thus as 6 increases from to a that portion of the curve is traced out which begins at A and passes on through P to an indefinite distance from the origin. When 6 is greater than a, r is negative, and is at first in- definitely great and diminishes as 6 increases from a. to tt. Since r is negative we measure it in the direction opposite to that we should use if it were positive. Thus as 6 increases from a to TT that portion of the curve is traced out which EQUILATEEAL OR BECTANGULAR HYPERBOLA. 221 begins at an indefinite distance from C in the lower left-hand quadrant, and passes on through Q to A'. HA' is found by putting 6 = ir in (1) ; then r becomes — a(e + l), therefore HA' is in length = a (e + 1). As 6 increases from tt to 27r — a, r continues negative and numerically increases, and may be made as great as we please by taking 6 sufficiently near to lir — a. Thus the branch of the curve is traced out which begins at A' and passes on through Q' to an indefinite distance. As increases from 27r — a to lir, r is qjfain positive, and is at first indefinitely great and then diminishes. Thus the portion of the curve is traced out which begins at an indefi- nitely great distance from G in the lower right-hand quadrant and passes on through P' to A. The asymptotes CL and GL' are inclined to the transverse axis at an angle of which the tangent is - ; hence we have cos L GA = -77-5 — TiT = - . and cos LGA' = : that is, V(a + 6 ) c e LGA' = a. Thus as 6 approaches the value a the radius vector approaches to a position parallel to GL. Similarly as approaches the value 2ir — a the radius vector approaches to a position parallel to GL'. Equilateral or Rectangular Hyperbola. 268. If in the equation to the ellipse o^ 4- 6V = aV, we suppose b=a, we obtain af + y' = a^, which is the equation to a circle ; so that the circle may be considered a particular case of the ellipse. If in the equation to the hyperbola o'V — ^'a:* = — «°^" we suppose b = a, we have y' — a^ = — c^. We thus obtain an hyperbola which is called the equilateral hyperbola from the equality of the axes. Since the angle between the asymptotes, which = 2 tan"' - , becomes a right a angle when h = a, the equilateral hyperbola is also called the rectangular hyperbola. 222 EXAMPLES. CHAPTER XII. The peculiar properties of the rectangular hyperbola can be deduced from those of the ordinary hyperbola by making b = a. Thus since b" = a' (e' — 1) we have e' — 1 = 1, therefore e = \/2. The equation to the tangent is (Art. 220) yy — xx = — a". From Art. 227 PG = PG' = ^{rr). The equation to the conjugate hyperbola is, by Art. 242, y' — ti? = d'. Thus the conjugate hyperbola is the same curve as the original hyperbola, though differently situated. By Art. 248^ GP=CD, and therefore by Art. 259, CP and CD are equally inclined to the asymptotes. EXAMPLES. 1. The radius of a circle which touches an hyperbola and its asymptotes is equal to that part of the latus rectum which is intercepted between the curve and the asymptote. 2. A straight line drawn through one of the vertices of an hyperbola and terminated by two straight lines drawn through the other vertex parallel to the asymptotes will be bisected at the other point where it cuts the hyperbola. 3. A straight line cuts an hyperbola at P and p, and the asymptotes at Q and q : shew that PQ =pq. 4. If a straight line be drawn from the focus of nn hyperbola the part intercepted between the curve and the asymptote = -: ; — -. — 7, , where 6 and a are the angles made *' ^ sma + sm0 ° respectively by the straight line and asymptote with the axis. 5. PQ is one of a series of chords inclined at a constant angle to the diameter AB of a circle : find the locus of the point of intersection oi AP and PQ. 6. P is a point in a branch of an hyperbola, P' is a point in a branch of its conjugate, CP, CP', being conjugate semi- diameters. If S, S' be the interior foci of the two branches, prove that the difference of SP and S'P' is equal to the difference of AC and BC. EXAMPLES. CHAPTER XII. 223 7. If ar, y be co-ordinates of any point of an hyperbola, shew that we may assume x = a sec 6, y = h tan 6. 8. A straight line is drawn parallel to the axis of y meet- ing the hyperbola -^ — ?j = 1, and its conjugate, at points P, Q: shew that the normals at P and Q intersect each other on the axis of X. Shew also that the tangents at P and Q intersect on the curve whose equation is y* (ay — 6V) = 46V. 9. Tangents to an hyperbola are drawn from any point in one of the branches of the conjugate ; shew that the chord of contact will touch the other branch of the conjugate. 10. Find the equation to the radii from the centre to the points of contact of the two tangents in Example 9, and if these radii are at right angles, shew that the co-ordinates of the point from which the tangents are drawn are 11. Two tangents to a parabola include an angle a: shew that the locus of their point of intersection is an hyperbola with the same focus and directrix. 12. Shew under what limitation the proposition in Exam- ple 30 of Chapter X. is true for the hyperbola. 13. The ratio of the sines of the angles made by a diameter of an hyperbola with the asymptotes is equal to the ratio of the sines of the angles made by the conjugate diameter. 14. With two conjugate diameters of an ellipse as asymp- totes a pair of conjugate hyperbolas is constructed: prove that if one hyperbola touch the ellipse the other will do so like- wise ; prove also that the diameters drawn through the points of contact are conjugate to each other. ( 224 ) CHAPTER XIII. GENERAL EQUATION OF THE SECOND DEGREE. 269. We shall now shew that every locus represented by an equation of the second degree is one of those which we have already discussed, that is, is one of the following : a point, a straight line, two straight lines, a circle, a parabola, an ellipse, or an hyperbola. The general equation of the second degree may be written aa? + hxy + cy^ + dx + ey +/= ; we shall suppose the axes rectangular ; if the axes were oblique we might transform the equation to one referred to rectangular axes, and as such a transformation cannot affect the degree of the equation (Art. 87), the transformed equation will still be of the form given above. If the curve passes through the origin /= 0; if the curve does not pass through the origin / is not = 0, we may there- fore divide by/ and thus the equation will take the form a a? + h'xy + c'y^ + d'x + e'^ + 1 = 0. 270. We shall begin by investigating the possibility of removing from the equation the terms involving the first power of the variables. Transfer the origin of co-ordinates to the point [h, k) by putting x = x' + h, y = y' + k, and substituting these values of X and y in the equation ax' + hxy + cj^ + dx + ey +/= (1); thus we obtain flMj" -I- bx'y + cy" + {2ah + bk + d)af+ (2ck + hh + e) y +/'=0 (2), where /' = ah!' + bhk + oli? + dh + ek +f. (3). CENTRAL CUBVES OF THE SECOND DEGREE. 225 Now, if possible, let such values be assigned to h and k as ■will make the coefficients of x' and y' vanish; that is, let 2ah+hk+d = 0, and 2ck+bh + e = 0; ^, , 2cd — be , 2ae — bd thus h = Y2 — A — 1 k = -i5 — -. — . b' — 4ac J' — 4ac It will therefore be possible to assign suitable values to h and k, provided b' — 4ac be not = 0. We shall see that the loci represented by the general equa- tion of the second degree may be separated into two classes, those which have a centre, and those which in general have 7iot a centre, and that in the former case b' — 4ac is not zero, and in the latter case it is zero. We shall first consider the case in which b' — 4iac is not zero, and consequently the values found above for h, and k are finite. Equation (2) thus becomes aa;'' + ij;y + cy"+/'=0 (4). Now if (4) is satisfied by any values x^,y^ of the variables, it is also satisfied by the values -x^,—y^. Hence the new origin of -co-ordinates is the centre of the locus represented by (1). Thus if 6' — 4ac be not = 0, the locus represented by (1) has a centre, and its co-ordinates are h and k, the values of which are given above. The value of/' may be found by substituting the values of A and k'va. (3) ; the process may be facilitated thus : we have 2ah + bkJrd = fi, 2c& -(- 6A + e = 0; multiply the first of these equations by A, and the second by k, and add ; thus 2aA' -h 2cA''' -I- 2bk}i + dA -h ek = 0, or 2f-dK-ek-2f=f:i; ,, , ., , dk-'rek . cd^ + a^-led therefore / =/-h_^— =/-h ^a.^^, ■ We shall retain/' for shortness. T.C.S. 15 226 : TRANSFORMATION OF THE EQUATION. 271. We may suppress the accents on the variables in equation (4) of the proceding Article and write it a^-\-lxifJt-cy'^+f' = ^ (5). This equation we shall further simplify, by changing the directions of the axes. (Art. 81.) Put x = x' cosd — y sin 6, y ~x'wsi6 + y' cos 6, and sub- stitute in (5) ; thus x* (a cos'^ + c sin'^ + 6 siii'^ cos 0) + 3/" (a sin" 5 + c cos'^ — & sin cos 6) -\-x'y' {2 (c- a) sin 0cos0 + b {cos' 6 -sin' 6)} +/' = 0...(6). Equate the coefficient of x'y' to zero ; thus 2 (c - a) sin ^cos e +.b (cos'^ - sin'^) = 0, or (c - a) sin 20 + b cos20 = Q ; therefore tan20 = (7). a—c Since can always he found so as to satisfy (7), the term involving x'y can be removed from (6), and the (equation becomes x'' {a cos' 5 + c sin' ^ + J sin ^ cos 6) +_y" {a sin"^ + c cos'5 - J.sin ^ cos 0) +/' = 0, or Ax" + By"+f'=0 (8), where ^ = ^ {a + c + (a — c) cos 20 + 6 sinS^j, Ii=l{a + c —(a - c) cos 2d-h sin 20}. Since tan:20 = — — , a — c a — c cos 2g= ,„. . and sin 20=- Hence A=i[a+c + >/{b' + {a-cy\l ^=i[a-|-c-V{6' + (a-cfl]. CENTKAL CURVES OF THE SECOND DEGREE. 227 We may suppress the accents on the variables in (8) and write it — j-i ^ ji y (1) If A, B, and /' have the same sign, the locus is im- possible. (2) If A and B have the same sign and/' have the con- trary sign, the locus is an ellipse of which the semi-axes are respectively \/(-2)'-V(-S)- ^'^■^^'■'^ The locus is of course a circle if A = B. (3) If A and B have different signs, the locus is an hyperbola. (Art. 211.) We have supposed in these three cases that /' is not = ; if /'= 0, and A and B have the same sign the locus is the origin ; if /' = 0, and A and B have different signs the locus consists of two straight lines represented by y=^\/{-i)"'- From the values of A and B we see that AB = ^ -—^- . Hence A and B have the same sign or different signs according as 6* — 4ac is negative or positive. 272. Hence we have the following summary of the results of the preceding Articles of this Chapter. The equation CM?' + hxy + cy' + da! + ey+f=0 represents an ellipse if 6" — 4ac be negative, subject to three exceptions in which it Tepresents respectively a circle, a point, and an impossible locus. If V — 4ac be positive, the equation represents an hyperbola subject to one exception when it represents two intersecting straight lines. 273. We may notice that the equation found in Art. 271, 15—2 228 CENTRAL CURVES OF THE SECOND DEGREE. tan 29 = , has an infinite number of solutions : for if 2a a — c be one value of 20 which satisfies the equation, then if 2^ = 2a + mr, where n is any integer, the equation will be satisfied. But these different solutions will all give the same position for the axes. For the values of are comprised in ?i7r the expression a + -^- , and by ascribing different values to n we obtain a series of angles each differing from a by a multiple of ^ , and the only changes that will arise from selecting different values of n are that the axes of x and y in one case may occupy respectively the positions of the axes of y and x in another, or the positive and negative directions of the axes may be interchanged. The denominator in the value of cos 20 and of sin 20 in Art. 271 may have either sign; but the sign must be the same in both in order that the relation tan 26 = may a — c •' hold. 274. It appears from the former part of Art. 271, that by turning the axes through an angle 6 the equation aa?-{-bxy-\-cy*+f' = Q becomes aV + b'x'y + cy'^ + /' = 0, where a =\{a + c+ {a — c)co&20 + h sin 26], b' = {c- a) sin 20 + b cos 20, c = ^ {a + c — {a — c) cos 20— b sin 20]. Hence a' + c' = a + c; and 6" - 4a'c' = {(c - a) sin 26 +b cos 20]* - (a + c)" + {(a - c) cos 20 + b sin 26^ = (a-c)'+6'-(a + c)'' = &»-4ac. Thus the expression b' — 4iac has the same value whether it be formed from the coefficients of the general equation of the second degree before or after the axes have been shifted. The same remark applies to the expression a + c. CURVES OF THE SECOND DEGREE WITHOUT A CENTRE. 229 Hence we conclude that if the curve represented by the general equation ax^+bxy + cy' + dx + ey +f=0 be a rect- angular hyperbola, a + c = ; for if the curve were referred to its transverse and conjugate diameters as axes this relation would hold, and therefore, as we have just seen, it must always hold whatever be the axes. 275. We have next to consider the case in which 6' — 4ac IS zero/ We cannot now as in Art. 270 remove the terms involving the first power of the variables fi;om the general equation, but we can still simplify the equation as in Art. 271, by changing the direction of the axes. Let the equation be ax' + bxy + cy' + dx + ey +f= (1); put x = x' cos 6 — r/ sin 6, y = x' sin d + y cos 0, then (1) becomes ic" (a cos' d + c sin' d + bsind cos 6) + y" (a sin" e + ccos^0-b sin 6 cos 0) + xy' {2 (c - a) sin cos ^ + .6 (cos" d - sin' &)\ + a;'(dcos0 + esin^) + y (ecos0-dsin0) +/=0 (2). Now let tan 16 = , then the coefficient of xy in (2) vanishes, and as in Art. 271 the coefficients of aj" and y" are ^[a + c + V{(«— c)' + 6'}]. One of these coefficients must therefore vanish since their product is - — j — , which, by hypothesis, = ; suppose the coefficient of a;" = 0, thus, by suppressing accents on the variables, (2) may be written Cy' + Dx + Ey+f=Q (3). If D be not = 0, this may be written and thus the locus is a parabola. (Art. 125.) If Z) = 0, then (3) represents two parallel straight lines, or 230 CURVES OF THE SECOND DEGREE WITHOUT A CENTRE. one straight line, or an impossible locus, according as E'' is greater, equal to, or less than 4 Gf. Hence if 6" — 4ac = the equation VLX- + hxy + cy^ + dai + ey +/= represents a parabola subject to three exceptions, in which it represents respectively two parallel straight lines, one straight line, and an impossible locus. By combining this result with those stated in Art. 272, we have a complete account of the general equation of the second degree. 276. We have shewn in Art. 270, that when 6' — 4ac is not = 0, the general equation of the second degree represents a central curve ; we shall now prove that when b" — 4cac = the curve has not a centre except when the locus consists of two parallel straight lines. If a curve of the second degree have the origin of co-ordinates for its centre, no term involving the first power of either of the variables alone can exist in the equation. For if possible suppose that the origin of co-ordinates is the centre of the curve ax^ + bxy + cf + dx + ey +f= (1), and let x^, y, be the co-ordinates of a point on the curve, and therefore —x^.—y^ co-ordinates of another point on the curve; substitute successively in (1), then ax^' + bxj/^ + cy^ + dx^ + ey^+f=0, ax^ + K^i + cy^ - dar, - ey, +/= ; therefore, by subtraction, 2((fo,-l-eyJ = (2). Now unless d and e both vanish, (2) can only be true when {x^, y,) lies on the straight line dx + ey = Q. But the centre of a curve is a point which bisects every chord passing through it ; hence the origin of co-ordinates cannot be the centre of the curve (1) unless both d and e vanish. 277. Suppose then that we have an equation ax^ + bxy + cif + dx + ey+f= (1), PROPERTIES OF THE COEFFICIENTS. 231 in which 6" — 4ac = 0. Here a and c cannot both be zero, for then b -would also be zero, and (1) -would not be an equation of the second de^ee ; -we shall suppose that a is not zero. Now if the curve denoted by (1) had a centre, and -we took that' centre as the origin of co-ordinates, the terms involving the first power of x and y would vanish by Art. 276. But from Arts. 270 and 274 it follows that when 6' — 4ac = 0, we cannot in general make these terms vanish by changing the origin or the axes. The only exception that can arise is when the nume- rators in the values of h and k in Art. 270 vanish, so that the values of h and k become indeterminate, and the two equations for determining them reduce to one; see Algebra, Chapter X7. We have then 2ae — bd ="0, so that e = 5- . Hence, by sub- stituting for c and e, the equation (1) becomes b'li^ , bd ,. „ aa' + bm/ + ^+dx+^y+f=0, thati.,-.. a(-+iy-+'Z(-+i9+/=« (')■ Equation (2) will furnish two values of a; 4- ^ , so that if these values are possible the locus consists of two parallel straight -lines. la this case any point on the straight line which is psurallel to these two and midway between them wiU be a centre. Thus the result enunciated in the beginning of Art. 276 is demonstrated. 278. We may observe that relations similar to those obtained in Art. 274 hold when the axes of co-ordinatee are oblique. For suppose the equation ax' +bxy + cy^ +/' = to be referred to rectangular axes, and let the axes be trans- formed into an oblique system inclined at an angle, w ; sup- pose moreover that the new axis of x coincides with the old axis of x. We have then ta put (Art. 84) x = x' + tf' cos s), y = y' sin a> ; substitute these values in the above equation and it becomes a'x" + b'x'y' + cY+f = 0. 232 PROPEKTIES OP THE COEFFICIENTS. where a' = a, h' = 2a cos Q) + 6 sin w, c =a cos' (u + 6 sin w cos (» + c sin' w ; thus 6'' — 4a'c' = (6' — 4ac) sin' w, and a' + c' — 6' cos o> = (a + c) sin' « ; 6" — 4a'c' so that — ^T — = 6' — 4ac, sin' 0) , a' + c' — V cos 6) and :— = = a + c. sin &> Therefore, by means of Art. 274, we conclude that for any system of axes, rectangular or oblique, the expressions 6" — 4a'c' J a' + c' — 6' cos o) . , , , , , — ^-5 and T—r, remain uncnang'ed when the sin to sm at axes are changed. These results are very important, because as we have seen, the curve will in general be an ellipse, parabola, or hyper- bola according as the former expression is negative, zero, or positive ; and a rectangular hyperbola if the latter expression be zero. These results may be obtained by another method, which will be found instructive. Suppose that the axes of x and y are inclined at an angle X; and let us determine the points of intersection of the curve ax'+hxy + cy'' = g (1), and the circle a? + 2xycos\ + f = 7^ (2). Combining (1) and (2) we obtain g {a?-\- 2xy cos \ + ^) = r" (aai* + bxy + c^) ; that is (g — r'a) a;' + {^g cos X — r'6) xy+{g — r'^c) y' = 0. This is a quadratic equation for finding - . Solving the quadratic we find that the expression under the radical sign is r* (b" — 4ac) — ir'g (6 cos \ — a — c) — 4^^ sin* X, TRACING A CURVE OF THE SECOND DEGREE. 233 If this expression vanishes the two values of - are equal ; this indicates that the circle (2) touches the curve (1) : and hence we may draw the important inference that the squares of the semi-axes of the curve (1) are numerically equal to the values of r* given by the equation ^b'—4ac . . bcos\ — a — c . . . /.> *• ^?^ -^'^—^^ *^ =0 (*>• Now suppose the axes of co-ordinates transformed into another system inclined at the angle \', and let (1) become ax'- + b'x'y' + c'y" = g. ; then the quadratic equation jt" — 4a'c' . » h' cosX' — a — & , . „ ,.y ^^I^'--^'-^— ^?T ^^ =« (^) has the same geometrical meaning as (4), and the roots will therefore be the same. Hence (4) and (5) must coincide, and therefore 6" — 4ac _ 6'" — ia'c ,„, sin'X sin'X' , JcosX — a — c b' cos\' — a — c' >„. and :-j- = ;r-j-7 (7). sm X sm \ In fact if we divide — 4£r' by either member of (6) we obtain the numerical value of the product of the squares of the semi- axes of the curve. Similarly if we divide 4g times either member of (7) by the corresponding member of (6) we obtain the numerical value of the sum or of the difference of the squares of the semi-axes, according as the curve is an ellipse or an hyperbola. 279. We shall now shew how to trace a curve of the second degree from its equation without transformation of co-ordinates; the axes may be supposed oblique or rect- angular. Let the equation be ax* + bxy + cy^ + dx + ey+f=0 (1). 234 TRACING A CURVE OF THE SECOND DEGEEF Solve the equation with respect to y ; thus = aa; + y8 + |^I-(x'+2;ja; + g)|' (3); , h n s iie — 2cd e' — icfi, where a = -2^. ^ = -13' -P^^ITi^' S^J^to- I. Suppose b^—iac negativej and write —ftiOj; , ; thus (3) becomes y = wc + P±[-(ji{x* + 2px + q)]^ (4). Now gf + 2px-¥q = {x+py'+q—p^; if then, g — p' be positive, the quantity under the radical is negative and the locus impossible ; if q —p^ = 0, the locus is the point deter- mined by x = —p, y = ax + ^-; .if q—^-he negative, we may put {x+py + q-f={x+p + ^{p^-q)]{x+p-^(p^-q)\ = (x — ', D, G', G. Since G G' is parallel to the chords which D'D bisects, DD' and GO' are conjugate diameters. GG' is a known quantity since the ordinates of and G' are known. DD' is also a known quantity since the abscissae and ordinates of D and D' are known. The angle between GG' and DD is known from the equation to DD ; the axes of the ellipse may therefore be found (Arts. 193, 195). 23G TRACING AX HYPERBOLA. n — AjCLG II. Suppose b* — 4ac positive ; put fi for — j-^ — ; thus equation (3) becomes y = aa; + /3+{/t(ar'+2pa; + 2)|* (7). Now x^ + Ipx + q = {x-\- pf+ q —p' ; if then q—p'he posi- tive, the quantity under the radical is always positive, what- ever positive or negative value be assigned to x. The curve therefore extends to infinity. Also it may be shewn as before, that the straight line y = ax + ^ is a diameter of the curve ; but it never meets the curve, because the quantity Ji«'+2pa;+g; or {x +pf + q—p^ cannot vanish. Hence the curve consists, of two unconnected branches extending to infinity, and is therefore an hyperbola. If q —p' = 0, (7) becomes y = ax -{■ ^ ± \f fi {x + p). The locus now consists of two intersecting straight lines. If q—p* be negative we may as before write (7) in the form y = aa; + ;S.'+ {/i (a; — '/)(« — S)}'. Hence x may have any value, positive or negative, except those between 7 and h ; thus the curve consists of two unconnected branches extending to infinity, and is therefore an hyperbola. We shall be assisted in drawing an example of this case by ascertaining the position of the asymptotes. The equation to the curve is y = oa; +;S + {ji4(a;' + 2px + g-)}* ; therefore y = ax + fi ±x V/* (l + + j) • Expand by the Binomial Theorem ; thus = ax + ^±^fi. {x+p) + &c. The terms included in the &c. involve negative powers of X, and may therefore be made as small as we please by suf- TRACING AN HYPERBOLA. 237 ficiently increasing on; hence from the nature of an asymptote the required equations to the asymptotes are y = ax + ^+^//i{a;+p), and y = aa; + /3 - V/t* i^e+p)- Hence we can draw the asymptotes, and therefore the axes, for they bisect the angles between the asymptotes. The intersection of the asymptotes is the centre, and thus the situation and form of the hyperbola are known. We may observe that the tangent of the angle between the asymptotes is, by Art. 41, l + u' — fj. l+a'-fj, substitute for a and u their values and we obtain — -. a + c The expression q -p- = i (b'-4gc)' ' this vanishes when (e* — 4c/) (6" — 4ac) — (be — 2cd)' = 0, and therefore when (i" — 4ac)/+ oe" + cd^- bed = 0; so that if this relation holds the locus ■ represented by (1) consists of two intersecting straight lines. We have hitherto supposed that c is not zero, and as 6' — 4ac cannot be negative if c be zero, it was not necessary to advert to the possibility of c being zero while considering the first case. But as c may be zero consistently with 6' — 4ac being positive, we must now examine the consequences of supposing c zero. The equation (1) may be solved with respect to x instead of with respect to y. Hence it will be found on investigation that the results hitherto obtained, when ¥ — 4ac is positive, are certainly true provided that a and c are not both zero; the latter case requires further examination. Suppose then a = and c = ; thus (1) becomes bxy + dx + ey +/= ; by chang- ing the origin this can be put in the form bxy +/' = Q, where/' = -:^-^ — ; the curve is therefore an hyperbola with the new axes for its asymptotes, except when b/—de = 0, and then it becomes two intersecting straight lines. "When o = 238 TRACING A PARABOLA. and c = 0, the expression (V — 4ac)/+ oe' + cd* — bed reduces to 6 {hf— de); thus we conclude that when b" —4sac is positive the equation (1) always represents an hyperbola, except when (6* — 4ac)/+ae'' + c£Z'' — 6ed = 0, and then it represents two intersecting straight lines. III. Suppose V — 4ac = 0, then (2) becomes y = -^~ ± ^ {2 {be - 2cd) x+e'- 4c/}*, which may be written y = ax + ^±— {p'x + q')^, ^here a = -l, /3 = -|^. p' = 2 {be - 2cd), g' = e* - 4c/. If p' be positive, the expression under the radical is positive or negative, according as x is algebraically greater or less than — ^; if p' be negative, the statement must be reversed. In both cases the curve extends to infinity in one direction only and is therefore a parabola. The straight line y = ax+ fi is a diameter, bisecting all ordinates parallel to the axis of y, and meeting the parabola at the point for which x^—^. ■P If p' = 0, the equation becomes y = ax + ^± ^^ ; this equation represents bwO' parallel straight lines if q is positive, and one straight line ii q' = 0; if g' is negative, the locus is impossible. We have hitherto supposed in considering the third case that c is not zero ; if c = 0, then 6 = 0, since 6' — 4ac = ; hence a and c cannot both be zero, for the equation (1) is supposed to be of the second degree. As before, we may solve equation (1) with respect to x, and thus determine the peculiarities which occur when c = 0. We have found for example when c is not zero, that the locus will consist of EQUATION OF THE SECOND DEGREE. 239 two parallel straight lines, when he — 2cd = 0, and e' — 4!cf is positive ; in like manner, if a be not zero, we can shew that the locus will consist of two parallel straight lines when bd — 2ae = 0, and cf — 4a/ is positive. By means of the re- lation 6' — 4ac = 0, it is easily shewn that the second form of the conditions coincides with the first when a and c are both different from zero. When a=0 the first is the neces- sary form of the conditions, but we see that the second form will then also hold. When c = the second is the necessary form, though the 'first will then also hold. Hence we shall include every case by stating that both forms of the conditions must hold. Similarly the conditions under which the locus will con- sist of one straight line, or will .be impossible, may be in- vestigated. 280. We wiU recapitulate the results of the present Chapter with respect to the locus of. the equation oar* + bayy + cf + dx + ey +/= 0. I. If 6' — 4ac be negative, the locus is an ellipse admitting of the following varieties : (X) c-= a, and ^r- = cosine ©f the angle between the axes ; locus a circle (Art. 104). (2) («* - 4ef)-'^' ^Aac) - (he - 2adjf positive ; locus im- possible. (3) (e' - 4c/) (J' - 4ac}- ^e - IcSif = ; locus a point. II. If 6*— 4ac be positive, the locus is an hyperbola, except when Q? — 4ac) f + ae' + cd^ — bde = 0, and then it con- sists of two intersecting straight lines. III. If V — 4fflC = 0, the locus is a parabola, except when be-2cd = 0, and&d-2ae=0; and then it consists of two parallel straight lines, or of one straight line, or is impossible, according as e* — 4c/" and d" — 4a/ are positive, zero, or negative. 240 EXAMPLES. CHAPTER XIII. EXAMPLES. 1. Find the centre of the curve a? — ixy + %* — 2ow! + iay = 0. 2. Find the centre of the ellipse 3. Find what is represented by aa^ + 2bxy + cif = 1, when b' = ac. »4. Find the locus of the centre of a circle inscribed in a sector of a given circle, one of the bounding radii of the sector remaining fixed. 5. In the side AB of a triangle ABC, any point P is taken, and PQ is drawn perpendicular to AG: find the locus of the point of intersection of the straight lines BQ and GP. 6. BE is any chord parallel to the major axis AA' of an ellipse whose centre is G ; and AD and GE intersect at P: shew that the locus of P is an hyperbola, and find the direction of its asymptotes. 7. Tangents to two concentric ellipses, the directions of whose axes coincide, are drawn from a point P, and the chords of contact intersect at Q : if the point P always lies on a straight Une, shew that the locus of Q will be a rect- angular hyperbola. 8. Find what form the result in the preceding Example takes when two of the axes whose directions are coincident are equal. 9. Prove that an hyperbola may be described by the intersection of two straight lines which move parallel to themselves while the product of their distances from a fixed point remains constant. EXAMPLES. CHAPTER XIH. 241 10. Two straight lines are drawn from the focus of an ellipse including a constant angle ; tangents are drawn to the ellipse at the points where the straight lines meet the ellipse : find the locus of the intersection of the tangents. 11. Find the latus rectum of the parabola (y — a;)* = ax. 12. Shew that the product of the semi-axes of the ellipse j^-4ixif + 5a?=2 is 2. 13. Find the angle between the asymptotes of the hyper- bola jry = bx* + c. 14. Find the equation to a parabola which touches the axis of iC at a distance a, and cuts the axis of y at distances /3, ff from the origin. 15. If two points be taken ia each of two rectangular axes, so as to satisfy the condition that a rectangular hyper- bola may pass through all the four, shew that the position of the hyperbola is indeterminate, and that its centre describes a circle which passes through the origin and bisects all the straight lines which join the points two and two. 16. Two straight lines of given lengths coincide with and move along two fixed axes in such a manner that a circle may always be drawn through their extremities : find the locus of the centre of the circle, and shew that it is an equilateral hyperbola. 17. A variable ellipse always touches a given ellipse, and has a common focus with it : find the locus of its other focus, (1) when the major axis is given, (2) when the minor axis is given. 18. Draw the curve y^ — oxy + Ga;' — 14a; -l- 5y + 4 = 0. 19. Draw the curve a;' + y* — 3 (a; -1- y) — jry = 0. 20. Find the nature and position of the curve f-Sxy + 25a:' + &cy - iZcx + 9c' = 0. 21. The equation to a conic section is ax'+^bxy+cy' =1 : shew that the equation to its axes is xy {a — c) =b {a? — y'). 16 242 EXAMPLES. CHAPTEB XIII. 22. The locus of the vertices of all similar triangles Trhose bases are parallel chords of a parabola will in general be another parabola ; but if any one of the triangles toiich- the parabola with its sides, the locus becomes a straight line. . 23. A series of circles pass through a given point 0, have their centres in a straight line OA, and meet another straight line BG. Let M be the point at which one of the circles meets the straight line OA again, and let JV be either of the points at which this circle meets BG. From M and JV" straight lines are drawn parallel to BG and OA respectively, intersecting at P. Shew that the locus of F is an hyperbola which becomes a parabola when the two straight lines are at right angles. 24. The chord of contact of two tangents to a parabola subtends an angle /8 at the vertex : shew that the locus of their point of intersection is an hyperbola whose asjmaptotes are inclined to the axis of the parabola at an angle ^ such that tan = ^ tan ;3. 25. Determine the locus of the middle points of the chords of the curve aaf'+ 2hxy + cy* + Sear + ^fy + gr = 0, which are parallel to the straight line xsind—y cos 6 = 0; and hence find the position of the principal axes of the curve. 26. Shew that the equation («* - o*)" + (/ - o")' = a* represents two ellipses. 27. AB and AG are given in position, and BG is of constant length : shew that if PB and PG be drawn making any constant angle with AB and AG the locus of P is an ellipse. 28. A number of parabolas whose axes are parallel have a common tangent at a given point: shew that if parallel tangents be drawn to all the parabolas the points of contact will lie on a straight line passing through the given point. 29. If on one of the longer sides of a rectangle as major axis an ellipse be described which passes through the inter- section of the diagonals, and straight lines be drawn firom EXAMPLES. CHAPTER XIII, 243 any point of that part of the ellipse which is external to the rectangle to the extremities of the remote side, they will divide the major axis into segments which are ia geometrical progression. 30. A series of ellipses have their equal conjugate dia- meters of the same magnitude, one of them being common to all while the other varies in position : shew that tangents drawn from any point in the fixed diameter produced will touch the ellipses at points situated on a circle. 31. TP, TQ are tangents to a central conic section, and the chord PQ is produced to meet the directrices at It and S' : shew that RP.R'P : RQ.R'Q :: TP* : T(^. 32. In any conic section if PQ, PR make equal angles with a fixed chord PK, and QR be joined, shew that QR wiU pass through a fixed point for all positions of PQ, PR. 16—2 ( 244 ) CHAPTER XIV. MISCELLANEOUS PROPOSITIONS. 281. We shall give in this Chapter some miscellaneous propositions for the most part applicable to all the conic sections. To find the equation to a conic section, the origin and axes being unrestr^ted in position. Let a, h be the co-ordinates of the focus; and let the equation to the directrix he Ax + By + G=0. The distance of any point («, y) from the focus is {(a; — af + (y — b)Y, and the distance of the same point from the directrix is Ax + Bi/+G Let e be the excentricity of the conic section; then if {x, y) be a point on the curve, we have, by definition, ,(«-„).+ (,-Vl!-lM^^ („; thewfore (.-»)■+ fa- })■ - ^^^J^^t "'' (2). We see from (1) that the distance of any point on a conic section from the focus can be expressed in terms of the first power of the co-ordinates of that point whatever be the origin and axes. This is usually expressed by saying the distance of any point from the focus is a linear function of the co-ordinates of the point. 282. It will be seen by examining the equations to the conic sections given in the preceding Chapters that any conic section may be represented by the equation rf = mx + na?. The origin is a vertex of the curve and the axis of a; an TANGENT TO A CUBYE OF THE SECOND DEQBEE. 245 axis of the curve ; m is the latus rectum ; in the parabola « = ; n is negative in the ellipse and positive in the hy- perbola. In the circle m is the diameter of the circle and w = -l. 283. To find the equation to the tangent at any point of a curve of the second degree. Let the equation to the curve be aa?+bxy + cy' + dx + ey +/= (1), the axes being oblique or rectangular. Let x', y' be the co-ordinates of the point, x", y" the co-ordinates of an adjacent point on the curve. The equation to the secant through these points is y-y'=|i^(^-^') (2). Since {x', y') and {x", y") are on the curve, cm;" -1- hx'i/ + cy'* + dx'+ ey' +f = 0, ax"" + bx"y" + cy'" + dx" + ey" +f= ; therefore a (x"" - x*) + b {x"y" - x'y') + c (y"^ - y") + d{x"-x') + e(2/'-y) = 0. or («" - x) {a (x" + x) + by" + d] therefore vlnl.^- '^^f^JWY'l^ ' X —X c{y+y) + ox+e Hence (2) may be written , a{x' + x')+by" + d , ,. V— V = -r-r, A f^T (x — x). ^ ^ c(y" +y) + bx' + e ^ ' Now in the limit x' = x and y' = y' ; hence the equation to the tangent at the point (a;', y) is , 2cm; 4- 62/ + d / ,v V—V— — -^ — -t — r^i (x—x\ " " 2cy +bx +e ^ ' 246 NOBMAL TO A CUKVK OF THE SECOND DEGREE; This equation may be simplified ; we have by reduction y {2cy' + lx' + e)+sc {2ax + hy + d) = y (2cy' + biB' + e)+x' [iaaf + by' + d) = 2 irc, and d" + e" = d= + e'. Hence the normal at the origin will meet the new chord at the same distance from the origin as it met the original chord, that is, will meet it at the same paint. Since this is true whatever he the directions of the axes, it follows that all the chords intersect at the same point. 287. By comparing Arts. 154, 204, and 264, we see that the polar equation to any conic section, the focus being the pole and the initial line the axis, is r = = ^r , where ^ 1 + e cos I = half the latus rectum. We shall use this in proving the following proposition : The semi-latus rectum of any conic section is an harmonic Tnean between the segments made by the focus of any focal chord of that conic section. Let A'SP = 6. see the figure to Art. 158 ; therefore SP = =— a . 1+e cos a Suppose PS produced to meet the curve again at P'; I therefore SP' = 1 +ecos(7r+^)' ^, , 1 1 1+ecos^ 1-ecos^ 2 therefore ^ + gp^ ^ + 1 = T' which proves the proposition. 288. The polar equation to the tangent to a conic sec- tion, the focus being the pole and the initial line the axis, is (Arts. 205, 265) I - = ecos^ + cos(a-5) (1), where a is the angular co-ordinate of the point of contact. TWO TANGENTS TO AN ELLIPSE. 249 Similaxly the polar equation to the tangent at the point whose angular co-ordinate is ^8, is -=ficose + cos()3-5) (2). At the point where these tangents meet, we have cos (a - 5)= cos (/8-^. Now we cannot have a — = ^ — 0, since a and /S are by supposition different ; we therefore take a— 6 = 6 — ^, there- fore 6 = ^. Thus the two tangents (1) and (2) meet at the point whose angular co-ordinate is — = — . For example, suppose the conic section an ellipse ; let ASP = a, ASQ = y3, and let the tangents at P and Q meet at T; then AST='!-^; theiekre PST=^^ = QST; that is, the two tangents drawn from any paint to an ellipse subtend equal angles at either focus. 250 TWO TANGENTS TO AN HYPERBOLA. Similarly the two tangents drawn from any point to a parabola subtend equal angles at the focus. With respect to the hyperbola we have to distinguish two cases. We have shewn in Art. 231, that from any point included between the asymptotes and the curve, two tangents can be drawn both meeting the same branch of the curve, but from any point included within the supplemental angles of the asymptotes two tangents can be drawn meeting different branches of the curve. If now the two tangents from a point meet the same branch of an hyperbola, it may be shewn as in the case of the ellipse, that they subtend equal angles at either focus. We will consider the case in which the tangents meet different branches. Let T be a point from which tangents TP, TQ are drawn to different branches of an hyperbola. Let A8P== a ; and let the angle which Q8 produced through 8 makes with .4£i be /9 ; then /8 is an angle greater than TT, and ASQ = /3 — ir. Thus the equations to TP and TQ will be respectively - = ecQs0 + cos (a — ^), - = c cos 6 + cos{fi — 0). TWO TANGENTS TO AN HTPEBBOLA. 251 At the point T where the tangoDts meet, we have cos {oL-G)- cos (fi-B). We may therefore take 6 = — ^ — , that is, we have Z. - " 2 as the angle which TS produced makes with AS; thus AST = 7r- 2 therefore TSP=nr-^-^. TSQ = ^-^; therefore TSP + TSQ = -ir ; that is, the angle which one tangent subtends at either focus is the supplement of the angle which the other tangent sub- tends at the same focus. 289. We have given in Art. 120 the definitions of a pole and polar with respect to a given circle. The same defini- tions are used generally substituting conic section for circle. If then the equation to the curve be aa? + bxtf + cy* + dx + eff +f = 0, the equation to the polar of (x, y') is (Art. 283) X (2ax' + by' + d)+y {2cy' + hx' + e) + dx + ey + 2/= 0. The equation just given always represents a straight line at a finite distance from the origin except when both 2ax' + hy' ■\-d=Q, and 2cy' + ia;' + c = 0. But if x' and y satisfy these relations they are the co-ordi- nates of the centre of the curve ; see Arts. 270 and 276. Hence strictly speaking there is no polar corresponding to the centre of a conic section ; this fact is frequently expressed by saying that the polar of the centre is the straight line at infinity. See page 74. 290. If one straight line pass through the pole of another straight line, the second straight line will pass trough the pole of ilie first straight line. 252 POLE AND POLAE. Let («', y') be the pole of the first straight line, and therefore the equation to the first straight line X (2aa;' +bi/' + d)+y (2c/ + bx' + e)+dx' + ey' + 2/= 0. . .(1). Let (x", y") be the pole of the second straight line, and therefore the equation to the second straight hne X (2ax"+ by"+ d) +y{2cy" + bx" + e) + dx"+ ey"+ 2/= 0...(2). Since (1) passes through {x", y") we have «" (2ax' + by + d) + y" {2cy' + bx + e) + dx + ey' + 2/= 0, that is, x (2oa;"+ by"+ d) + y'{2cy"+ bx"+ e) + dx"+ ey"+ 2/= ; hence (2) passes through («', y). 291. The intersection of two straight lines is the pole of {he straight line which joins the poles of those straight lines. See Art. 122. 292. If a quadrilateral ABCD be inscribed in a conic section, of the three points E, F, G, each is the pole of the straight line joining the other two. Let E be the origin ; EA, ED the directions of the axes of X and y ; and let the equation to the conic section be aa!'+bxy + cy'+dx+ey+f=0 (1). QUADBILATEEAL IN A CONIC SECTION. 253 Also suppose EA = h, EB = h', ED = k, EG=k'. The equation to ^Cis | + |, = 1 (2); the equation to BD is jr'+j.= 1 (3); SO 1/ the equation to AD is r + t = 1 (4); the equation to CB is r5 + r'=l (^)- From (2) and (3) it follows that the equation -G+^')+K^f)=2 («) represents some straight line passing through O. But from (4) and (5) it follows that (6) represents some straight line passing through F. Hence (6) must he tlie equation to FG. Suppose in (1) that y = ; then we have the quadratic asf +dx+f= ; and the roots of this equation are h and h'; hence h + h' = , hh' = -^; therefore r + r- = — ->• Simi- a a h h f 1 1 1^ 1 « ^^^'lc + k'=-r Hence (6) becomes dx-\-ey + 2f=Q. But this, by Art. 289, is the equation to the polar of the origin ; therefore FG is the polar of E. Similarly EG is the polar of F. Hence, by Art. 291, G is the pole of EF. 293. To determine the form of the general equation to a conic section when the axes are tangents. Let ax' + hxy + cy' + dx + ey+f=0 (1) be the equation to the conic section. To find where the curve meets the axis of x, put y = in the above equation ; thus aa^ + dx +f= 0. or 254 CONIC SECTION REFEBRED TO TANGENTS AS AXES. If tbe axis of a; is a tangent to the curve it must meet the curve at only one point (see Art. 171) ; hence the roots of the above quadratic must be equal ; therefore (£' = 4a/ (2). Similarly that the axis of y may be a tangent to (1) we must have e' = 4c/ (3). Substitute the values of a and c from (2) and (3), then (1) becomes d'a;' + 4dfx + ey + ^tefy + ^hfxy + 4/" = 0, or {dx + ey + 2/)' + (4&/- 2de) an/ = 0, (d , e , ,V ■ W-de p d^ 1 «.__! 2&/-de _ ^ 2/~ h' '2f~ k' 2/* "'*' thus we obtain for the required equation (| + |-l)V^ = 0. By putting successively x and y = 0, we see that h is the distance from the origin to the point where the curve meets the axis of x, and k is the distance from the origin to the point where the curve meets the axis of y. If it be required to determine a conic section which touches two given straight lines at given points, and also passes through another given point, we may assume the last written equation to represent it, so that the straight lines to be touched are taken as the axes of x and y; then by putting the co-ordi- nates of the additional given point in the equation we find a single value for /*. Thus there is only one conic section satisfying the data. 294. Suppose the equation (M-i)+'-^=o (1) to represent a parabola. Then, by Art. 280, PARABOLA BEFERRED TO TANGENTS A3 AXES. 255 therefore u=0, or u = hk J( fj. = 0, (1) becomes ^ + 1 - 1 = ; this equation repre- sents the straight line joining the points of contact of (1) with the axes. 4 If fi.= —j-r,vfe have from (1), (f+l-)"-S i^y. .W„r.|,sy(g)+|.I, therefore yj + y/|-±l. We may write this remembering that the radicals may be positive or negative. Thus (3) is the equation to a parabola referred to two tan- V ^ gents as axes. 295. We may notice the form of the equation to the tangent to the parabola yi+yi=^ w- The equation to the secant through («', y) and (x", y") is Since {of, y') and (x", y") are on the parabola, we have 256 snnLAR curves. therefore ^--"-/-' = -^y"-/y'; „nH y"-y'-'J y"-^^ ^f+^/s' V^ ^/y" + ^/y' x"-x' V«"-V«'Va'"+V«' 'Jh'^x+^Jx" Hence the equation to the secant may be written y-y=- yj—LJlJL (x - x'\ Hence we have for the equation to the tangent at («', y) y ^ _ y' *'' _i Similar Curves. 296. Definition. Two curves are said to be similar and similarly situated when a radius vector drawn from some fixed point in any direction to the first curve bears a constant ratio to the radius vector drawn from some fixed point in a parallel direction to the second curve. Two curves are said to be similar when a radius vector drawn from some fixed point in any direction to the first curve bears a constant ratio to the radius vector drawn from some fixed point to the second curve in a direction inclined at a constant angle to the former. The two fixed points are called centres of similarity. 297. If two curves are similar, so that a pair of centres of similarity exists, then an infinite number of pairs of centres of similarity can be found. For, suppose 0, 0' to denote one pair of centres of simi- larity; and let OP, OQ be radii vectores of the first curve, and O'P', (y Q' the corresponding radii vectores of the second curve, so that the angle PO Q = the angle P'O' Q , and WP' ~ Tfa ■ S'^PP°se any point S taken and joined to ; ALL FABABOLAS ABE SIMILAR. 257 then make the angle P'O'S' = the angle POS, the angles being measured in the same direction, and take (yS' so that /y Of' f)' p' _ „ = yyp : then S and S' shall be centres of similarity. For join 8P, 8Q, S'F, S'Q' ; then the triangles SOP, 8' OP' are similar; and so also are the triangles SOQ, S'O'Q'. Hence it easily foUows that the angle QSP = Q'S'P' ; qp CQ and that -^rp. = sr^- > and thus the proposition is established. 298. All parabolas are simitar curves. Let 4a be the latus rectum of a parabola, and 4a' the latus rectum of a second parabola. The polar equations of these curves, the foci being the respective poles, are 2a , _ 2a' ^^ITcosl' *" ~l+cos^" Hence, if 6 = &, we have — , = — . Thus any two para- bolas are similar, and the foci are centres of similarity. 299. To find the conditions which must hold in order that the curves ai^ + hxy + cf + dx + ey ■{■ f =- Q (1), a'a?+b'xy+c'f+d'x+e'y+f=Q (2), may be similar and similarly situated. Suppose Qi, k), (h', k') the respective centres of similarity ; for a; and y in ( 1 ) put A + r cos 5, and k+r sin d respectively ; we shall thus obtain a quadratic in r which may be written Lr^ + Mr + N=0 (3). For X and y in (2) put h' + r' cos 6, and y -1- r' sin B re- spectively ; we shall thus obtain a quadratic in r which may be written ZV»-|-Jf'r' + jr' = (4). Now that the curves may be similar and similarly situated, we must always have r = Xr, where X is some constant quan- tity ; thus (4) becomes \''i'r» + \ilf'r-HiV^'=0 (5). T. c. s. 17 258 CONDITIONS OF SIMILARITY. Since (3) or (5) will give the values of r, these equations must be identical; thus \'L'~\M'~N' ^"^• Since neither N nor N' involves 6, we deduce as a neces- sary condition that y-, must be constant whatever may be. Put for L and L' their values ; then a cos'^ + 6 sin ^ cos ^ + e sin'^ ^ ^ ,_. -, — „-,,,. — 2 Z-. — r^^-Ta = * constant = ii say. . . ..(7) ; a cm 6 + bsm0 cos^ + c sm*0 '^ •' ^ ■" therefore (a—fia') cos'^ + (b—ftb') sinfl cos 0+(c—iic') sin'5 = 0. Since this is to be true whatever may be, it follows that d-y-^ <^>- Hence we have arrived at (8) as necessary conditions, in order that (1) and (2) may be similar and similarly situated. We have stiU to ascertain whether these are sufficient to ensure the similarity. The direct method would be to exa- mine if h, k, h', k' can be so chosen as to make (6) hold ; but the following method is more simple. The equations (1) and (2), by means of (8), may be written ax* + bxy + cy' +dx + ey +/= 0, ax' + hxy -I- c^ + /* {d'x + e'y +/') = 0. I. Suppose b* — 4ac = ; then each curve is in general ;i parabola, and therefore the curves are similar ; also their dia- meters are parallel so that the curves are similarly situated. See Art. 279. This conclusion is subject to the exceptions that may arise when either locus instead of a parabola, be- comes one or two straight lines, or impossible. II. Suppose 6' — 4ac not = 0. We may then by changing the origin of co-ordinates for each curve reduce the equations 10 the form ass' + bxy + cy' -H/, = 0, ax* + bxy + cy* +/, = 0. CONDITIONS OF S Tltm.All tTY, 259 By expressing these equations in polar co-ordinates, they give j>= zli a cos*^ + 6 sin 5 cos 5 + c sin'^ ' j.>* = ZLlx . a cos'^+ftsin ^cos^-1- c sin'f^' ' Thus, if d=ff, we have -7 = constant. Hence the curves r are in general similar and similarly situated. This conclusion is subject to the exceptions that may arise when either locus instead of a curve becomes two straight Hues, or a point, or impossible. 300. Next, suppose we require the curves (1) and (2) of Art. 299 to be simUar withoid the limitation of being simi- larly situaied. For x and y in (1) we put respectively A + rcos^, A; +r sin ft For X and y in (2) we put respectively A'+/cos(^ + a), A' + r'sin(^ + a), where a is some constant angle at present undetermined. Pro- ceed as in Article 299 ; instead of equation (7) we shall now have g co s'ff -^ & sin ^ cos g + c sin'^ a' cos' (^ + a) + h' sin (6 + a) cos (0 4- a) + c sin'C^ + a) = a constant = fi say. This may be written a cos* d + hsxk.6co%6 + c sin'ff _ A cos'^ -I-.B sinfl cos^ + C siu' 5 ~ '*' where A=a' cos'a + c' sin*a + b' sin a cos a, B=2{c'- a) sin a cos a + b' (cos'a - sin'a), C=a' sin' a+c cos'a — 6' sina cos a. That the curves may be similar we must have a b c 17—2 260 AREA OF A POLYGON. A. + C Hence each of these ratios must equal ; ^ a + c therefore -r- = i — ; — i- • n» AS therefore • ' {A + Gf- (a + cy AC {A + Gf ac (a + c)' ' AG ao {A+Cf (a + cy £'-4AG h'-iac (A + Gf (a + c)'- A+G = = a' + c', B'-4>AG = = 6" - 4a'c', -r—, r-s- = b'-4uc And therefore Hence, But and 5* - 4^ = 6" - 4a'c', (Art. 274) ; therefore -r~, jr-- — -. ^^ {a' + c'Y {a + cY This relation must therefore hold, in order that the given curves may be similar. From the results obtained in Art. 278 it is easy to derive an instructive verification of the condition of similarity just demonstrated. It will be seen that similar conic sections have the same excentricity. Area of a Polygon. 301. In Art. 11 we have given an expression for the area of a triangle in terms of the co-ordinates of its angular points : we shall now investigate the corresponding expression for the area of any polygon,. Let the angular points of the polygon taken in order be («i. ^i). K. y.). •••(«,; y-); *ake any point {pi, y) within the polygon and draw straight lines to the angular points of the polygon, thus dividing the polygon into triangles having a common vertex at («, y). Then by Art. 11 the numerical values of the areas of these triangles are respectively ABEA OF A POLYGON. 261 2 1^ (y. - yi) + *i (y - y.) + «'. (2/1 - y)} • 2 ■ ^(2/3- y.) +a',(y-ya) +«s (2/,-y)j-, 2 -^(j/n-^^i) + «,.-. (y-yJ+i^n (y«-.-y)|. 2 1* (yi - y«) + ^» (s' - 2^1) + «'i (y. - y)} • Let us ossMTwe, for the present, that the sum of these ex- pressions will give the axea. By addition x and y disappear, and we obtain 2 W (y.-y.) +'^. (yi- y.) + a^. (y,-y*) + •■• + *»-i (yn- - y») + «.. (y^i - yi)| • By multiplying out this expression may be written thus : 2 Y'<2/t - «iy. + ^^ly. - a'^3 + — + «,y-t - *„-.y« + «.y, - a^-y,} ■ The expression may also be written thus : 2 [vi K-a:J + y,(a', -«i) + ya (a^.-^J + - + y«-i («« - «.-.) + y»(«i - O} • 302. We now proceed to examine the admissibility of the assumption made in the preceding Article. Suppose that the polygon has no re-entrant angle. We must then shew that the expressions for the areas o£ the triangles used in the preceding Article are aU of the same sign; for unless this is the case we do not obtain a correct numerical value of the area of the polygon by adding these expressions. The required result may be obtained by the aid of a principle which we have already applied ; see Arts. 54 and 215. 262 AKEA OF A POLYGON. Consider the expression given for the axea of the first triangle in the preceding Article. The ezpression will retain the same sign for all positions of {x, y) which are on the same side of the straight Hne passing through {x^, y,) and (x,, y,). Similarly the expression given for the area of the second triangle in the preceding Article will retain the same sign for aS. positions of (x, y) which are on the same side of the straight line passing through (ar,, y,) and (a;,, y,). Thus if the two expressions have the same sign for one position of {x, y) within the polygon, they will have the same sign for all such positions. But by trial we can ascertain that the two expressions have the same sign -when « = „ {^i + a^,) and 1 ... y = - (y^ + yj : the two expressions will in fact be found then to coincide. Thus the two expressions have the same sign for all positions of {x, y) within the polygon. Similarly the expressions for the areas of the second and third triangles have the same sign. And so on. Thus the assumption made in the preceding Article is justified. 303. We will now briefly illustrate the method by which it may be shewn that the expressions obtained in Art. 301 for the area of a polygon hold even when the polygon has re- entrant angles. A Suppose, for example, we have a quadrilateral figure ABCD, with a re-entrant angle at B. Through B draw a straight line parallel to the axis of x, and take a point h on this straight line, such that AbCD is a quadrilateral figure without a re-entrant angle. Let the co-ordinates of .4 be a:,, y,; let those of 5 be HOMOLOGOUS TEUNGLES. 263 x^,y,; and so on. Let the abscissa of b be x. Then we know that the area of Ab CD is numerically expressed by 2 [^1 (y* -y.) + ^(i/,- y^ + a^a (y. - y*) + ^< (y, -y.)\- Now as X increases this expression becomes algebraically greater since y^ — y^ is positive; and as x increases we see from the figure that the area increases : hence it follows that the expression is positive. Put x = x^ + h, so that h = Bb. The expression then becomes 2 1^. ^y* - y and as nA(y, — y,) is obviously equal to the area of ABCb, it follows that the other part of the expression is equal to the area of ABCB. 304. Although the results given in Art. 301 are not of great importance, yet the reasoning in Arts. 302 and 303 is very instructive. The method of Art. 303 may be appHed whatever be the form of the figure, with slight modifications which do not affect the principle. Homologotis Triangles. 305. In Art. 76 we have spoken of homologous triangles; we will here give another property relating to such triangles. Suppose ABC. A'B'C, A"B"C" three triangles such that any two of them are homologous ; and suppose moreover that AB, A'B, A"B' meet at a point : then the three centres of homology will lie on a straight line. For consider the triangles AA'A" and BB'B'. By sup- position AB, A'B', and A"W meet at a point: therefore, by Art 76, the intersections of corresponding sides of the triangles lie on a straight line; that is the intersection oiAA' and BB', of A A" and B'B', and of A" A and B"B lie on a straight line. 264 EXAMPLES. CHAPTER XIV. And conversely if the three centres of homology lie on a straight line the sides AB, A'B, A"B" meet at a point; so also do BG, EG, B'G"; and GA, G'A', G"A". This also follows from Art. 76. 306. It may be easily shewn that if we take the equa- tions to the sides of two triangles as in Art. 76, then the equations l"u + TOU + WW = 0, lu + m"v + nw = 0, lu-\-mv + n"w = will determine a ttird triangle such that any two of the triangles are homologous, and that any three corresponding sides meet at a point. EXAMPLES. 1. Straight lines are drawn through a fixed point : shew that the locus of the middle points of the portions of them intercepted between two fixed straight lines is an hyperbola whose asymptotes are parallel to those fixed straight lines. 2. Through any point P of an ellipse QPQ* is drawn parallel to the major axis, and PQ and PQ' each made equal to the focal distance SP : find the loci of Q and Q'. 3. In the given straight lines AP, AQ are taken variable points p, q, such that Ap zpP :: Qq : qA ; shew that the locus of the point of intersection of Pq and Qp is an ellipse which touches the given straight lines at the points P, Q, 4. TP, TQ are two tangents to a parabola, P, Q being the points of contact; a third tangent cuts these at p, q respectively : shew that -=^ + -f=^ = 1. 5. TP, TQ are equal tangents to a parabola, P, Q being the points of contact : if PT, QT be both cut by a third tangent, shew that their alternate segments will be equal. 6. From a point are drawn two straight lines to touch a parabola at the points P and Q; another straight line touches the parabola at B and intersects OP, OQ at S and T: EXAMPLES. CHAPTER XIT. 265 if V be the intersection of the straight lines joining PT, QS, crosswise, 0, R, V are on the same straight line. 7. From an external point two tangents are drawn to an ellipse : shew that an ellipse similar and similarly situated will pass through the external point, the points of contact, and the centre of the given ellipse. 8. A and B are two similar, similarly situated, and con- centric ellipses ; C is a third ellipse similar to A and B, its centre being on the circumfereace of B, and its axes parallel to those oi A OT B : shew that the chord of intersection of A and G is parallel to the tangent to B at the centre of C. 9. The straight line joining any point with the inter- section of the polar of that point with a directrix subtends a right angle at the corresponding focus. 10. If normals be drawn to an ellipse from a given point, the points where they cut the curve will lie on a rectangular hyperbola which passes through the given point and has its asymptotes parallel to the axes of the ellipse. 11. If CM, MP are the abscissa and ordinate of any point P, on the circumference of a circle, and MQ is taken equal to MP and inclined to it at a constant angle, the locus of the point Q is an ellipse. 12. Having given the equation to a conic section oar' + 21x1/ + y'+f=0, find the locus of the intersection of normals drawn at the extremities of each pair of ordinates to the same abscissa. 13. Any two points P, Q are taken in two fixed straight lines in one plane such that the straight line PQ '\& always parallel to a given straight line ; P, Q are severally joined with two fixed points H, R : find the locus of the intersection ofP.H'and QR. 14. The tangent at any point P of a circle meets the tan- gent at a fixed point A at T, and T is joined with B the extremity of the diameter passing through A : shew that the locus of the point of intersection of AP and BT is an ellipse. 266 EXAMPLES. CHAPTER XIV. 15. The polax equation to a conic section from the focus being ccos0=b, shew that the equation to a straight line which cuts it at the points for which d = a. and /S respectively, ■ 1 a J. fa '^ + ^\ ' IS — c COS = cos ^r^ sec r \ 2 / 2 16. Chords are drawn in a conic section so as to subtend a constant angle at the focus : prove that the locus of the foot of the perpendicular drawn from the focus on the chord is a circle, except in a particular case when it becomes a straight line. 17. If SP, 8Q be focal distances of a conic section in- cluding a constant angle, shew that PQ touches a confocal conic. 18. Having given two fixed points through which a conic section is to pass, and the directrix, find the locus of the corresponding focus. 19. The focus and the directrix of an ellipse are given ; through the former a straight line is drawn making with the latter an angle whose sine is the excentricity of the ellipse. Find the locus of the points where this straight line meets the curve, the excentricity being variable. 20. A series of conic sections is described having a com- mon focus and directrix, and in each curve a point is taken whose distance from the focus varies inversely as the latus rectum : find the locus of these points. 21. Two conic sections have a common focus S through which any radius vector is drawn meeting the curves at P, Q, respectively. Shew that the locus of the point of intersection of the tangents at P, Q, is a straight line. Shew that this straight line passes through the intersection of the directrices of the conic sections, and that the sines of the angles which it makes with these straight lines are in- versely proportional to the corresponding excentricities. 22. A straight line is drawn cutting an ellipse at the' points P,p; let ^ be either of the points at which the same straight line meets a similar, similarly situated, and concentric EXAMPLES. CHAPTER XIV. 267 ellipse : shew that if the straight line moves parallel to itself, J^^Q -Qp is constant. 23. In two straight lines OX, OY, which intersect at 0, take OA = a, OB = h : shew that the centres of all the conic sections which touch the straight lines at A and B lie on the straight line ay = bx. 24. About two equal ellipses whose centres coincide, and whose major axes are inclined to each other at a given angle an ellipse is circumscribed : if A and B be the semi-axes of the circumscribing ellipse, a and h the semi-axes of the equal ellipses, and 2a the inclination of their major axes, then will aV + A'B'' = {AV + BV) cos'' a + {AW + B\b') sin" a. Hence shew that about the two equal ellipses a similar ^ ellipse may be circumscribed. 25. Two similar ellipses have a common centre and touch !■ Gw" + 2A'vw + ^Bwu + 2C'uv = 18—2 276 EQUATION TO A CONIC SECTION. will generally represent any assigned conic section, if the constants A, B, C, A', S, C are properly determined. For suppose we divide the equation by one of the constants as C, there are then five independent constants left. Now let S denote any assigned conic section; take five points on S and substitute the co-ordinates of the five points successively in the above equation; we shall thus have five equations for determining the five constants. Suppose a, b, c, a', b' the values thus determined, then the equation aw' + bv' + cu? + 2aW + 2b' wu + 2uv = represents a conic section which has five points in common with S, and which therefore coincides with 8. (Art. 307.) 314. The method of the preceding Article, although im- portant and instructive, is not satisfactory, because we have not shewn that the five equations from which the constants are to be determined are consistent and independent. There may be exceptions to the theorem, and we therefore use the word generally in the enunciation. If the three straight lines Tneet at a point, then the curve denoted by the equation always passes through that point, and the equation in this case will not represent any assigned conic section. If the three straight lines are parallel, u, v, w take the forms Ix + my+p, Ix + my + p', Ix + my+p", and the equation takes the form \{la; + myy + fi,(lx + my) +v = 0, which represents two parallel straight lines, and thus will not represent any assigned conic section. With these exceptions, however, the theorem is universally true, as we shaU now shew by another demonstration. Since the straight Unes are not all parallel, two of them at least will meet; suppose m = and d = to be these two, and take their directions for the axes of ^ and x respectively; then M = becomes x = 0, and v = becomes y=0; also w = may be written Ix + my + m = 0. We have then to shew that the equation Aaf + Bf + C{lx + my +ny + 2A'y{lx + my+n) + 2Fx{lx + my + n) + 2C'xy = (1) EQUATION TO THE TANGENT. 277 wUl represent any assigned conic section by properly deter- mining the constants A, B, Suppose aa?' + 2Ja!y + c/+ 2dx + 2ey-\-f=Q (2) to be the equation to the assigned conic section. Arrange the terms in (1) and equate the coefficients of the corresponding terms in (1) and (2) ; thus Cr?=f, A'n + Gmn = e, B'n+ Gin = d, B+Gm^ + 2A'm = c, Glm + A'l + B'm + C' = b, A+GP+2B'l=a. These equations determine successively G, A', B, B, G', A. As the given straight lines do not meet at a point, n is not zero ; hence the values found for G, A',... are all finite and determinate. Thus (1) is shewn to coincide with (2), and the required theorem is demonstrated. 315. We will now investigate the equation to the tan- gent at any point of the curve represented by Av^ + Bi^ + Cw' + 2A'vw + 2Bwu + 2G'uv = 0. Let u, v', w be the values of u, v, w respectively at one point of the curve, and u", v", w" their values at another point of the curve. Then the equation to the straight line joining these two points may be put in the form A{u-u'){u-'vi') + B{v-v'){v-v") + G{w-w')[w-w") + 2A' (v-v'){w-w")+2B (io-w')(u-u")+2G' (u-u'){v-v") = Au'+ Bv' + Gu^+ 2A'vw + 2B'wu + 2 G'uv. For this equation is really of the first degree in the variables u, V, and w, and therefore represents some straight line ; more- over the equation is satisfied at the point {u, v, w'), and also at the point {u", v", w"), and therefore it represents the straight line which passes through these two points. Now suppose the point («", v", w") to, move along the curve until it coincides with the point {u, v', w). Then the ■secant becomes ultimately the tangent at {u, v, w'), and the equation to this tangent is Auu' + Bm' + Gwtt}' + A' {yw' + wv') + B' (wu + uw) -l-C"(«D'-hW) = 0. 278 EQUATION TO THE TANGENT. 316. As a particular case of the preceding Article sup- pose that A' , S, and C are zero. Then the equation to the curve is ^M''+5t)'+(7w'=0 (1); and the equation to the tangent at («', v, w') is Auu'+Bmf+Cww' = (i (2). Hence we can find the condition which must hold in order that a proposed straight line may toiLch the curve denoted by (1). Let the equation to the proposed straight line be \u + iiv + vw = (3). If (3) denotes the equation to the tangent at {u, v, w'), we find l3y comparing (3) with (2) that Au' ^ Bv' _ Cw' A. fi V ' Let r denote the value of each of these fractions ; then , Xr , U'T r vr "=Z' " = £' '" = c These values must satisfy (1) since {u, v', w') is a point on the curve ; thus A^ B^ C this is therefore the required condition. 317. The investigation of Art. 315 may be modified in special cases by using a different form for the equation to the secant. For example suppose that A, B, and C are zero. Then the equation to the curve is A'vw + Bwu + C'uv = 0, which may be also put in the form ^'-.^+^:=o (1). The equation to the straight line which passes through EQUATION TO THE TANGENT. 279 the points («', v', w) and (u", v", w") on the curve may be put in the form I It r / ./ T / // — "• MM VV WW For this equation is of the first degree in the variables u, v, w, and therefore represents some straight line; moreover the equation is satisfied at the point (m , v', vf) and also at the point (m'', v", w"), and therefore it represents the straight line which passes through these points. Therefore the equation to the tangent at (u, v, w) is Au B'v C'w - ,„, —2+-rr+^T = (2). Hence we can find the condition which must hold in order that a proposed straight line may toitch the curve denoted by (1). Let the equation to the proposed straight line be Xu + /j,v-\- vw=0 (3). If (3) denotes the equation to the tangent at (v', v', w'), we find by comparing (3) with (2) that W /it)" vw' ■ From these relations and (1) we obtain as the required condition VC^'X) + V(-B» + '/{C'v) = 0. 318. To express the equation to a conic section which touches the sides of a triangle. Let m = 0, v = 0, w=0 be the equations to the sides of a triangle ; then any conic section may be represented by the equation Au- + Bv'' + Gw^ + 2A'vw+2Fwu + 2G'uv = (1). To find where this conic section meets the straight line m = 0, we must put u=0 ; thus (1) becomes £y' + Cw'+24W = (2). 280 CONIC SECTION TOUCHING Now from (2) we obtain by solution two values of - , say — =li^, and — =/tj. The equation v=fi,jto represents some straight line passing through the intersection of v = 6, and w = 0. Hence since (1) is satisfied by those values of x and y which make simultaneously ti = and v—fi^ = 0, the inter- section of the straight lines m = and v — fi^w = is a point on (1). Similarly the intersection of m = and v — fi^w = is a point on (1). Hence the straight line m = will meet (1) at two points, and therefore will not be a tangent to it, unless the straight lines v — fi^w = 0, and v—fi^w=0, coincide. Hence that u = mav touch (1) we must have fi = jj, , and therefore Similarly that v=0 may touch (1) we must h.-d,ve B'^=CA; and that w = may touch (1) we must have G"'=AB. From these three relations we see that A, B, and C must have the same sign, because the product of each two is positive. Also the sign of A, B, and C may be supposed positive, because if each of them were negative we could change the sign of every term in (1), and thus make the coefficients of u', v^, and w" positive. We may therefore put A=l\ B = m\ G=n-, thus A'=±mn, B'=±nl, C'=±lm. Hence (1) becomes ZV + mV + wW + 2mnvw + 2nlwu ± 2lmuv = (3). We shall now examine the ambiguity of signs that appears in this expression. I. Suppose all the upper signs to be taken. The equa- tion may then be written (lu + mv + nwf = 0. This is the equation to a straight line, or rather to two coincident straight lines. II. Suppose the lower sign to be taken twice and the upper sign once ; we have then three cases, {lu-^mv—nwf=Q, or (Zm - mjj + nw)' = 0, THE SIDES OF A TRIANGLE. 281 or (— lu + mv + nwY = 0. Each equation represents two coincident straight lines. IQ. Since then the forms in I. and II. represent straight lines, we see by excluding these cases from (3), that if a curve of the second degree touch the straight lines M = 0, v=0, w = Q, its equation must take one of the forms Pu^ + inW + nW — 2mnvw — 2nlwu — llmuv = 0. . . (4), Z V + mV + nW — ^mnvw + 'i.nlwu + 2linuv = . . . (5) , ZV + mV + nW + 2mnvw — 2nlwu + 2lmuv = 0. . . (6), IV + m'y' + nW + 2mnvw + 2nlwu — 2lmuv = ... (7). These four forms may also be written V(?(t)+ >/{mv)+ V(«w) =0 (8) from (4), a/{—Iu)+ >J{mv) + ^{nw) =0 (9) from (5), ^{lu) + ^{-mv) + s/{nw) =0 (10) from (6), V(Z«) + VC"*") +V(-nM')=0 (11) from (7), which may be verified by transposing and squaring, so as to put the equations in a rational form. 319. It is easy to verify the proposition that the curve represented by the equation 'J{lu) + V(»««) + V(«M') = cannot cut the straight lines m = 0, v=0, w = 0. For sup- pose the above equation satisfied by the co-ordinates of a point; then these co-ordinates must make lu, mv, and nw, all positive, or all negative. Suppose lu is positive; then for any point on the other side of m = 0, the expression lu becomes negative, and thus the co-ordinates of such a point will not satisfy the equation unless both mv and nw are also negative. But if the curve cuts the straight line m = 0, there will be points on both sides of m = lying on the curve, and it will be possible to change the sign of u without changing the signs of V and w. Hence the curve cannot cut the straight line M = 0. Similarly it cannot cut the straight lines v = 0,w = 0. 282 CONIC SECTION TOUCHING THE SIDES OF A TIUANGLE. The same mode of proof will shew that the curves repre- sented by equations (9), (10), and (11), of the preceding Article cannot cut the straight lines u = 0,v = O,w = O. 320. The forms in equations (5), (6), and (7) of Art. 318 may be derived from (4) by changing the sign of one of the constants. Thus, for exampje, (5) may be derived from (4) by changing the sign of I. In the following Article we shall use (4) as the equation to a conic section touching the sides of a triangle; i.t will be found that we might have used (5), (6), or (7). We shall see in a subsequent Article, a case in which it is necessary to distinguish the forms. See Arts. 324, 325. 321. Equation (4) of Art. 318 may be written (lu — mvY + nw {nw — 2mv — 2lu) = (1). If we combine this with w = 0, we deduce that lu — mv=0 (2); hence we can interpret the last equation; it represents a straight line passing through the intersection of m = and v = Q, and also through the point where the straight line w=0 meets the curve (1). It may be shewn as in Art. 310, that nw— 2mv — 2lu = (3) represents the tangent to (1) at the other point where (2") meets it. Similarly we can interpret mv — nw = Q (4)^ lu — 2nw — 2mv = (5), nw — lu = Q (6), mv — 2lu — 2imj = (7). The intersection of (3) with w = 0, of (5) with m = 0, and of (7) with v = will lie on the straight line lu + mv + nw = 0. The straight line lu + mv = passes through the intersec- tion of M = 0, and v = 0, and also through the intersection of (3) with w = 0; hence its position is known. EQUATION TO THE TANGENT. 283 Similarly the equations mv + nw = 0, and nw + lu = 0, can be interpreted. 322. We will now investigate the equation to the tan- gent at any point of the curve represented by A^u+B^/v + G>^w = (1). We might clear this equation of radicals and so obtain the form already considered in Art. 315, and then express the equation to the tangent at any point. Or we may proceed thus: The equation to the straight line passing through the points {u, v, w) and (u", v", w") on the curve may be put in the form A(u — u) B{v — v) C(w — w) _ . •Ju + V" V 2^ + V^ V *" + Vw' For this equation is of the first degree in the variables u, V, w, and therefore represents some straight line; moreover the equation is satisfied at the point («', v, w) and also at the point (m", v", w"), and therefore it represents the straight line passing through these two points. Now suppose the point [u", v", w") to, move along the curve until it coincides with the point {u, v', w). Then the secant becomes ultimately the tangent at {u', v', w') ; and the equation to this tangent is A{u-u') B(v- v') C(w-w') _ Vm' V^'' Vm*' , , . Au Bv Ciu _ ,„. thatis • -^, + -r-' + -T-' = (2). i^U */v tjw ' Hence we can find the condition which must hold in order that a proposed straight line may touch the curve denoted by (1). Let the equation to the proposed straight line be Xm + /i« + vw = (3). If (3) denotes the equation to the tangent at (ti', v' , w') we find by comparing (3) with (2) that A ^ B ^ G 284 CIRCUMSCEIBED CIRCLE. From these relations and (1) we obtain as the required condition \ fJL V 323. To find the eqimtion to the circle described round a triangle. It will he convenient in this and the two following Articles to use the form a;cosa + y sina— |j = as the type of the equation to a straight line; we shall therefore put a, /3, 7 for M, V, w respectively (Art. 71). Let a = 0, ;S = 0, 7=0 be the equations to the sides of a triangle; then, by AJi;. 309, l^y + mya+na^ = (1) will represent any conic section described round the triangle; hence by giving proper values to I, m, n, this equation may be made to represent the circle which we know by geometry can be described round the triangle. We might proceed thus : in (1) write for a, /S, 7 the expressions whidi they represent, then equate the coefficient of xy to zero, and the coefficient of ar* to that of y'; we phall thus have two equa- tions for determining y and -j; and with the values thus obtained (1) will represent the required circle. We leave this as an exercise for the student, and adopt another method. The equation to the tangent to (1) at the intersection of a = 0, and y3 = 0, is, by Art. 310, Z/8 + ma = (2). Let A, B, C denote the angles of the triangle opposite the sides a = 0, /3 = 0, 7 = 0, respectively; by Euclid, iii. 32, the tangent denoted by (2) must make an angle A with the straight line a = 0, and an angle B with the straight line /3 = 0. Suppose the origin of co-ordinates within the triangle, then the equation to the straight line passing through the intersection of a = and j8 = 0, and making angles A and B respectively with these straight lines, is a3in5-h/8sin^ = (3). INSCRIBED CIRCLE. 285 Thus (2) must coincide with (3); therefore we have I sin A o- -1 1 wi sin 5 — = ^ — 5 . Similarly, — = -. — ^ . m sm B n sm G Thus the equation to the circle described round the tri- angle is fiy sin J. + 7a sin jB + a^ sin C = 0. 324. To find the equation to the circle inscribed in a triangle. Suppose the origin of co-ordinates within the triangle ; then for all points on the circle a, /3, 7 are negative quantities (see Art. 54f). Now the equation to the circle must be of one of the forms (8), (9), (10), (11) given in Art. 318; the first is the only form applicable, namely, ■ V(Za) + V{'«/8)+VK) = (1), which is equivalent to v'(-Zj)-1-V(-«'/3) + V(-«7)=0 (2). The other forms are inapplicable, because they would introduce impossible expressions. We have then to deter- mine the values of I, m, and n. If we put a = in (1), we Q fh 71 obtain — = — ; thus —is the ratio of the perpendiculars drawn 7mm to the sides yS = 0, 7 = 0, respectively, from the point where the circle meets the straight line a = 0. Let r be the radius of the circle ; then we know from geometry that the perpen- C G dicular from this point on yS = is r cot -^ sin C or 2r cos' ^ ; a similar expression holds for the perpendicular on 7 = 0. l 2 Hence — = ^ • Similarly - = — . cos 2 Therefore the required equation is A B G cos -H- V* + cos -a v'/3+ cos 2 Vt = 0- 286 ESCBIBED CIRCLE. 325. To find tlie equation to the circle which touches one side of a triangle and the other two sides produced. Let the circle be required to touch the side opposite to the angle A and the other two sides produced. Suppose the origin within the triangle ; then for all points comprised between the side a = and the other sides produced, o is positive and yS and 7 are negative. Hence by Art. 318, the form of the equation to the circle must be VC- h) + V(wi^) + VC*^) = 0. Hence, as before, by considering the point where the circle meets the straight line a = 0, we have oTT — .-0 5-a. cos" —^ — sm'' — , cos-" -^- n ii L . ii £ and - = cos" — ^ — sm' -5- cos" Hence the required equation is A HO cos -5- V (- a) + sin 2 VyS + sin - V7 = 0. Similarly the equations to the other two circles may be written down. 326. The results in Arts. 312 and 321 which hold for any conic section, will of course hold for a circle inscribed in, or described about, a triangle respectively. We have only to use the values of I, m, n, found in Arts. 323... 325. 327. Many applications have been made of the method of abridged notation to express the equations to circles deter- mined by various conditions. We will give some of these applications as specimens, and the student will have no diffi- culty in applying the same methods to other examples. 328. If the equation to one circle, expressed in a rational form, be denoted by S = Qi, the equation to any other circle can be expressed in the form 8-\-\% + ii^ + irf=Q, by pro- perly choosing the constants X, fj,, and v. This result follows from the known form of the equation to a circle in the com- mon co-ordinates; see Arts. 88, 104, 110. Thus, to take the NINE-POINTS CIRCLE. 287 most general supposition, let the equations to two circles be in common oblique co-ordinates K{a?-^ 2x1/ cos ca + y^ -{■ Lx -i- My + If = 0, k (a;' + 2xy cos (i3-\-'i^ + lx + my + 7i =0. Denote the first equation hj S=0; then the second equa- tion is equivalent to S+ -J- {Ix + my + n) -Lx-My — N = a, which we may denote hy S + u = Q. Here u is an expression of the first degree in x and y, and so will be identical with \a + /tyS + vy, S we determine X, /x, and v suitably. If jSr=0 and S +'!\,i+ fi/S + vy = be the equations to two circles, Xa + /i/S -t- vy = will be the equation to the radical axis of the two circles ; see Art. 110. Since aa. + b0 + cy is a constant, by Art. 73, we may instead of >Si + (\a + fj0 + vy) use S+{a3 + b0 + cy) {la. + m^ + ny) or /S -I- (a sin ui -1-/3 sin 5 -I- 7 sin C){l^ + m^ + ny), provided we properly determine the constants in each case. 329. To express the equation to the circle which passes through the middle points of the sides of the triangle of reference. Let a = 0, /3 = 0, 7=0 be the equations to the straight lines which form the triangle of reference ; see Art. 78. Assume for the required equation j87 sin A+ya.&m.B+afi sin G + {asaLA -)-j8sin5 4-7sin C)(Za + m/3 + «7) = 0; see Arts. 323 and 328. At the middle point of the side BC we have = 0, and y3 _ sin C 7 ~ sin .B ' 288 NINE-POINTS CIRCLE. substituting in the assumed equation we obtain sin A sin C ' « • ^ /»»* sin C . N „ + 2sinC ■ p +n)=0, V sin if / sin B or sin^-r2 (msinG + Msin5) =0. But sin A = siaBcosG+ cos 5 sin C; thus sin C (2m + cos B) + sin B (2n + cos G) = 0. In a similar manner we obtain two analogous equations ; and from the three equations we deduce Z = — g cos A, m = — -= cos B, n — — -^ cos C. Z A ^ Hence the required equation is ySy sin A + yn sin i?-|- a/3 sin G — -^ (asm A + 13 BinB + y sin 0) (a cos^+yS cos5 + 7 cos (7) = 0. The radical axis of this circle and the circle described round the triangle of reference is therefore determined by a cos ^ + j8 cos 5 + 7 cos 0= 0. 330. To express the equation to the circle which passes through the feet of the perpendiculars from the angles of the triangle of reference on the opposite sides. Assume for the required equation (87 sin A +rfx sin B + a^ sin G + (a sinj4 + /8sin£-l-7sinC)(Za-f-m)8 + 7i7) = At the foot of the perpendicular from .4 on 5(7 we have = 0, and— = j:; substituting in the assumed equation we obtain sin J. cos 0, /cos (7 . _ , . _\ /mcos(7 . \ „ o — + Dsm5 + sm C 5- + m =0, cosi^ \cosB /\cosB J or sinJ.cos5cos C+sin.4(mcos(7 + wcos5) = 0; therefore (»" + h cos£J cos C + (n + „ cos(7j cos B=0. NINE-POINTS CIRCLE. 289 In a similar manner we obtain two analogous equations; and from the three equations we deduce Z = — 2 cos w4, m = — 2 cos -B, n = — ^ cos (7. Hence the required equation is ;87 sin A+'yasinB + aj8 sin C — g(asin^+/3sin5 + 7sinC)(acosJ. +/8cosjB + 7COs(7) =0. Thus the circle is the same as that considered in the pre- ceding Article. 331. Let denote the intersection of the perpendiculars from the angles of a triangle on the opposite sides. Then it is known that the circle which passes through the six points specified in the preceding two Articles also passes through the middle points of OA, OB, and OG. The circle is called the nine-paints circle. See Appendix to Eiiclid. It is easy to shew that the circle which passes through the six points specified in the preceding two Articles also passes through the middle points of OA, OB, and 00. For consider the triangle OBC. The perpendiculars from the angular points on the opposite sides meet these sides respec- tively at points which coincide with the feet of the perpen- diculars from the angles A, B, G on the opposite sides ; thus we know that the circle considered in the preceding two Articles passes through these points: hence it also passes through the middle points of OB and OC, as well as through the middle point of BO. Similarly the circle also passes through the middle point of OA. is a centre of similitvde of the circle described round the triangle ABO and the nine-points circle of the triangle ; see Art. 119. For, as we have just seen, the three radii vectores drawn from to the circumfei-ence of the former circle, namely OA, OB, OG, are respectively double the radii vectores drawn in the same direction to the latter circle; and it is easy to shew that the same ratio will hold for any cor- responding radii vectores. See Example 58 of Chapter xiv. - " ■ 19 290 EADICAL AXIS. 332. To investigate the conditions which must hold in order that the general equation of the second degree may re- present a circle. Let the equation be ia» + M^" + AY + 2i'/37 + 23/ V + 2iV'«y3 = 0. Let A denote the area of the triangle of reference ; then, by Art. 73, aa + 6;3+C7=-2A; therefore aa' = — 2Aa — (6y8 + cy) a. Similarly 6/3= = - 2A^ - (07 + aa)/3 ; and cy'=— 2A7 - (aa + 10) 7. Substitute for a', yS*, and 7* in the general equation, and it becomes \a c / Then, by Arts. 323 and 328, we see that the necessary and sufficient conditions in order that this equation may represent a circle are 2L'-'^-— 2M'-—-— 2N'-—-~ b c £__^_ "' ^ a b c ' that is, 2L'bc -M (vu' — ww) = 0. This equation then represents the secant passing through the two given points. Hence the equation to the tangent at the point («', v, w) is uv + vu — 2ww' = 0. Suppose — = fi, then from the equation to the curve — ; = — ; similarly if -^ = p.", then — r, = — , . Thus the equa- w /J, ' •' w '^ w /J, ^ tioD to the secant may be written or u + fiiJi"v— Qj,' + /j,")w = 0; and the equation to the tangent may be written u + 11^-2/^10=0. 339. Next take the equation Pu^ + mV = n^w°. This may be written (nw + mv) (nw — mv) = Pu'. Hence by Art. 337 nw + mv = and nw — mv = are tangents to the conic section represented by the equation, and w = is the equation to the corresponding chord of contact. Since these two tangents meet at the point of intersection of i; = and w = 0, it follows that this point is the pole of m = 0. Similarly we may write the equation in the form {nw + lu) (nw — lu) = mV, and infer that the point of intersection of u = and w = is the pole of i; = 0. CONIC SECTIONS IN CONTACT. 297 Hence it follows that the point of intersection of m = and r = is the pole of w = 0. See Art. 291. 340. The following is a particular case of the preceding Article, a^+0' = ny. (See Art. 71.) Suppose the straight lines o = 0, )8 = 0, at right angles; then a' + 0' is the square of the distance of the point {x, y) from the intersection of a = and ^ = 0. Hence the above equation represents a conic section which has 7 = for its directrix, and the inter- section of a = and /3 = for its focus. The straight lines «7 — a = and 717 + a = are tangents to the conic section, touching it at the extremities of the focal chord /Q = 0: also these tangents meet on the straight line 7 = 0; hence, the tangents at the extremities of any focal chord meet on the cor- responding directrix. Also the above tangents meet on the straight line o = 0, which by supposition is at right angles to /3 = 0; hence, the straight line which joins the focus to the intersection of tangents at the extremities of a focal chord is at right angles to that focal chord. 341. If M = and i) = be the equations to two conic sections which meet at four points, then u+lv = will repre- sent any conic section which passes through the four points of intersection. This will be obvious after the proofs given of similar propositions. Also if w = and w' = be the equations to two straight lines, u + Iww' = will represent any conic section passing through the four points at which the lines w = and w' = Q meet the conic section m = 0. Also u + lw' = will represent a conic section passing through the points of intersection of the conic section m = 0, and the straight line w = 0. This conic section will have the same tangent as w = at the points where u — Q and w = intersect ; we might anticipate this would be the case from observing the interpretation of the equation u + Iww' = 0, and supposing the straight line w' = to approach the straight line w = 0, and ultimately to coincide with it. We may prove it strictly by taking one of the points where m = meets w = for the origin, and the straight line w = for the axis of a;; thus u becomes of the form Aa^ + Bxy+Gif + Dx + Ey, 298 CONIC SECTIONS IN CONTACT, and we can see, by Art. 283, that and Aoc' + BxyJrCf + Dx + Ey + ly' = have the same tangent at the origin. Also by giving a suitable value to I the equation u + Zw' = may be made to represent the two straight lines which touch the conic section m = at the points where it intersects the straight line w = 0. This may be inferred from Art. 293 ; the equation w = is equivalent to t + V — 1 = 0, and the fx V \^ equation m = is equivalent to ( T + 't: "" 1 ) + /^^ ~ ^- Thus by taking Z=— 1 we have u + l'w'' = fjusy; and the equation xy = denotes the two tangents to the conic section m = at its points of intersection with the straight line w = 0. 342. Pascal's Theorem. The three intersections of the opposite sides of any hexagon inscribed in a conic section are on one straight line. Let r = 0, s = O,t = O,u = O,v = O,w = 0,he the equations to the sides of a hexagon which is inscribed in the conic sec- tion S = 0. Let the hexagon be divided by a new straight line ^ = into two quadrilaterals, one of which has for its sides the straight lines obtained by equating to zero succes- sively, r, s, t, , and the other the straight lines obtained by equating to zero successively, u, v, w, ^. Now we know that if a, b, I, m are appropriate constants, the equation to the conic section may be written in the forms as^ + brt = and lv = muw — brt. The right-hand member of this equation vanishes when u and r simultaneously vanish, and when u and t simulta- neously vanish; also when w and r simultaneously vanish, and when w and t simultaneously vanish. Since the left- hand member is identically equal to the right-hand, the left- hand member must also vanish in these four cases ; that is, one of its two factors ^ and as—lv must vanish in each of PASCAL'S THEOKEM. 299 these four cases. By construction, ^ = represents the straight line joining the point determined by r = and w = 0, ■with the point determined by f = and u = 0; and thus we see that as—lv = is the straight line joining the intersection of M = and r = with that of i = and w = 0. But the straight line as — lv = obviously passes through the intersection of s = and v = 0; therefore the three points determined respec- tively by M = and r = 0,t = and w = 0, s = and v = 0, lie on a straight line. It is to be observed that if six points be connected by straight lines in different ways, as many as sixty figures can be formed which may be called hexagons in an extended sense of that word. Thus for six given points on a conic section there will be sixty applications of Pascal's Theorem. 343. Let s = be the equation to a conic section, and u = 0, v = 0, w = 0, equations to three straight lines ; then s—Pw' = 0, 8 — mV = 0, s — nW = 0, represent curves of the second degree touching the proposed conic section. By pro- perly choosing u, v, w, I, m, n, we may make each of the last three equations represent a pair of straight lines touching « = 0. (See Art. 341.) Thus, if there be a hexagon circum- scribed round the conic section s = 0, the equations s-fM'=0...(l), s-mV = 0...(2), «-j!W = 0...(3), may be taken to represent the six sides of the hexagon. By combining (1) and (2) we obtain s_Pm'_(s-7»V) = 0, or (mv-lu){mv + lu) = 0...{ii), for the equation to a pair of straight lines which pass through the intersections of (1) and (2). Similarly {nw — mv){nw+mv) —0 (5) represents a pair of straight lines which pass through the in- tersections of (2) and (3). And {lu — nw)(lu + nw) = (6) represents a pair of straight lines which pass through the in- tersections of (3) and (1). The six straight lines which we have obtained may be 300 bkianohon's theorem. arranged in four groups, each containing three straight lines which meet at a point, namely, mv — lu = 0, nw — mv= 0, lu — nw = 0, mv + lu = 0, nw + mv = 0, lu — nw = 0, mv + lu=0, nw — mv = 0, lu + nw= 0, mo — lu = 0, nw + mv = 0, lu + nw = 0. This result is consistent with Brianchon's theorem ; if a hexagon he described dbowt a conic section the three diagonals which join opposite angles meet at a point. For suppose that a hexagon is described round a conic section, and let its angular points be denoted by A, B, G, D, E, F. By properly choosing u, v, w, I, m, n, we may make equation (1) denote the straight lines AB and DE, equation (2) denote the straight lines BG and EF, and equation (3) denote the straight lines GD and FA. We will now examine what straight lines are determined by equations (4), (5), and (6). Equation (4) determines the two straight lines which pass through the intersections of the straight lines determined by (1) and (2); and as the signs of I and m are at present in our power we may take them so that mv — lu = shall repre- sent the straight line BE, and then m,v + lu = will represent the straight line joining the point which is common to AB and EF with the point which is common to BG and DE. Simi- larh as the sign of n is still in our power, we may take it so that nw — mv = shall represent the straight line GF, and then nw + mv = will represent the straight line joining the point which is common to BG and FA with the point which is common to GI) and EF. One of the two straight lines represented by (6) is AD, and the other is the straight line joining the point which is common to DE and FA with the point which is common to GD and AB; it is however not obvious how we axe in general to discriminate between these two straight lines. Thus the proof of Brianchon's theorem is not perfectly satisfaictory, and accordingly we shall give another proof by which the theorem is deduced from that of FascaL Let the angular points of the hexagon be denoted as before by the letters A, B, C, D, E, F. Let the straight line beianchon's theorem. 301 be dra\m which passes through the points of contact of the conic section and the tangents AB, BG ; also let the straight line be drawn which passes through the points of contact of the conic section and the tangents DE, EF; and let P denote the point which is common to these two straight lines. Then P is the pole of BE; see Arts. 103, 120, 289. In the same way we may determine the pole of OF which we shall denote by Q, and the pole of AD which we shall denote by R. By Pascal's theorem P, Q, and R lie on a straight line ; hence OF, BE, and AD meet at a point, namely, at the pole of the straight line PQR; see Art. 291. For further information on the subject of this Chapter the student is referred to Salmon's Gmiic Sections. EXAMPLES. 1. Shew that if a — c:a' — c'::b:b', a circle may be described through the intersections of the two conic sections ax' + hxy + cy' +dx + ey +f = 0, a'x' + h'xy + cY + d'x + e'lj +/'= 0. Find also the condition that a parabola may be described passing through the origin and the points of intersection of these curves. 2. Two conic sections have their principal axes at right angles : shew that a circle will pass through their points of intersection. 3. The equations to two conic sections are AY + 2Bxy + Cx* + lA'x = 0, ay- + 2bxy + ci? + lax = 0. Shew that the straight lines joining the origin with their points of intersection will be at right angles to each other if d{A+G)=A'{a + c). 4. An ellipse is described so as to touch the asymptotes of an hyperbola : shew that two of the chords joining the points of intersection of the ellipse and hyperbola are parallel. S02 EXAMPLES. CHAPTER XV. 5. If 0/8 = c' be the equation to an hyperbola (Art. 71), then O/S = 0, a' — ;8' = 0, a' — n'0' = 0, are the respective equa- tions to the asymptotes, the axes, and a pair of conjugate diameters, n being any constant. 6. The straight lines which bisect the angles of a triangle, meet the opposite sides at the points P, Q, R, respectively: find the equation to an ellipse described so as to touch the sides of the triangle in these points. 7. From any point two straight lines are drawn, one in- clined at an angle a, the other at an angle ^ + «, to the axis of a parabola: shew that another parabola may be described which shall pass through the four points of intersection, whose axis is inclined at an angle 2a to that of the given parabola. 8. Prove that the equation to the conic section which passes through the point (h, k), and touches the parabola if = Ix at the vertex and at an extremity of the latus rec- tum, is (y" -lx){k- 2Kf = (y - 2a;)"(A;'' -Ih). Shew that it is an ellipse or hyperbola according as the point (A, k) is within or without the parabola. 9. A conic section touches the sides of a triangle ABC at the points a, h, c; and the straight lines Aa, Bb, Cc, intersect the conic section at a', h', c: shew that (1) the straight lines Aa, Bb, Co pass respectively through the intersections of Bo' and Cb', Ca' and Ac', AV and Ba, (2) the intersections of the straight lines ah and a'b', bo and b'c, ac and a'c', lie respectively on AB, BO, CA. 10. A conic section is described round a triangle ABC ; straight lines bisecting the angles of this triangle meet the conic section at the points A', B, C, respectively: express the equations to A'B, A'C, A'B. 11. If a conic section be described about any triangle, and the points where the straight lines bisecting the angles of the triangle meet the conic section be joined, the intersections of EXAMPLES. CHAPTER XV. 303 the sides of the triangle so formed with the corresponding sides of the original triangle lie on a straight line. 12. Interpret the equation (M-')(M-')^-'^=»^ find how many parabolas can be drawn through four given points. 13. If M = 0, « = 0, w = represent the sides of a tri- angle, shew that the sides of any triangle which has one angle on each side of the former may be represented by M + wi; + — = 0, — l-v + lw = 0, mu + T + w = 0, m n I where I, m, n are constants. Find also the relation which must hold between I, m, n, in order that the straight lines joining corresponding angles of the two triangles may meet at a point. 14. A circle and a rectangular hyperbola intersect at four points, and one of their common chords is a diameter of the hyperbola: shew that another of them is a diameter of the circle. 15. ACA' is the major axis of an ellipse ; P is any point on the circle described on the major axis; AP, A'P meet the ellipse at Q, Q': shew that the equation to QQ' is {a" + F) 3^ sin + 2b^x cos - 2aV = 0, the ellipse being referred to its axes, and being the angle AGP. If an ordinate to P meet QQ' at R, the locus of R is an ellipse. 16. The locus of a point such that the sum of the squares •of the perpendiculars drawn from it to the sides of a given triangle shall be constant, is an ellipse ; and if the constant be so chosen that the ellipse may touch the side opposite to the angle A at D, then CD : BD :: b" : c'. 304 EXAMPLES. CHAPTER XV. 17. With the notation of Art. 323, shew that the equation to the straight line through G and the centre of the circle is a cos B = fi cos A. 18. Suppose in Art. 323 that D is the middle point of the arc AB ; then the equations to BD and AD are respectively a sin + 7 (sin A + sin B) = 0, /3 sin C + 7 (sia A + sin B) = 0. 19. In Art. 318, equation (4), if A', B', C be the points of contact of the triangle and conic section, shew that the equation to A'B' is lu + mu — nw = 0. 20. In the figure of Art. 292, suppose m = the equation \jo AG,v = the equation to BD, and «; = the equation to EF, and that iV + mV — ii'w' = represents a conic section passing through A, B, C, D : then express the equations to the tangents at A, B, C, D, and also to the straight lines AB, BG, CD, DA. Shew also that the straight line FG passes through the intersection of the tangents at A and B, and of those at G and D. 21. Express by the aid of Art. 323 the equation to the circle described round the triangle formed by the straight lines Hence deduce the last proposition of Art. 146. 22. Give a geometrical interpretation of equation (1) in Art. 310, and shew that it is a particular case of the theorem, in Art. 336. 23. Interpret the last equation in Art. 323: deduce the following theorem ; if from any point of the circle which circumscribes a triangle, perpendiculars are drawn on the sides of the triangle, the feet of the perpendiculars lie on one straight line. 24. If ellipses be inscribed in a triangle each with one focus on a fixed straight line, the locus of the other focus is a conic section passing through the angular points of the triangle. EXAMPLES. CHAPTER XV. 305 25. Three conic sections are drawn touching respectively each pair of the sides of a triangle at the angular points where they meet the third side, and each passing through the centre of the inscribed circle: shew that the three tangents at their common point meet the sides of the triangle which intersect their respective conies at three points lying on a straight hne. Shew also that the common tangents to each pair of conies intersect the sides of the triangle which touch the several pairs of conies at the above three points. 26. With the angular points of a triangle AB C as centres, and the sides as asymptotes, three hyperbolas are described, having A', B', C as their vertices respectively: prove that if A A' sin -^ = BB' sin ^ = C(7' sin ^ , the intersections of each pair of hyperbolas lie on the axis of the third. 27. The necessary and sufficient. condition in order that the equation /a' + m^ + wf = may represent a rectangular hyperbola isZ + m + n = 0. The necessary and sufficient condition in order that Ipf + mya + nayS = may represent a rectangular hyperbola is I cos A + m cos .B + n cos 0=0. 28. Shew that V(?a) + V(«»y8) + V(«7) = represents in general an ellipse, parabola, or hyperbola according as Imn ( - + X "*" "") ^^ positive, zero, or negative ; where a, 6, c denote the lengths of the sides of the triangle formed by a = 0, /3=0, 7 = 0. 29. Shew that Z/87 + 7717a + nayS = represents in general an ellipse, parabola, or hyperbola according as Pa* + mV + nV - 2.lmdb - 2mnbc - 2nlca is negative, zero, or positive. 30. Express the equation to the circle which is con- centric with the inscribed circle of the triangle of reference, and passes through the angular point A, T.c-s. 20 306 EXAMPLKS. CHAPTER XV. 31. Find the fourth point of intersection of the conic sections Ivw + tnvm + nuv = 0, and Xma + vrivm + vluv = 0. 32. Shew that the equation to the radical axis of the circles inscribed in a triangle and circumscribed about it is A Ti G a cosec A cos* --• + /8 cosec B cos* «- + 7 cosec G cos* -^ = 0. 33. Find the equation to the diameter of the curve Iffy + my a + noifi = which passes through the point of in- tersection of the straight lines /S = and 7 = 0. 34!. Find the equation to the tangent to the curve V(fc) + '/{m^) + V(«7) = 0, which is parallel to the straight line 7 = 0; and thence shew that the centre of the curve is determined by « = ^ = "^ ^ mc+nb na + k lb + ma ' 35. Employ the method of Art. 332, and the result given in Example 29 to find the condition which determines whether the general equation La? +M^ + iVV + 2i'/S7 + 2M'y:L + 2N'afi = represents an ellipse, parabola, or hyperbola. 36. A conic section passes round a triangle, and the tangent to the curve at each angular point is parallel to the opposite side of the triangle : shew that the cui-ve is an ellipse. 37. OP, OQ are tangents to an ellipse at P, Q, and asymptotes of an hyperbola ; MS is a common chord parallel to PQ : shew that if PR touches the hyperbola at R, QS touches it at iS; also if PS, QB intersect at U, then OU bisects PQ. 38. If t, M, V, w be linear functions of m and y, shew that the equation to the tangent at the point {i, u', v', w') to the conic section given by tu = vw is tu' + ut' = vvi + wv. E XAMP LES. CHAPTEE XV. 307 39. Ifa = 0,/3 = 0.7 = 0,^ + | + ^=0, -5^+^ + 2 = 0. — + r-H — =0,De the equations to the sides of a hexagon which circumscribes a conic section, shew that «i (Vs - V.) + o,fe - Kc,) + a,(6,c, - &,c.) = 0. 40. ABC is the triangle of reference; D, E, F are the middle points of the sides: express the equations to the straight lines which bisect the angles of the triangle DEF. 41. Express by means of abridged notation the equation to the ellipse which touches the sides of a triangle at the middle points of the sides. 42. From a point P two tangents are drawn to a conic section meeting it at the points M and iV respectively ; the straight line through P which bisects the angle MPN meets the chord MN a,t Q; any chord of the conic section is drawn through Q : shew that the segments into which the chord is divided by the point Q subtend equal angles at P. 20—2 ( 308 ) CHAPTER XVI. SECTIONS OF A CONE. ANHARMONIC RATIO AND HARMONIC PENCIL. Sections of a Cone. 344. We shall now sliew that the curves which are included under the name conic sections, can be obtained by the intersection of a cone and a plane. Definition. A cone is a solid figure described by the revolution of a right-angled triangle about one of the sides containing the right angle, which remains fixed. The fixed side is called the axis of the cone. Let OR be the fixed side, and OHC the right-angled triangle which revolves round OH. In order to obtain a cone such as is considered in ordinary synthetical geometry, SECTIONS OF A CONE. 309 we should take only & finite straight line 00; but in analy- tical geometry it is usual to suppose OG indefinitely prodiiced both ways. A. section of the cone made by a plane through OH and OG will meet the cone in a straight Une OB, which is the position OG would occupy after revolving half way round. Let a section of the cone be made by a plane per- pendicular to the plane BOG; let APhe the section, A being the point where the cutting plane meets OG; we have to find the nature of this curve AP. Let a plane pass through any point P of the curve, and be perpendicular to the axis OH ; this plane will obviously meet the cone in a circle DPE, having its diameter DE in the plane BOG. Let MP be the straight line in which the plane of this circle meets the plane section we are considering, M being in the straight line DE. Since each of the planes which intersect in the straight Hne MP is perpendicular to the plane BOG, the straight line MP is perpendicular to that plane, and therefore to every straight line in that plane. Draw AF parallel to ED, and ML parallel to OB ; join AM. Let AM =x, MP = y, OA = c, HOG = a, OAM= 6 ; the angle AML will be equal to the inclination oiAM to OB, that is, to TT — ^ — 2a. „ MD sin MAD sm6 ^, j. ,,„ xsmd Now -rr-r = -■ — ^ttTa- = ; therefore MD = . MA sin MDA cos a cos a EM=FL = FA-AL = '2.csaicL-AL; AL _ sin AM L ^ sin(7r-g-23) AJI~siiiALM~ .fir \ ' sm^2+«j therefore ^^ = '^ ''"cos a ^" ' , r rrir o • a;sin(^ + 2a) therefore EM = 2c sm a ^^ -. cos a But, from a property of the circle, MP' = EM. MD ; a;sin6( x sin { 6 + 2a) ) therefore if = ■{ 2c sin a \ . " cosa ( cos a ) 310 SECTIONS OF A CONE. If we compare this equation with that in Art. 282, we see that the section is an elUpse, hyperbola, or parabola, accord- sin^ sin (0+ 2a) . ,. ... ., . ing as 3 ' is negative, positive, or zero, that is, according as 6+2x is less than ir, greater than w, or equal to tt. Hence if AM is parallel to OB the section is a parabola, HAM produced through M meets OB the section is an ellipse, HAM produced through J. meets OB produced through the section is an hyperbola. If c = the section is a point if ^ + 2a is less than tt, two straight lines if ^+ 22 is greater than tt, and one straight line if ^ + 2a = TT. The section is also a straight line whatever c may be, if ^ = or tt. The equation above obtained may be written r = sin sin (0 + 2a) J 2c sin a cos a cos'a I sin (^ + 2a) -^}; suppose + 2a to be less than tt, so that the curve is an ellipse; then by comparing this equation with the equation y' = -t (2aa; — a?), -we have „ _ 2c sin a cos a V _ sin sin (0 + 2a) ""sin {0 + 2^' a' ^"a ' _, „ c sin 2a ,, c'sin'asin^ Thus 2a= ■ ,/i . o M P = • fa . a \ • sin [0 + 2a) sin (0 + 2a) ., ,_, y _ cos'g - {sin'(g + a) - sin'a} _ cos'(g + a) Also 6 •"• X ^~ ~^ — o ^~ Q ■ a COST a cos a If we suppose in the figure on page 308 that AM is pro- duced to meet the cone again at A', then 2a = AA', as might have been anticipated ; also b may be shewn to be a mean proportional between tbe perpendiculars from A and A' on the axis Off. Similar results may be obtained when the curve is an hyperbola. 345. An ellipse of given excentricity can always be ob- tained &om a given cone by properly choosing the cutting SECTIONS OF A CONE. 311 plane. For we have the equation cos'(^ + a) =e*cos'a, in •which a and e are given, e being less than unity. Now it is manifest that there must exist a value of d between and TT ^ — a which satisfies this equation, and also a value of Q between „ — a and tt — 2a. From a given cone we cannot obtain an hyperbola of given excentricity unless the given quantities are such that e' cos'' a is not greater than unity. 346. Art. 344 admits of great extension. We may first give a more general definition of a cone. If a straight line move so as always to pass through a fixed point and a fixed curve the surface generated is called a cone. The fixed point is called the vertex, and the fixed curve the directrix. If a cone be formed with any conic section as directrix any plane section of the cone will he a conic section. The demonstration will be similar to that in Art. 344. Let be the vertex, and instead of the circle with BG as a diameter let there be any conic section for directrix. The plane EPD is to be taken parallel to the plane of the direc- trix, so that the curve EPD will be a similar conic section. The plane OBG may be any fixed plane passing through the vertex, so that it will not be necessarily perpendicular to the plane EPD. Now an equation of the second degree will hold between MP and MD, because the curve EPD is a conic section; and MD bears a constant ratio to AM; therefore an equation of the second degree holds between MP and AM. Aiid MP is always parallel to a fixed direction. Therefore the cui-ve AP is a conic section. 347. In consequence of the extension of the definition of a cone it is necessary to have a special name for the par- tictilar cone considered in Art. 344; and accordingly it is called a right circular cone. The word circular indicates that the directrix is a circle; and the word right indicates that the straight line drawn from the vertex to the centre of the directrix is at right angles to the plane of the directrix. 312 SECTIONS OF A CONE. An oblique circular cone is a cone in which the directrix is a circle, but the straight line drawn from the vertex to the centre of the directrix is not at right angles to the plane of the directrix. When the word cone occurs in mathematics the student will often have to determine from the context whether the word is used in the general sense of Art. 346, or is used as an abbreviation for right circular cone. 348. The case of an oblique circular cone deserves sepa- rate consideration. Let be the vertex of the cone ; EPD a section parallel to the plane of the directrix, which is therefore a circle. Let AP be a section made by any plane. Let i^ be the intersection of these two planes; and JED that diameter of the base which bisects Pp. Let M be the point of bisection, and MA the intersection of the plane PAp and the plane EOD. Then MP is always parallel to a fixed direction, but is not necessarily at right angles to AM. Proceeding as in Art. 344 we have MP' = EM.MD. Now EM = FA — AL. Also the ratio of AL to AM is con- stant, and so is that of MD to AM. Thus finally we obtain MP^ = \ . AM — fjb . AM\ where \ and ft, are constants, which involve FA and the sines and cosines of the angles MAD, MDA, AML. It is easy to shew that in a certain special case the section is a circle. Suppose the plane OED perpendicular to the plane of the directrix ; and suppose the plane MAP perpen- dicular to the plane OED: then MP is at right angles to SECTIONS OF A CONE. 313 AM. liet AM produced meet OE at A'; then the section ■will be a circle provided MP^ = AM. MA', that is provided AM. MA' = EM. MD. This requires the triangles AMD and A' ME to be similar ; thus the angle MAD must be equal to the angle MEA', and the angle MDA equal to the angle MAE. Such a section of an oblique circular cone is called a svh-contrary section. 349. Conversely, suppose we have a given conic section, and we require to form an oblique circular cone which shall contain the conic section. Refer the conic section to axes consisting of a diameter and the tangent at its extremity. The angle between these axes will determine the angle AMP of the preceding figure. Then X and fj. will have known values, so that we have two equations for finding four unknown quantities, namely, FA and the angles MAD, MDA, AML. Hence the problem is indeterminate ; and will remain indeterminate even if one condition is introduced. Such a condition, for example, might be the following: let MA produced meet at Q the plane through parallel to the plane of the directrix ; and let -4 ^ be required to have a given value. Suppose AM produced to meet OE at A'; then OQ MD , OQ LA AQ~MA' ^"""^A'Q'MA- .V. f OQ' • MD.A L therefore AQ.A'Q " ~MA'~' ' The right-hand expression is what we have denoted by /j,; thus when the conic section is given, and also AQ, it follows that OQ is known. Anhamumic Ratio and Harmonic Pencil. 350. We will now give a short account of anharmonic ratios and harmonic pencils, which are often used in investi- gating and enunciating properties of the conic sections. 314 ANHARMONIO RATIO. Let there be four straight lines meeting a1 a point ; then if any straight line ADGB be drawn across the system, -Tri -^ TTri '^^ be a Constant ratio. AG JJG Suppose the point where the straight lines meet ; then AB sin ^05 AG waAOO AU' sin ABO' AG AO' smAGO' therefore AB _ sin AOB sin AGO AG~ sinAOG' sinABO' „. ., , BB sin DOB sin DGO Similarly _ = -j-^^..-^-^^ ; , . AB DB _sxaAOB sin DOB tneretore Jc^7)C- sin AOG^ sin DOG' Now suppose any other straight line A'D'G'B' drawn across the system, then since AOB and A' OB' are the same angle, and so on for the other angles, we have AB AC 4? ~dg' A'B VB' 'AG' '■ D'C" which proves the proposition. HAHMONIC PENCIL. 315 Similaxly we can prove that each of the following is a constant ratio AB CB AG BG AD ■ 'GD'^^'^'AD'^BD' 351. Definitions. Any four straight lines meeting at a point form a pencil. A straight line drawn across a pencil is called a trans- versal. The four points at which the straight line meets the pencil form a range. A f.v , ^ ,. AB DB AB CB Any one of the constant ratios -j^^^q, AB^'GD ' —Tj^ -T- D/S is called an anharmonic ratio of the pencil. The pencil is called harmonic if AB . DC — AD . BC, that is, if the rectangle formed by the whole straight line {AB) and the middle part {DC) is equal to the rectangle of the other two parts {AD), {BC). 352. The harmonic pencil is so called because it divides any transversal harmonically. For since AB .DC= AD . B C, ~I'n~~nr>' *^^* ^^' ^ ^® ^^^^ ^^' ■^^' ■^"^' *^® ^^^> s®*'*>^^) and third quantities respectively, the first is to the third as the difference of the first and second is to the difference of the second and third. When the pencil is harmonic one of the three constant ratios of the pencil is equal to imity. We shall sometimes select one of the anharmonic ratios of a pencil, and confine our attention to it, and shall then speak of the selected ratio as the anharmonic ratio of the pencil. 353. Suppose OA, OB, OC, Oi) form an harmonic pencil; if we take any new origin 0', and join O'A, O'B, O'G, OD, these four straight lines form a new harmonic pencil ; for the transversal ABGD is cut harmonically. 316 HAKMONIC PENCIL. 354. The anharmonic ratio of a pencil is not altered if the transversal meet the straight lines of the pencils produced, instead of the straight lines themselves. Suppose OA, OB, 00, OD to be a pencil, and let a transversal A'BG'iy meet three straight lines of the pencil, and the fourth AO produced at A'. The angles A' OB', AOB are supplemental; and so are AOD, A'OD'; and so on. Hence any anharmonic ratio formed on ABCD is equal to the corresponding ratio formed on A'B'C'D'. 355. Suppose AB.CD = AD.BG, so that OA, OB, OC, OD form an harmonic pencil. By the last proposition A'B' C'B' _ AB GB _ A'ly ■ G'J)' ~ ad'- GB~ ' therefore OA', Off, 00', OD' form an harmonic pencil. Similarly OC', OB, OA, and BO produced through will form an harmonic pencil. Thus from one harmonic pencil by producing the straight lines through the vertex, we can derive four other harmonic pencils. 356. The straight lines whose equations are a = 0, ^ = 0, a — A;/8 = 0, a + AyS = form an harmonic pencil. Let OM be the straight line a = 0, ON the straight line /3 = Q, OP the straight line a-k0 = Q, OQ the straight line HABMONIC PENCIL. 317 Let a transversal meet the pencil at mpiiq; then (Art. 70) sinPOM ^ Bin QOM sin POiV" sinQOJV' therefore sin POM sin QOi V s,m POJS" sin QOM~'' therefore (as in Art. 350) £^ . 2^ = 1 ; ^ ^ pn qm therefore pm .qn = pn. qm. The same result will follow if we draw the transversal in a diflferent position. The harmonic pencil is so formed that its outside straight lines are always one of the two a = and /S = 0, and one of the two a — kfi=0 and a+kfi = Q. 357. The anharmonic ratio of the four straight lines k = 0, /3 = 0, a-kfi = Q, a + k'^=Q, is p. For as in the preceding Article we have sin POM sin PON = k. sin QOM sin QON = 4'; therefore, by Art. 351, j-, expresses the anharmonic ratio. 318 PROPERTIES OF A CONIC SECTION. 358. Article 356 will also hold if the equations to the straight lines heu = 0,v = 0,u — kv = 0, and u + kv = 0. Tor, by Art. 57, we have m = Xa, v = fi^, where X and /j, are con- stant quantities; hence the equations u — kv = and u + kv = may be written \a — kfi0 = and \a + k/ifi = 0, or a — k'^ = and o + A'/S = 0, where kf =■—. Hence Article 356 becomes immediately applicable. 359. The four straight lines EB, EG, EG, EF, in Art. 75, form an harmonic pencil ; for their equations are M = 0, w = 0, lu — nw = 0, lu + nio = 0. By symmetry FB, FA, FG, FE, will also form an har- monic pencil. Also GD, GC, GF, GE form an harmonic pencil, for their equations are respectively Zm — mD=0, m'j—nw = Q, lu—mv — {mv — nw) = 0, lu — inv + mv — nw = 0. 360. A straight line drawn throvgh the intersection of two tangents to a conio section is divided harmimieally by the curve and the chord of contact. Befer the curve to the tangents as axes; its equation will be of the form (Art. 293) (!+|-0'+'^^=° w- Suppose a straight line drawn through the origin, and let its equation be (Art. 27) f=l- <^)- Thus the distances from the origin of the points of inter- section of (1) and (2) wiU be the values of r found from the equation fir mr ,\* , . ^ PB0PEBTIE3 OF A CONIC SECTION. 319 U+A-W+'*'^ = ° (^)- If r and r" be tlie roots of the equation, we have P^P-KM) <*'■ Also the equation to the chord of contact is M-i=« (^)- Hence for the distance (r,) of the point of intersection of (2) and (5) from the origin, we have the equation Ir, . mr, , 1 I m ,„. f^-f = '' °V. = A + -^ («)• 2 11 From (4) and (6) we have — = — + -^ , thus r, is an har- ' 1 '■ ' monic mean between r' and r". Since LMNO is divided harmonically, if from any point in AB we draw straight lines to L, N, and 0, these straight lines o with AB form an harmonic pencil. A particular case is that in which the point in AB is the intersection of the tangents at N and L, which we know wUl meet on AB produced. (See Arts. 103, 185.) 320 PROPERTIES OF A CONIC SECTION. 361. Let A, B, C, D be four points on a conic section, and P any fifth point. Let a denote the perpendicular from P on AB, /3 the perpendicular from the same point on BG, 7 on CD, 8 on DA. Then by Art. 336 we know that wherever P may be, ay bears a constant ratio to fiS, Now AB . a = twice the area of the triangle PAB = PA. PB. sin APB; , , PA. PB. am APB therefore a = -j-s • Similar values may be found for ;8, y, 8. Thus PA.PB.PC.PD . .TjD • m>n ■-, ^„ ail -4-P-B • sin CPD AB . (JD bears a constant ratio to PA.PB.PC.PD . „p^ .„ „p . ^^-^ smBPC.BmDPA, therefore ^-^ nn/^ '^^"r>p T is constant, that is, the pencil sm BPC . sm DP A drawn from any point P to the four points A, B, C, D, has a constant anharmonic ratio. EXAMPLES. 1. Different elliptic sections of a right cone are taken all perpendicular to one plane which contains the axis of the cone: if the elliptic sections have equal major axes, shew that the locus of the centres is an ellipse. 2. If two spheres be inscribed in a right cone so as to touch the plane of any section, the points of contact of the plane with the spheres will be the foci of the conic section, and the intersections of this plane with the planes of contact of the spheres and the cone will be the directrices of the conic section. EXAMPLES. CHAPTER XVI. 321 3. Find the locus of the foci of all the parabolas which can be cut from a given cone. 4. Shew that a given hyperbola cannot be cut from a given cone unless the vertical angle of the cone is greater than the angle between the asymptotes of the hyperbola. 5. Shew that the latus rectum of any section of a given cone varies as the perpendiculax from thp vertex of the cone on the plane of section. 6. A conic section circumscribes a triangle, and at each angular point the tangent, the two sides of the triangle, and the perpendicular on the opposite side form an harmonic pencil: determine the equation to the conic section. 7. If the equations to, the three diagonals of a quadri- lateral be w = 0, V = 6, to = 0, shew that the equations to the four sides may be put in the form lu + mv + nw = 0, — lu + mv + nio = 0, lu—.mv + mo = 0, lu + mv — nw = 0. 8. In the diagram of Art. 360 suppose a straight line Onl to be drawn meeting the curve at n and I : then shew that the straight lines Nl and Ln intersect on AB. T. c. S. 21 ( 322 ) CHAPTER XVII. PROJECTIONS. 362. In the preceding Chapters we have carried on our investigations chiefly by the aid of co-ordinates; there are various other methods by which we may discover and de- monstrate theorems relating to the Conic Sections : we shall now explain one of these, which is called the method of projections. 363. There are two kinds of projection which may be called respectively orthogonal projection and conical pro- jection: we proceed to consider the former. 364. Definitions. From any point let a perpendicular be drawn on a fixed plane ; the intersection of the perpen- dicular with the plane is called the orthogonal projection of the point on the plane. The plane on which the perpen- dicular is drawn is called the plane of projection. The orthogonal projection of any line straight or curved is the locus of the orthogonal projection of every point in that Hne. 365. We shall use the word projection as equivalent to the term orthogonal projection, until the contrary is specified. 366. The projection of a straight line is in general a straight line. Q R Let PQ be a straight line. From any point P in the straight line draw Pp perpendicular to the plane of projection. PROJECTION OF A STRAIGHT LINE. 323 meeting it at p. Let a plane pass through PQ and Pp, and let its intersection with the plane of projection he pq, and draw Qq in that plane parallel to Pp. Then pq is a straight line, by Euclid, XI. 3: and we shall shew that pq is the projection of PQ. Take any point R in PQ; and in the plane QPp draw 2ir parallel to Pp, meeting pq at r: then Rr is perpendicular to the plane of projection, by Euclid, XI. 8, so that r is the projection of R. If the given straight line be perpendicular to the plane of projection, its projection is a point. 367. The length of the projection of a straight line is equal to the length of the original straight line multiplied by the cosine of the angle between the straight line and its pro- jection. Let PQ be a straight line, ^g its projection; draw Pm parallel to pq meeting Qq at m. Then pq = Pm = PQ cos QPm. And by the angle between PQ and pq is meant the angle between one of these straight lines as PQ, and any straight line Pm parallel to the other. Thus QPm, is the angle be- tween PQ and pq. 368. The projections of parallel straight lines are them- selves parallel straight lines. Let there be two parallel straight hues : denote them by PQ and RS. Let p denote the projection of P, and r the projection of R. The plane QPp is parallel to the plane SRr, by Euclid, XI. 15; the intersections of these planes with the plane of projection are parallel by Euclid, XI. 16; and these inter- sections are the projections of PQ and RS by Art. 366. The angle between PQ and its projection is equal to the angle between RS and its projection by Euclid, xi. 10. 369. Let the boundary of any plane figure be projected; then the area of the prqjected figure is equal to the area of the 21—2 324 AREA OF A PBOJECTION. original figure multiplied by the cosine of the angle hebvoeen the two planes. First, suppose the figure to be a triangle having one side in the plane of projection. Let ABO be the triangle, having the side AB in the c plane of projection. Let c be the projection of C. Draw OD perpendicular to AB, and join cD. Cc is pei-pendicular both to Ac and Dc ; thus AD' = AG'-CD' = Ac' + Cc' - {Do' + Cc') =Ac'-Dc'; therefore the angle ABc is a right angle. Now the area of ABo = „ AB . Be ; and the area of ABC = IaB.BC: therefore area of ABc Be ^_ r=777T= cos CZ)c; area of -4^(7 DG and CBc is the angle of inclination of the planes. Next, suppose the figure to be any triangle. Let ABC be any triangle. Let the plane of ABC meet c a b the plane of projection in the straight line ab. Let 7 denote PROJECTION OF A TANGENT. 325 the angle between the planes. Join aB. Then, by the former case, area of projection oiaCb = area of aCb x cos 7, area of projection of aBh = area of aBb x cos 7; therefore, by subtraction, area of projection of aGB = area oi aCBx cos 7. This shews that the proposition is true for any triangle which has one angular point in the plane of projection. Hence it is also true for the triangle aAB. And therefore, by subtraction, it is true for the triangle ABG. Next, suppose that the area is any plane rectilinear figure. Then the figure may be decomposed into triangles, and as the proposition is true for each triangle, it is true for the whole figure. Lastly, suppose that the area is bounded by curved lines. We may inscribe any rectilinear polygon in this figure, and the proposition will be true of the polygon; and by suflS- ciently increasing the number of sides of the polygon, and diminishing the length of each side, the area of the polygon can be made to differ as little as we please from the area of the figure. Thus we may admit that the proposition is also true for the area bounded by the curved lines. 370. The projection of the tangent at any point of a curve is the tangent at the corresponding point of the projection of the curve. Let P and Q be two points on a curve ; let p and q be their projections. Then the straight line through p and g is the projection of the straight line through P and Q. Let Q move along the curve to F ; then the limiting position of the secant through P and Q is the tangent at P to the curve : and as Q moves to P along the curve, q moves to p along the projection of the curve, and the limiting position of the secant through p and q is the tangent at p to the projection of the curve. 371. The projection of a circle is an ellipse. 326 PROJECTION OF A CIRCLE. Let C be the centre of a circle ; let 5(7B' be that diameter which is perpendicular to the intersection of the plane of the circle and the plane of projection. Let AA' be the diameter at right angles to BB'. Take any point P on the circumference of the circle, and draw Pif perpendicular to AA'. Then CM' + PM'=CP''= CA\ Now, suppose the projections of C, P, M to be denoted by c, p, m respectively. Then cm is parallel and equal to CM, a,nd jpm = PM cosy, where 7 is the angle of inclination of PM to its projection. Thus (^,my+^^CA^ ^ ' cos 7 Let cm = x, pm=i/, CA=r: then cos 7 Now 7 is the same for every ordinate ; see Art. 3G8; and cm and mp are at right angles by the reasoning in the first part of Art. 369. Thus the above equation represents an ellipse having r for the major semi-axis, and r cos 7 for the minor semi-axis. The straight line A GA' is either the line of intersection of the plane of the circle and the plane of projection, or is parallel to this line. In the former case m and M coincide, and 7 is the angle of inclination of the two planes; in the latter case 7 is equal to the angle of inclination of the two planes by Euclid, xi. 10. AREA OF AN- ELLIPSE. 327 It is obvious that the centre of the circle is projected into the centre of the ellipse. 372. Conjugate diameters of an ellipse are the projections of diameters of a circle which are at right angles to each other. For if diameters of a circle are at right angles to each other, each diameter is parallel to the tangent at the extremity of the other diameter. Hence, by Arts. 368 and 370, the projection of each dia- meter is parallel to the tangent to the projection of the circle at the extremity of the other diameter. Therefore, by Art. 191, the projections of diameters of the circle which are at right angles to each other are conjugate diameters of the ellipse. 373. The area bounded by two radii of a circle which are at right angles to each other, and the corresponding arc of the circle, is a quarter of the area of the circle. Therefore, by Arts. 369 and 372, the area bounded by two conjugate semd- diameters of an ellipse and the corresponding arc of the ellipse is one quarter of the area of the ellipse. 374. To find the area of an ellipse. Let a and b be the major and minor semi-axes of the ellipse; therefore, by Art. 371, the ellipse can be obtained by projection from a circle of radius a; and the cosine of the angle between the plane of the circle and the plane of the ellipse is - . The area of the circle is known to be tto'; see Trigonometry. Hence, by Art. 369, the area of the ellipse is - x 7ra*, that is iTcA. 375. It is now easy to see that certain properties may be immediately inferred to belong to the ellipse from the fact that they belong to the circle ; for example, the results of Arts. 182, 184, 185, and 194 may be thus obtained. Such properties are called projective properties. Also Art. 203 may be thus obtained ; for it is obvious by Euclid, III. 31, that if a chord and a diameter of a circle are 3-28 CONICAL PROJECTION. parallel, the supplemental chord is parallel to the diameter of the circle which is at right angles to the first. 376. Again, Art. 208 may he obtained by projection. For by Euclid, iii. 35, if two chords of a circle intersect, the rectangles contained by their segments are equal. Denote the chords by PQp and QOq, so that PO x Up = QO x Oq. Let CD be a radius parallel to OP, and GE a radius par- allel to OQ. Then as CZ>= CF, we have POxOp _ QOx Oq GD^ ~ GE- '' Now when the circle is projected into an ellipse, since VO is parallel to GD, the projection of PO bears to the projection of GD the same ratio as PO bears to GB; and in like manner the projection of Op bears to the projection of GB the same ratio as Op bears to CB. A similar remark applies to the projections of QO and GE, and to the pro- jections of Oq and GE. Hence the property of the ellipse follows from that 6f the circle by projection. 377. It will be instructive for the student to apply the method of projections to the following examples: 20, 38, 42, 50 of the Examples attached to Chapter ix, and 2, 3, 9, 19, 23, '1^, 25, 81, 32, 43, 52 of the Examples attached to Chapter X. 378. We proceed to consider the other kind of projection which is called conical projection, and sometimes perspective projection. 379. Definitions. The conical projection of any point on a given plane is the intersection of the plane by a straight line drawn from a fixed origin through the point. The conical projection of any line, straight or curved, is the locus of the conical projection of every point in that line. Or we may put the definition thus : if every point in a line, straight or curved, be joined with a fixed origin, the assemblage of these joining straight lines will constitute a cone, and the intersection of the cone with any given plane is called the conical projection of the line on the plane. SIMILAB PROJECTIONS. 329 The fixed origin is called the vertex, the given plane is called the plane of projection ; when the projected line lies in a plane, that plane is called the original plane. 380. It is obvious &om our definition that the shadow formed on any plane by a figure when light falls on it from a point, is the conical projection of thfe figure corresponding to the point as vertex. By the single word projection in the remainder of this Chapter is to be understood conical projection. 381. The projection of a straight line is in general a straight line. For the projection of a straight line is the intersection of two planes, namely, the plane of 'projection and the plane passing through the given straight line and the vertex. If the given straight line passes through the vertex, its projection on any plane not passing through the vertex is a point. 382. The projection of the tangent to a curve at any point is the tangent to the projection of the curve at the corresponding point. This may be established in the manner of Art. 370. 383. Projections on parallel planes of the same figure with the same vertex are similar. Let denote the vertex. Let P, ^, ^ be the projections of three points of a figure on any plane; p, q, r the pro- jections of the same points respectively on a parallel plane. Join PQ, QR,pq, qr. Then by Euclid, VI. 4, PQ^OP^^OQ^QR pq Op Oq qr Also the angle PQR = the angle pg-r by Euclid, xi. 10. In this way we can shew when the projections are recti- linear figures that they are similar ; and the proposition may be extended to curvilinear figures in the manner exemplified in Art. 360. 330 PROJECTIVE PROPERTIES. 384. It follows from our definitions that if two plane sections be made of a cone, each curve thus obtained may be considered as the projection of the other. Now it appears from Art. 349, that any conic section may be projected into a circle by a suitable adjustment of the cone and the plane of projection. And thus it will follow that certain pro- perties may be inferred to be true for the conic sections when they have been shewn to be true for the circle. Such properties of a figure as may be inferred to be true for the projection of the figure are called projective properties. It is not possible to give a brief definition of these properties; it will be seen that they relate to the positions of points and straight lines, and not to the magnitudes of lines. 385. For an example of projective properties we may take the theory of poles and polars. Thus the properties of Arts. 101 and 102 being demonstrated for a circle, may be inferred to be true of any conic section by Arts. 382 and 384. Again, Pascal's Theorem and Brianchon's Theorem might be demonstrated for the circle, and then inferred to hold for any conic section. Of course the method would be ad- vantageous only in the case in which it would be easier to demonstrate the property for the circle than for a conic sec- tion generally. 386. But the great advantage of the method of pro- jection arises from the fact that by properly choosing the projecting cone and the plane of projection, we are able to simplify the theorem we wish to establish by substituting some particular case instead of the general enunciation : this we shall now proceed to explain. 387. It is obvious from our definition that every point has its projection unless it lies in a plane through the vertex parallel to the plane of projection ; and then the straight line from the vertex through the point never meets the plane of projection. Hence we may say that -points in the plane through the vertex parallel to the plane of projection have no projections; this is usually expressed, by saying that such points a»-e projected to infinity. PROJECTION OF AN ANGLK 331 The plane through the vertex parallel to the plane of projection may be called the vertex plane. 388. If two straight lines intersect in the vertex plane their projections are parallel straight lines. The projections cannot meet because the point at which the original lines meet is projected to infinity by Art. 387. 389. Conversely, if the projection of a point is at an infinite distance, that point must be in the vertex plane. If the projections of two or more points are at an infinite dis- tance, those points must be in the vertex plane ; if the points are known to be also in another plane, they must be in the intersection of this plane and the vertex plane, so that they must be in a straight line. 390. Ani/ aTigle may he projected into an angle of as- signed magnitude. Let A denote the angular point ; let a plane be drawn parallel to the plane of projection meeting the straight lines which form the angle at B and C. On BG in the plane parallel to the plane of projection describe a segment of a circle containing an angle equal to the given angle. Join A with any point on this segment and take any point on the joining straight line for the vertex. Then the angle BAG will be projected into an angle of the assigned magnitude. 391. In the preceding investigation, the jplane of pro- jection may be any plane which does not coincide with the plane of the angle, and is not parallel to it : we may, there- fore, take the plane of projection such that the corresponding vertex plane shall pass through an assigned straight line, that is, so that an assigned straight line shall be projected to infinity. If we require a second angle to be also projected into an angle of assigned magnitude, we must determine in the manner employed a second segment of a circle ; and when these segments intersect, the point of intersection may be taken as the vertex. 332 pascal's theorem. 392. Any quadrilateral may be projected into a parallelo- gram having a given angle. Draw the third diagonal of the quadrilateral; see Art. 75. Take the plane of projection such that this straight line is projected to infinity : then the projection of the quadrilateral is a parallelogram, for the opposite sides of the projection do not meet. Then apply Art. 390. 393. As an example of the application of projections we will take the following theorem: a quadrilateral is inscribed in a conic section; tangents are drawn at the angular points, thus forming a second quadrilateral: the diagonals of both quadrilaterals intersect at a point. Project the figure so that the inscribed quadrilateral may be a rectangle; see Art. 392. The sides of a rectangle in- scribed in a conic section are parallel to the axes. Hence, by symmetry, the diagonals of this figure and of that formed by the tangents at the angular points will intersect at a point. 394. Any conic section may he prelected into a circle, and a given straight line in the plane of the conic which does not intersect the conic section may he at the same time projected to infinity. This has been shewn in Art. 349. 395. The following demonstration of Pascal's Theorem has been given by the method of projections : Project the conic into a circle, so that the quadrilateral formed by two pairs of opposite sides may become a parallelogram; see Arts. 392 and 394: the theorem then reduces to a simple property of the circle, namely, if a hexagon be inscribed in a circle, and two pairs of opposite sides be parallel, so is the third pair. This simple property may be established by elementary geometry. This demonstration is however not complete. For in Art. 394, there is the limitation that the straight line which is projected to infinity does not intersect the conic section, and thus Pascal's Theorem is only established for such figures as conform to this restriction. Writer.s who use the method of projections are accustomed to asswme that theorems which EXAMPLES. CHAPTER XVII. 333 axe demonstrated under certain limitations will hold when those limitations are removed. For example, a straight line which really does not meet a curve, is called an idecU secant, and treated in effect as if it did meet the curve. But it would he beyond the scope of an elementary work hke the present to discuss the grounds on, which the assumption is based. EXAMPLES. The following Examples may be treated by the method of projections: 1. CP and CD are any two conjugate semi-diameters of a given ellipse ; tangents to the ellipse at P and D meet at T: shew that the triangle CPT is equal to the tri- angle CBT. 2. CP and CD are any coiyugate semi-diameters of a given ellipse ; K is an}' point in PD, and CL is the semi- diameter parallel to PD : shew that the triangle KCL is of constant area. 3. CP and CD are any conjugate semi-diameters of a given ellipse; PQ is a chord drawn parallel to a fixed straight line : shew that DQ will be parallel to a fixed straight line. 4. If from the extremities of the axes of an ellipse any four parallel straight lines be drawn, the points at which they cut the curve will be the extremities of conjugate diameters. 5. CP and CQ are any two semi-diameters of an ellipse; from P a straight line is drawn parallel to the conjugate to CP, meeting CQ at M; from Q a straight line is dra^vn par- allel to the conjugate to CQ, meeting CP at iV: shew that the triangles C'PJi and CQF are equal 6. PQ is any diameter of an ellipse; B, S any two points on the curve ; let PR and QS, or these straight lines 334 EXAMPLES. CHAPTER XVII. produced, meet at M, and FS and QR at T: shew that TM is parallel to the diameter conjugate to PQ. 7. If parallelograms which circumscrihe an ellipse have their areas constantly equal to n times that on the major and minor axes, all the angular points of the parallelograms lie on two ellipses similar to the given one, and having their axes to those of the given ellipse as V (ji' + m) ± Vl^' — n) to unity. 8. If a parallelogram be inscribed in the inner of two similar, concentric, and similarly situated ellipses, and its sides be produced to meet the outer, and the adjacent points of intersection belonging to each pair of parallel lines be joined, shew that the quadrilateral figure formed by producing these joining straight lines will be a parallelogram, having its comers situated on a third ellipse, similar to the two former, and independent of the original parallelogram. ( 335 ) ANSWERS TO THE EXAMPLES. CHAPTER I. 8. The co-ordinates of D are J {x^ + x^ and \ (y, + y^. The co-ordinates of G are -^(a;, + x, + x^) and ^(y, + y,+ y^. 10. Let r and be the polar co-ordinates of C. Then the angle .4 0C= the angle BOG ; at 6- $,= 6,- 6 ; thus e = \{0,+ 0^). Again, from the knowh expriession for the area of a triangle (see Trigonometry, Chapter xvi.),' triangle AOB = ^r\r^ ain {0^~ 6 j), triangle AOG = \r^r sin {6 - fl,), triangle BOG=\r^r sin {6- 6). Thus r^r, sin (&,- «,) = r^r sin {6 - 6,) + r,r sin (fl,- 6) = r(r, + r,)sinj(^,-e,)j therefore r (rj+ rj = 2rir, cos J {6^ — 6^. CHAPTER III. 1. (1) y-l-2a! = l. (2)a!=2. (3) y=a;. (4) »=Q. 2. y-4 = -3(a:-4), y- 4 = ^(o;- 4). 3. 2,-1 = (73 -2) as, y-\ = -UZ + 2)x. 4. 2^ = 2!, y = -x. 5. y = -j^x, a; = 0. 6. 90«, a; = -J, y = |. 7. 60". 8. 45°. 9. y=^(x-a). 10. y = a;. 11. 2^2. J{a' + b') " a + b a' b a b 15. (1) The origin. (2) Two Straight lines, y = x«D.dy = -x. (3) Two straight lines, a: = and a:-(-y = 0. (4) The axes. (5) Im- possible. (6) Two straight lines, x = and y = a, 16. (1) Two 336 ANSWERS TO THE EXAMPLES. CHAPTER III. straight lines, a; = a and y = 6. (2) The point {a, 6). (3) The point (0, a). 17. The straight lines y = a; and y = Sx. 19. 4y = 5«, and 3i/ + 2,x—20 = 0. 20. Let a be the length of the side of the hexagon; the equations are, to AB, y = 0; AC, yJ3 = x; AD,y = x^3; AE,x = 0; AF, y + xJ3 = 0;, ^P,y=J3{x-a); BD, x==a.; BE, y+ J3{x-a).= 0) BF, y.J3 + x-a^0; CD, y + xJ3=2aJ3; CE, yJ3 + x=3a; CF, 2y = aJ3; DE, y = a^3; DF, yJ3-x = 1a; EF, y-x^3 = aJ3. 21. If (a;,, 2/,), (a:., y^), (a;,, y,) bp the angujaj: points, the co-ordinates of £C ^ fl? 1J -^ V» the point midway, between the first and second are ' * -, -^ — ; similarly the co-ordinfites of the point niid,way- between the second and third points are known ; and then the required equation can be found by Art. 35. 22. ^,— ^tano,. 24. - + |=1, - = ? : tansrent of the ansle between them. — ^ — ,,- . 29. The a h' ° ar-V points whose ab^issae are a + rj{f^ + h'). and, a— -rj^a'+h'). 31. y(-g'-4^0_ 35_ 9Q0_ ^ Jf(fl) = gives a system A + G ■ X / o of straight lines through the origin ; sin 3d = gives the three straight lines y = 0, y-xJ3; y = — xJ3. 40. The second pair of straight lines bisect the angles included by the first pair. 44. Let ABC be the triangle ; take A for the origin and straight lines through A parallel to the two given straight lines as axes ; let a;,, y, be the co-ordinates of B, and a;,, y, those of C. T?hen it may be shewn that the equations to the three diagonals are aj.-a;, .'--.'• x, x. from these equations it may be shewn that the three diagonals meet at a point. 45. Take as origin and use polar equations to the given fixed straight lines. 46. Let a;, be the abscissa of the point of intersection of the two straight lines ; then the area of the triangle is J{Cj-c,)x,. 47. This maybe solved by Art. 11. ANSWERS TO THE EXAMPLES. CHAPTEBS IIL IV. 337 Ovwe may use the result of the preceding Example ; for by dra'vr- ing a figure we shall obtaia three triangles to which the preceding Example applies, and the required area is the difference between two of these triangles and the third. The resiilt is \2{m^-m,) 2(m,-jn,) 2(m,-m,))' which may also be written thus ., {"i K- '"a) + g. ('"i- "»3) + C3 ("»,- m,)}' 2 (to,- m,) (m, - m,) (m,- to,) That sign should be taken which gives a positive result. It wUl be seen that the numerator vanishes if the three points of inter- section of the straight lines lie on a straight line ; the denominator vanishes if any two of the straight lines are parallel. CHAPTER IV. Of V iC f/ 1. - + f- = -; + n- 7. Since the required straight line is a b a parallel to that considered in Example 5, we may assume for its equation ocos.4— jScos ^+A = 0, where k is some constara to be determined. Now at the middle point of AB, we have <» = „ sin £, P = ^siaA; therefore ^ sin .B cos ^ - q sin .4 cosB + k=0; thus & is determined. 13. Assume for the equation Xu+n.v+vw=0; then since the straight line passes through the first point, XZ + /t7?n- j^ = 0, and since it passes through the second point, XZ' + /ii«i' + vn' = 0. From these two equations find the ratios of X, /x, v; thus we obtain for the required equation (m»' - m'n) v, + {nV - n'l) v + (Imf- I'm) w = 0. 14. ab{u—v) + c(b + a)w = 0. 15. Assume for the required equation la+m^+ny = 0; at the ceuti-e of the inscribed circle o=j8=yj thus l + m + n=0; at the centre of the circumscribed circle a, )3, y are proportional respec- tively to cos it, cos 5, cosC; thus ?cos^ +nicos£ + ?icosC = 0. Hence the required result may be obtained. T. C. S. 22 338 ANSWERS TO THE EXAMPLES. CHAPTER IV. 18. To GP, 2ntt;-nMJ = 0; \.o BP, 2lu-2mv + nw = ; to AQ, lu — 2mv + 2nw -0; to £Q, lu—2mv = 0. 26. Take a = 0, ^ = 0, y = to represent the sides of the triangle A'B'C; then the equations to BO, CA, AB will be respec- tively p + y=Q, y + a = Q, a + jS = 0. Then the equation to AA' will be j8 — y = 0, so that A A' is perpendicular to BO. 27. The equation to 00' is ;8-y = 0; take j8-y-Xo = for the equation to the straight line drawn through D. Then it will be found that the equation to OF is ^ - y — X (a — y) = 0, and that the equation to O'E is /3 — y — A. (a + ^) = 0. Thus at the point P we have j3 = — y. The same relation holds at the point Q. 2g la + mfi + wy '- V(^'+ "*'+ "■'" ^"*'* cosji — 2nl cosB — 2lm cosC) I' a + m'P + n'y ~ J{1"+ m,"+ n"— 2m'n' cos A - 2n'l' cos B — 2l'm' cos C) ' 30. See Ex. 29 and page 72. 31. Denote the triangle by PQR ; the area is n-n- — 5 where p is the perpendicular from P on QB. The length of this perpendicular is known from. Example 30 ; and sin P, sin Q, and sin It are known from page 70. 32. a\ + bii. + cv=0. 33. ^X + m/t + ray = 0. r ly n/ 34. We must have — r — = — — — identical with X /JL g-d' . aa + bP-{aa'+bp') X ev this gives aX + 6/1 + cv = 0. 35. A point. 36. We shall find that AH le , AF lb ■ ; and AO Ic + na' AB Ib + ma' , triangle AFF Pbe triangle ABO (Ic y na) {lb + ma) ' . , triangle DBF 2ahclmn triangle ABO ~ {lc + na)(U> + ma)(td> + mc) ' ANSWERS TO THE EXAMPLES. CHAPTERS IV. V. VI. 339 38. Divide hj •/; tlma we have a quadratic in -: then as in Art. 60 we obtain {F' - AB) {B'- BC!)= {FB - BE)', that is AD' + BE'+CF'-ABG- iDEF= 0. 39. See Art. 9 and xii. of Art. 78 : thus we get the first form. Also (a, - a,) sin .4 + (;S, - )8,) sin .B + (y, - •/,) sin C = ; transpose the last term and square; thus we express {a^—a^{fi^—p^ in terms of (ai^-a^', (j3,-)3j)', and (y,-7j)', and so obtain the second form. To obtain the third form from the first we put _ fez^^ /(/?. - ft) sin .B + (y, - y.) sin g\ for (a,-a.)^ and make a sin ^ ( / similar substitution for (|8i— ft)'. CHAPTER V. 1. (^ + yy = a' {a^ -■,/). 3. y'" = cx' 2-J. 4, 2/" sin' o = ioKc'. 6. By Art. 83, we have sin(o>-o) sin(iij-jS) , sin o , sin^ TO = ? -, n- > !-i, m=-. — , n =-. . sin 10 sm <■> sm o> sm lu CHAPTER VI. 1. (1) Co-ordinates of the centre 2 and - 2, radius 3. (2) Co-ordinates of the centre - 3 and |, radius ^. 2. The first straight lino meets the circle at the points (-4, 3) and (3, -4); the second at the points (0, -5) and (-5, 0); the third touches it at the point (-4, - 3). 3. a? + y'=hx + ky. 5. a?-x(a^ +a;')+^-y{y' +y")+«:') = h'k'. 22—2 340 ANSWERS TO THE EXAMPLES. CHAPTER VI. 9. 2y+3a: = 0. 14. a^ + y* -xy-hx-lcy = Q. 15. Inclination of axes 120° ; co-ordinates of the centre each =/t ; radius = ^ 16. Inclination of axes 60° j co-ordinates of the centre each = ^ ; radius =-^2. 17. sd'+i/'+xy J2-9^Q. ,„ . . -, ,. -.r. J(h' +le'-2hk cos m) " " " 2 sin CD 23. !c'+y' = a(x+-^; r=-^cos(fl-^V 27. A circle. 28. Use the equation in Example 26. 29. Using polar co-ordinates, we have r + ^{r' + l'-2rlcose)=/ir' + l'-2rlcos(^-eyf , irhere I is the length of a side. Keduce and we get |V3r-2Zcos(e-^)}'=0; thus the locus is the circle circumscribing the triangle. 30. sin' a + sin'/S -I- sin'y -I- ... = cos'a-i-cos'/S-i- cos'y-1- ... and sin2a-^ sin2/3 -I- sin 2-y-)- ... =0. 32. If the perpendiculars are both on the same side of the straight line the locus is a circle ; if on different sides the locus consists of two straight lines. 33. A circle. 34. A circle. 36. Solve the quadratic in r ; it will be found that r=2a cos 6 or — a sec 6 ; thus the locus consists of a straight line and a circle. 38. Take the extremity of the diameter as the pole ; it will follow from Example 37, that the tangent at F is represented by the equation 2c cos' a = r cos (2a — ff), and the tangent at Q by the equation 2ccos')8 = r cos (2j3-6). These tangents meet at T, so .., , ., , cos (2a- e) cos(2j3-6>) , ^,. , ,, at that point we have ^-5 '- = ' ;„ — - : from thiswe shall cos' a cos'jS find tan 6 — ■= ^ , so that if C be the centre of the circle 2 cos p cos a _ c sin (B + = -j — t; • A — U a—c 342 ANSWERS TO THE EXAMPLES. CHAPTEK VIII. CHAPTER Vm. 1. y = 2x. 2. ^ = bax — a?. 3. The locus consists of two parabolas of which the centre of the circle is the common focus, and the directrices are the two tangents to the circle which are parallel to the fixed diameter. 4. The second curve is a parabola having its axis coinciding with the negative part of the axis of y ; the curves intersect at the origin and at the point a: = 4o, y = -4a. 5. y = x + a. 6. tan"' J. 7. y+x=ia. 8. At the point (9a, - 6a); length 8a J2. 9. y = 2a ^3, x = Za. 11. The abscissa of the required point is or 3a. 13. The curve is a parabola having its axis parallel to that of y, and its vertex at the point a; = J, y = \. The straight line is a tan- gent at the point x=\, y = Q. 20. Abscissa of required point is — ( —p- +y'), ordinate - ( — 7- + y'y, length of chord -^ (4a' + y}*. 22. Locus of Q,x = — 2a. Locus of Q', a;' = ay'. 23. Eefer the parabola to FT and the diameter at P as axes. See Art. 161. 25. See Art. 155. 27. Transform equation (1) of Art. 125 to polar co-ordinates, and we shall deduce r = 2a M^ . ^ sin" 28. Use the result of the preceding Example. 29. .= 2a^HL^:±4^H2i^>. 30. The locus is a cos'tf parabola; see Art. 147. 32. Jx+ Jy = J{a2 J2). 33. (]/-x'Y-8ax'J2 = 0. 34. se' + y'-x{a + x')-yy' + ax'=0. 37. Use the result of Example 5, Chap. vi. 41. The equation to one tangent can be written y = m,{x + a)-i — , (see Example 40), fli and that to the other y = (x + a") — a'm. By eliminating m we have for the required locus a; -I- a -1- o' = 0. This supposes the parabolas both to extend along the positive direction of the axis of a; ; if the second parabola extends along the negative direction the final result will be a; -t- a - o' = 0. 42. Take for the equation to the chord y = mx + n ; then to find the abscissa of the middle point of the chord we must take half the sum of the roots of the ANSWERS TO THE EXAMPLES. CHAPTER Till. 343 equation (mx + n)'= iax : so that the abscissa is — ","'" ' . Now m since the chord touches the parabola ^ = 8a(x-c) the equation (ma; + n)' = 8a{x- c) must have equal roots ; by means of this condition it can be she-vm that 5 — = c. 44. The equa^ tion to the normal at a point (x', y") is y-y' = — -^{x—x'). If the normal is to pass through a given point (A, k) -we have V v" ^-3/ = — -^{h-x'); also x' = j-. Thus we obtain a cubic equation for determining y', namely y"+ ia{2a — h)y' — 8a'k = 0. By Chapter ill. of the I'heory of Equations the sum. of the roots of this cubic equation is zero. The points of intersection of the parabola with a circle {x — h)' + {t/ — cf = 1* are found by combining the equations to the two curves. Thus we obtain (£-')"*<»-"''-'■• (-=) which is an equation of the fourth degree in y. By the Theory of Equations the sum of the roots is zero. If then three of the roots coincide with those already shewn to have a sum equal to zero, the fourth root is zero; and the corresponding point is there- fore the vertex of the parabola. 45. The tangents of the inclinations to the axis of x of the three normals that can be drawn through a point (x, y) are determined by the equation + - = 0. See Art. 135. Suppose to,, m,, m^ the roots of this cubic, then by Chapter iii. of the Theory of EqvMions «i, + »H,+ 7»3=0, mm^+mmi + m^m =2 , m^m^m^ = — ; Or a if two of the normals are at right angles we may put tn^m^ = — 1 i from these equations by eliminating m,, m^, and m^, we find y = (a;- 3a). 46. By the length is meant the length of the common chord ; by the breadth is meant the distance between the two tangents which are parallel to the common chord. ■^ ' • "7715 — r^ • 55. The equation y = mx + — repre- 344 ANSWERS TO THE EXAMPLES. CHAPTEES VIII. IX. sents a tangent to the parabola ; if this passes through the point (A, k) we have k = 7nh+ — ; also m = - — r , ■where {x, y) is any point on the tangent; thus Jc — - — 7 A = — ^ — j- ; this will give the first form of the equation. The second form may be deduced from the first; the student will see hereafter what suggested the second form; see Arts. 341 and 343. 56. The equation y* — iax represents the parabola; and the equation h/— 2ax=2ah represents the chord of contact; hence it follows that the equa- tion 4ax {ky — 2ax) = Idha^ represents some locus passing through the intersection of the parabola and chord; then see Art. 61. 57. x = , y = a( — + — ). If the equation to the third tangent is y = m^ H the required ordinate is /111 1 \ a I — + — + — + ). CHAPTER IX. 1. -T5. 3. y-\-ex—a; the intercept on the axis of Ot , , X a; = - ; and the intercept on the axis of y = a. 3. w + oe' = — . e e 4. The excentricity is determined by e*+ e° = 1. 5. y = -(a? + a); 1/ = — ,- ; the straight lines are parallel if 2e'= 1 . 6. y= — (« — oe); 2ae the abspissa of the point of intersection is = . 1 + e^ "■ y = -(l+e)(x-a): tan '^j -,. 8. — 7, gr- ^ _ / . -9 la 9. The co-ordinates of the point are x = ,. - ,„. , y = ,, ., — r^. 10. The co-ordinates of the point are x=-^, 1/ = —^. ANSWERS TO THE EXAMPLES. CHAPTER IX. 343 19. It will be found that the circle falls entirely without the ellipse if the inclination of the two parallel straight lines to the major axis be greater than tan"' -r-. 22. - cos <^+ rSin^= 1. 25. The co-ordinates of the required point are x=—r^ ~ , y=— r^ j-i ; the straight lines are parallel when e* + e' = 1. 28. x'+y'-x(ae + a:^-yy'+aex'=0. 30. If the point (7t, A) be between the directrices, the sum of the perpendiculars is . . ^.j — jjjj. ; if the point (h, k) be not between the directrices, the sum of the per- pendiculars is ± . ^„ — Tjj5^ , the upper or lower sign being taken according as A is positive or negative. 31. A circle having its centre at the centre of the ellipse and radius = a + b. 32. y = ±a;±^(a'+6'). See Art. 171. 34. Locus is the circle 3f+ i/'=a'+b' ; this maybe deduced from the second part of Example 33. 35. See remark on Example 55 of Chap. viii. 42. The first part of this Example may be solved by finding the equation to the straight line passing through the points of inter- section of the two ellipses. 45. a^+i^={a'+b')' (x + y). 46. Let h, k be the co-ordinates of an external point ; the equa- tion to the corresponding chord of contact is a?ky + l?hx = a%'; the equation to the straight line through {h, k) perpendicular to the chord is (y — k) b'h = a'k {x — h). We require that the latter straight line shall be a tangent to the ellipse ; the necessary condi- tion may be found by comparing this equation with the equation y = mx+ J{m'a^+b'); thus we shall obtain for the condition l^a'+ h'b" = A'A" (a'- bf. 48. a' {y'+ 2yk) + b' {x'+ 2xh) = 0. 51. Transferring the origin to the vertex of the ellipse the equar tion becomes y = m,{x — a) + ^{m'a'+ b') = Tnx - ma + two ( 1 + — ^ j : mx— ma + ma \\ +- — 5-^i , where c=(l-e)o. (. ma ) ^ ' 346 iNSWEES TO THE EXAMPLES. CHAPTERS IX. X. Expand the square root by the Binomial Theorem ; then ulti- mately ■when e = 1 and a is infinite, we have y = mx + — . 52. An ellipse. 53. The locus is an ellipse ; i{ A he the origin, A£ the axis of x, each of the co-ordinates of the focus is equal to half the radius of the circle. 54. -j^ . 55. Put a cos for x and 6 sin ^ for y in the preceding result (Art. 168) ; e'a then the greatest value is ot- • 57. Let P denote a point on the ellipse, and Q the centre of the circle inscribed in the triangle SPH ; then if y* be the ordinate of P it may be shewn that the ,. , ,, . , ,. , area of triangle iS!F2?' ey' ,. . radius of the circle -which = -. ; -, — . . . =- = ir^— ; this semipenmeter of tnangle 1+e is the ordinate of Q. Let x' be the abscissa of P, then it may be shewn that the abscissa of Q is ex' ; thus it will be found that the required locus is an ellipse. 58. Find the point at which S2 meets the normal at P ; also find the point at which SZ' meets the normal at P; it will then appear that the points coin- cide. 60. See Example 12 of Chapter vii. CHAPTER X. 1. xb (bx' - ay") + ya {ay" + bx") = a'i'. 2. Befer the ellipse to the diameter and its conjugate as axes. 3. See Art. 11. 8. r (as' sin' + ¥ cos' 0) = 2ab' cos 6. 9 and 1 0. TJse the re- sult of 8. 12. Kesult the same as that in Example 11. 13. They intersect -when fl = and when ^ = „ . 14. The equations to the tangents at the ends of the latera recta are (Art. 205) r (e cos 6 + sin ^) = a (1 — e*) ; r (sin 6 — e cob 6) = a (1 + ^; r (e cos 6 - sin 6) = o(l-e'); r(a,ia.6 + ecoa6) = -a{l + e'). The eqiiations to the tangents at the ends of the minor axis are rain6=b; rsin6 = -b. 15. A straight line through S. See Art. 205. 17. cos 6 = - :^i^ , r = o (1 ^- ee'). 18. Be- 1 + ee ^ ' ANSWERS TO THE EXAMPLES. CHAPTER X. 347 twccn - and^. 20. See Art. 208. 22. The sine of the a angle between the radius vector from the centre and the tangent P IS - , where p' {a' + b' — r') = aV by Art. 196; then the least value P' of ^ may be shewn to be when 2r' = a' + 6'. 29. It may be shewn that the axis of the parabola must coincide with one of the axes of the ellipse, hence the latus rectum wiU be either 2a' 26* -77-8 — iiv or -77-i — Us.- 31. An ellipse. 32. An ellipse. J{a'-i-b') J{a'+b') ^ ^ 35. Use the polar equations to PQ and pq ; see Art. 205. 38. Two of the sides of the parallelogram are determined by the equations - cos ^ + r sin i = ± 1, and the other two by the a b equations - cos i^'+ ^ sin <^'= ± 1 ; see Example 22 of Chap. ix. It may bo shewn that the diagomils of the parallelogram inter- sect at the centre of the ellipse ; then if the centre of the ellipse be joined with two adjacent corners of the parallelogram the triangle thus formed is one fourth of the parallelogram; and the area of the triangle is known by Example 7 of Chap. I. . bx'—ai/ , ,, ,. , m/+bx' .„ „, 41. The abscissa is 5—^, and the ordinate -^ . 42. The co-ordinates of the intersection of the tangents are found in Example 41 ; call them h and k, then use the second form given in Example 35 of Chap. ix. 44. The greatest value may be found by substituting for x and y' their values from Art. 168 ; it is o6(,y2-l). 48. An ellipse referred to its equal conjugate diameters. 51. This may be solved by means of Example 50. Or we may take the usual axes, then if x', y' be the . „ , ..■.,•„, a(aas'+bi/) , blax'+bu') co-ordinates of P those of Jf will be , . / ' and ' - „ - ; Qt -r fl + those of iVr will be ^^^^^ ^d ^-^^ ■ Hence the solu- tion can be completed. 52. See Art. 208. 348 ANSWERS TO THE EXAMPLES. CHAPTERS XI. XII. XIII. CHAPTER XI. 1. y'-3«'=-3a'. 2. A straight line. 7. See Arts. 178 and 228. CHAPTER XII. 4. Let a straight line be drawn through the focus meeting the hyperbola at P and p and the asymptotes at Q and q ; then it may ., , _ 2a(e'— 1) 2asin'a _ 2aBinasin0 be shewn that Pp = = — \ — 5^ = — = 53, Qq= — = y^ , ^ 1-e'cos'tf cos'a-oos'd' ^ cos' a- cos" fl ' and the required length is half the difference of Pp and Qq. 5. Take the centre of the circle as the origin, AB as the axis of X, and a diameter parallel to PQ as the axis of y ; then the locus ia given by the equation ^ =a? — a', and is therefore a rectangular hyperbola referred to conjugate diameters. 11. By Example 53 of Chapter VIII. we shall obtain tan a = ^'-^^ ^ ; thus (A + o)' tan' o = A"— 4aA; therefore (A + a)' sec' a = ^+ (A — a)'. 12. Both the diameters must meet the curve; it will be found that this requires the conjugate axis to be greater than the trans- verse axis. CHAPTER XIII. 1. The equation may be written (a; - 2y) (x — iy— 2a) = 0, and therefore represents two parallel straight lines; a straight line par- allel to them, and midway between them, will be a line of centres. h c 2. h = -^, k = -=. 3. Two parallel straight lines. 4. A parabola. 5. An hyperbola if the angle A is less than jr , an ellipse if it is greater than ^, a straight line if it is equal to ^ . 6. The equation to the hyperbola is a'y'=a'6'-4a6'a; + 36'a'; the asymptotes are determined by the equations oy = ± ( a: — =■ )"6 ^3. ANSWERS TO THE EXAMPLES. CHAPTER XIII. 349 8. The locus is then a straight line which coincides with the equal axes. 10. Use Art. 205. 11. ?^. 13. Tan"' - 4 6 U. {ay + X ^{mV- 2a/3/3'a! - a'(j8 + /3') y + aW = 0. 17. (1) A circle about the other focus of the given ellipse as centre ; (2) an ellipse about the other focus of the given ellipse as focus, and having the same excentricity as the given ellipse. 18. The equation is (y- Sx + l) (y-2a; + 4) = 0, and therefore represents two straight Unes. 24. Use the result given in Example 56 of Chap. viii. 26. The equation may be written ^af.+ y'+xi/J2-a'){!>f+ y'- xy J2 - o') = 0. 27. Take AB and AG as axes of x and y. Let the angle PBA and the angle PGA be each equal to a, and the angle £AC=u>. Let X and y be the co-ordinates of P; then j,^_ ysino) p^^ a:sina> sin a ' sin a And £C'= £P'+ CP'- 2BP. GP cos BPG. 28. Take the given point as the origin, the common tangent at that point as the axis of y, and the diameter through that point as the axis of X. Then the equation to the parabola will be of the form y'=4cx, and the equation to the other tangent y=mx + — , where m is con- stant for all the parabolas. Whatever be the value of c the point of contact is on the locus i^= 4:xm(y —mx), which is obtained by eliminating c ; that is on the locus (y - 2nui:y = 0. 29. We b' may take for the equation to the ellipse y'=-,(2ax-x'). Let o (x', y') be a point on it ; then the equation to one of the straight lines is y + 2b= ," x ; put y = 0, then x = — — ^ : this gives the length of one segment. The length of the segment at the other end of the major axis will be " 7 ~, l and therefore y'+ 26 the length of the third segment "^ . 30. Take one of 350 ANSWERS TO THE EXAMPLES. CHAPTERS XIII. XIV. the ellipses, and refer it to its equal conjugate diameters as axes, the axis of x being that which passes through the fixed point. Let G be the centre of the ellipse, P the point of contact of the tangent from the fixed point, FM the ordinate of F: then it may be shewn that Mis a, fixed point and MF a constant length. Tip P7?' and 57) = tto > 1^7 -Art. 158. 32. Refer the ellipse to rect- angular axes with F as origin and FX as the axis of x. The equation will be of the form aa^+ hxy + cy'+dx + ei/ = 0. Assume y=7nx + n for the equation to the straight line QB ; then the following equation will represent the two straight Unes FQ and FB, n In order that these two straight lines may be equally inclined to FK we must have b + = 0, so that n = — = — . Thus the n equation to QR becomes y = mx H j- — , and this is satisfied by x = — T and y = — —, whatever m may be. CHAPTER XIV. 2. Each locus is an ellipse. 4, 5, 6. Use tho. equation in Art. 294. 7. The equation to the ellipse is — + fi = 1 ; the equation to the chord of contact is — ^ + ^ = 1 : hence the a? y* xh yh equation -> + n = -5 + -ri represents some locus passing through the points of contact. 10. The equation to the hyperbola is{y-k) h'x = (sc - A) a'y. 12. Let y', y" denote the two ordi- nates which correspond to the same abscissa a;' ; then y'=-6a!'+ Jib'a^-ax"-/), y"=-bx'-J{b'x"-ax^'-/). The equations to the normals are, by Art. 284, ANSWERS TO THE EXAMPLES. CHAPTEB XIV. 351 {y -y')(ax'+by^ = {y'+ bx') {x - nf), and (y - y") {ax- + hy") = {f + bx') (x - x') ; by addition (a-6')a;'{y + 26x') + 6/=0...(l); by subtraction 6 (y + hx") - (o - 6') a/ = a; - x', therefore x' {i + lh'-a^^x-hy (2). Substitute the value of x' from (2) in (1) and the required equa- tion will be obtained. The locus is an hyperbola. 13. Locus a conic sectioo, which passes through H and E, and through the intersection of the fixed straight lines. 18. A circle haviug its centre on the straight line joining the two points. 19. Two loci, an ellipse, and a parabola. 20. A circle. 23. See Art. 293. 26. TJse the equation to the parabola given in Art. 294, and the equation to the circle given in Example 21 to Chap. TL 29. rsin2d = c. 30. x^y^ + y^x^=a'. 32. See Example 30 to Chap. z. 35. An ellipse. 37. In the first case the locus is a circle ; in the second case it is a straight line. 38. A circle having its centre at ^. 41. The equation to the parabola may be written y* = 4a (a; — o) + 4a' j the equation to the chord of contact is hy= 2a(as — o) + 1a{a + h); therefore the following equa- tion, represents some locus passing through the intersection of the parabola and the chord of contact, Ay — 2a (a: — o) (ky — 2a (a; — a)"! 44. , , ,«v — 2a{a: — o) f«v — 2a(a; — a))" y'=4a(a!-a) " „ . ^ rx +\— i -\ • " ^ ' 2a{a + h) (. a + h ) —^ +^ = \. 46. The equation is y' = ia{x - 8a). 60. The straight line — f = bisects the chord of contact, and is therefore parallel to the axis of the parabola ; if through the point (a, 0) a straight line be drawn making the same angle with the tangent at that point as the axis makes, the focus must be in this straight line : y{a+2h cos (sin tft — cos <^), hy = (6'— a') sin ^ cos <\> (sin ^ + cos ^). Similarly we can determine the co-ordinates of the point of inter- section of the normals at F and D ; denote this point by i?. Then express the area of the triangle CQli, which is one-fourth of the required area. 55. Take the centre of the square as the origin, and the axes parallel to the sides of the square. Then for the equation to the circle take ar'+y' = 20*, and for the equation to the conic take y'— a'= X (sc*— a"). The equation to the tangent to the circle at the point (a;,, y,) is a!a;,+ yy,= 2a". The equation to the tangent to the conic at the point (a^, y") is yy'-Aaa;'=a'(l-X). These equations must represent the same straight line. Hence elimi- nating. X and x^ and y, we shaU arrive at an equation which deter- ANSWERS TO THE EXAMPLES. CHAPTERS XIV. XV. 353 mines the required locua. It will he found that this equation may be -written {(a/'+y"- 2a')}{a'(x"+}/") - 2af'y"} = 0. 56. This follows from Art. 288. 57. Let a perpendicular be drawn from M on the tangent TQ, and let Ji denote the intersection of this perpendicular with SQ produced. Then SR = SQ+QR = 2a; and TR^TH. We have to find the value of the perpendicular from T on ,S'^; denote it by r; then r2a = twice the area of the triangle TSR. Let 2-+ JP+ Jy=0. 10. The equation to the conic sec- tion being iPy + mya. + na^ = 0, that to A'B is {m + n)a + ly = 0, that to A'G is {m + n)a + ip=0, and that to A'B" is (?» + »)o-i-(Z-i-w)/3— ny = 0. 13. Imn+l^Q, T /I o 23 354 ANSWEBS TO THE EXAMPLES. CHAPTER XT. a /(l+m,«) V(l + ,/(!+«».•) V(l + ■" ' or (1 + m,') ^y- «i.a;-^j ^y-m^ —^J K-m.) + ... = 0. 24. Suppose the focus iST is to lie on the straight line la + mP + ny = 0. "Let a, j8', y' denote the values of a, /3, y re- spectively for the other focus H of one of the ellipses. Then, by Art. 181, aa'=j3/3'=yy'=the square of half the minor axis. Hence, substituting in the given equation we obtain — + oj + -y = 0, that a. p y is, ip'y + my'a' + na'^ = 0. This shews that the locus of i? is a conic section passing through the angular points of the triangle. 25. It will be found that the conic sections may be repre- sented by the equations (1) ^y_a«=0, (2) ya-y8' = 0, (3) afi-y' = 0. Now, (1) may be written ^3 (y -H ^S - 2a) - (o - /S)' = 0, (2) may be written y (a -h y - 2^) - ()3 - y)' = 0, (3) may be written a (fi+ a-2y)~(y-a)' = 0} this shews that the tangents to the conic sections at the common point are given by y + )3-2a = 0, o-^y-2y3»0, fi+a-2y = 0; these three straight lines intersect respectively the straight lines a = 0, )3 = 0, y = 0, at three points which all lie on the straight line a + P + y=0. Again, (1) may be written /S(y-f4a-i-4/8)-(o-i-2)3)'=0, and (2) may be written a(y + ia + 4/3) — (fi+ 2o)'= ; and this shews that y + 4a + 4j3 = is a common tangent of (1) and (2), and this com- mon tangent meets y=0 at the point where /3-f a — 2y=0 meets it. And so on. 26. The equation to the first hyperbola is py=AA'*sai?-s ', similarly for the others. 27. See Art 274. 28 and 29. These may be solved by taking oblique axes coin- ciding with the sides of the triangle. Por instance, consider 29. ANSWERS TO THE EXAMPLES. CHAPTER XV. 355 "We have oa + 6j8 + cy==-a6sinC^. Thus the equation may be written cnafi - (Ifi + ma) (ab sin C + aa + bp) = ; and taking CA for the axis of x, and CB for the axis of y, -we have a = a; sin (7, P=ysaaG. Substitute for a and y3 and then to the equation in X and y we may apply the ordinary test ; see Arts. 272 and 278. '=°« 2 A 30. S= J {aa + bfi + cy)', where S denotes a' cos* -j; + . . . : see Art. 334. 31. u {mn'— rn'ri) = v {nV — n'l) = w (Im'— I'm). 32. Let iS, = be the equation to the inscribed circle, S^= the equation to the circumscribed circle, these equations not being necessarily in their simplest forms ; see Art. 110. Then, if A be a suitable constant, S^ — kS^ = will represent the straight line required. In this way we shall have a" cos* Tj- + yS' cos* ^ + y' cos* -^ - 2^y cos' -= cos' ^ — 2-ya cos' jr- cos' -^ — 2a/J COs' -^ COs' -^ Z A Z Z — k (fiysiaA + yasiaB + a^ainC) = (oa + 6jS + cy) (Za + m^ + ny), where I, m, n, are to be found. Then by comparing like terms we can find I, m, n. 33. It may be shewn that the equation — = -^-7 — represents a diameter ; for this equation represents a straight line passing through the intersection of the tangents at A and £, and through the middle point of AB, Hence the centre of the conic ... . , , nB + my ly+na ma + lB , ., section 13 determined by — — ->—= — = —: and then •'a b c ' the required equation can be found. It is P _ y . m(al—bm + , w = denote EF : then ■we may assume hi + mo = as the equation to FG, and lu + mv + nw = as the equation to FA. Then by Arts. 358 and 359, the equation to FB is lu + mv—nu) = Q. It may now be 358 ANSWERS TO THE EXAMPLES. CHAPTERS XVI. XVn. slie^ni that la — mv+nw = denotes EC, and that mv+nw—lu=0 denotes EB. 8. Join Nl cutting AS at ff ; join On and GL. Let 01 cut AB at m. Then GO, GN, GM, GL form an harmonic pencil ; and so also do GO, Gn, Gm, Gl : see Art. 360. Therefore, by the aid of Art. 354, it follows that Zffw is a straight line. Or the Example may be deduced immediately from Art. 292. CHAPTER XVII. 2. It is easily seen that the triangle KCL is the projection of a triangle of constant area in a circle. Since the area of a triangle is half the product of the base into the perpendicular from the vertex on the base, the result may be put in this form: the length of the perpendicular from G on PD varies inversely as the semidiameter parallel to PD. 8. This is to be considered in the first place ■with respect to coricentric circles and rectangles. Let G denote the centre of the circles, L a comer of the in- scribed rectangle, so that 2^ is on the circumference of the inner circle. Let r be the radius of this circle, and R the radius of the outer circle ; let x and y be the co-ordinates of L. Draw through L a straight line parallel to the axis of x meeting the outer circumference at M, and a straight line parallel to the axis of y meeting the outer circumference at K. Complete the rectangle of which LM and LN" are adjacent sides ; and let P denote the other comer of this rectangle. Then the abscissa of J/ is J{E'-j^), and the ordinate of If is J (I? -of); and these are the co-ordinates of P. Thus CP'= R'-a?+ E"-^ ='2R'-i'; so that the locus of i* is a concentric circle, the ladiua of which is independent of the original rectangle. THE END. CAUBBISOE: FBIIITKD by C. J. CLIT, U.A. tX THE nNinSBSZIT FBZBB. WOEKS BY L TODHUNTER, MX, P.R.S. Natural Philosophy for Beginners. In two parts. Fart I. The Propertieii of Solid and Fluid Bodies. With numeroui Examples. iSmo. 3t.6d. PartIL Sound, Light, and Heat. y.6d. Euclid for Colleges and Schools. 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