coRNea UNIVERSITY LIBRARIES Mathemalii£» LSbfwy CORNELL UNIVERSITY LIBRARY 3 1924 059 322 820 P^ Cornell University WB Library The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924059322820 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. CORNELL UNIVERSITY LIBRARY THIS BOOK IS THE GIFT OF Richard L. Sv/enson Interlaken, M.Y, MATHEMAUCS LiBRARY BRIEF COURSE IN ANALYTIC GEOMETRY BY J. H. TANNER PROFESSOR OF MATHEMATICS IN CORNELL UNIVERSITV AND JOSEPH ALLEN ASSISTANT PROFESSOR OF MATHEMATICS IN THE COLLEGE OF THE CITY OF NEW YORK NEW YORK •:• CINCINNATI •:■ CHICAGO AMERICAN BOOK COMPANY The Modern Mathematical Series, lucien augustus wait. iSenior Professor of Mathematics in Cornell University,) GENERAL EDITOR. This series includes the following works: BRIEF ANALYTIC GEOMETRY. By J. H. Tanner and Joseph Allen. ELEMENTARY ANALYTIC GEOMETRY. By J. H. Ta.nnkr and Joseph Allen. DIFFERENTIAL CALCUL0S. By James McMahon and Virgil Snyder. INTEGRAL CALCULUS. By D. A. Murray. DIFFERENTIAL AND INTEGRAL CALCULUS. By Viroil Sntder and J. I Hdtchinson. HIGH SCHOOL ALGEBRA. By J. H. Tanner. ELEMENTARY ALGEBRA. By J. H. Tanner. ELEMENTARY GEOMETRY. By James McMahon. Copyright, 1911, by J. II. TANNEU A.Nli .lOSEPH ALLEN. Ehtebbd at Stationerb' Hall, Lonuun. brief anal. 3eoh. W. P, 5 . PREFACE In their " Elementary Course in Analytic Geometry " the authors of this book endeavored to prepare a text suitable for the general student, as well as for the student who desires a definite scientific training, including preparation for advanced mathematical study. The success of the "Elementary Course" has been demonstrated by its adoption and continued use in many of the leading colleges and scientific schools. Since the publication of that text, however, there has been a growing desire for a somewhat briefer book preserving the same rigor of proofs and careful analysis, but omitting or shortening the less important details. The " Brief Course in Analytic Geometry " is prepared to meet this demand. The chief features of the " Elementary Course " which have won such universal approval have been included. These are : (1) An extended introduction to the method of the analytic geometry, — showing the value of the interpretation of the negative number by means of coordinates, the relation of the locus to the equation, and the use of the equation to find and demonstrate properties of the locus. (2) Introduction of the demonstration of general theorems by numerical examples, in order to make clear the method used before bringing in the added difficulties of literal notation and general proofs. (3) The use of some intrinsic properties of curves, to empha- size further the fact that the coordinates are a tool, an inter- mediate device, not essential to the properties which are studied by their aid. IV PREFACE (4) Rigorous proof of all the theorems within the scope of the book. (5) An abundance of carefully graded numerical exercises. In preparing the " Brief Course " the senior author partici- pated actively in the planning of the scope and genei'al arrange- ment of the work, but the junior author is responsible for the working out of most of the details. Cordial acknowledgment is due to Dr; A. B. Turner, College of the City of New York, who read both the manuscript and the proofs, and to Mr. G. M. Green, who prepared the answers. CONTENTS PART L PLANE ANALYTIC GEOMETRY Chapter I. — Introduction Algebraic and Trigonometric Conceptions LXTtOLE 1. Number 2. Constants and variables 3. Functions 4. Identity, equation, and root 5. Notation 6. The quadratic equation. Its solution 7. Zero and infinite roots 8. Properties of the quadratic equation 9. The quadratic equation involving two unknowns 10. Directed lines. Angles 11. Trigonometric ratios 12. Other important formulas . . 13. Orthogonal projection Chapter II. — Geometric Conceptions. The Point 1. coordinate systems 14. Coordinates of a point 16. Positive and negative coordinates 16. Cartesian coordinates of points in a plane 17. Rectangular coordinates ........ 18. Notation II. ELEMENTAKT APPLICATIONS 19. Distance between two points 20. Direction from one point to a second 21. Parallel lines. Perpendicular lines 22. The area of a triangle 23. The analytic method 24. To find the coordinates of a point which divides in a given ratio the straight line from one given point to another . 26. Analytic geometry Chapter III. — The Loons and the Equation 26. The locus of an equation 27. Illustrative examples : rectangular coordinates .... 28. The locus of an equation 29. Classification of loci .30. Construction of loci. Discussion of equations .... 31. The equation of a locus V VI CONTKNTS 32. !:!. 34. 35. 3H. 39. Kquation of a locus with known properties K(iuation of a locus traced by a moving point Kijuation of a circle : second method . General theorems I'oints of intersection of two loci Product of two or more equations Locus represented by the sum of two equations Fundamental problems of analytic geometry PAOF 43 ■1-1 4(! 47 48 50 51 53 Chapter IV. The Straight Line. Equation of First Degree Ax+By+C = 40. Introductory .... 59 41. Equation of straight line through two given points . . 69 42. Equation of straight line in terms of the intercepts ... 60 43. Equation of a straight line through a given point and in a given direction .... . . ... 63 44. Equation of a straight line in terms of tlie perpendicular from the origin upon it and the angle wliich that perpendicular makes with the x-axis . . 64 45. Summary 66 46. Keduction of the general equation Ax + By + C = to the standard forms. Determination of a, b, m, p, and a in terms of A, B, and C 67 47. To trace the locus of an equation of first degree ... 69 48. To find the angle made by one straight line with another . . 72 49. Condition that two lines are parallel or perpendicular . 73 50. Line which makes a given angle with a given line ... 76 51. The distance of a given point from a given line .... 81 52. Bisectors of the angles between two given lines ... 84 63. The equation of two lines 86 64. 55. 66. 57. Chapter V. — The Circle Special Equation of the Second Degree Ax^ + Ay'^ -\-2 Gx + 2 Fy -\- C = Introductory The circle : its definition, and equation In rectangular coordinates every equation of the form Ax^ + Ay'^ + 2 Ox + 2 Fy + C=0 represents a circle Equation of a circle through three given points . . 92 . 92 . 94 . 94 . 96 Tangents : illustrative examples 97 Equation of tangent to the circle x^ + y- = r' in terms of its slope 99 Equation of tangent to the circle in terms of the coordinates of the point of contact : the secant method . . 101 Equation of a normal to a given circle . ... 103 Lengths of tangents and normals. Subtangents and subnormals 105 Tangent and normal lengths, subtangent and subnormal, for the circle . . 100 Circles tlix'ough tlie intersections nfJ._wri_ir)vpn_pivpio= iat CONTENTS ABTICLB 66. Common chord of two circles 67. Radical axis ; radical center. The angle formed by two inter- secting curves Chapter VI. — Polar Coordinates 68. Introductory .... (59. Polar coordinates . 70. Applications 71. Standard equations of the straight line and circle Chapter VII. — Transformations of Coordinates 72. Introductory 73. Change of origin, new axes parallel respectively to the original axes 74. Transformation from one system of rectangular axes to another system, also rectangular, and having the same origin : change of direction of axes . . ...... 76. The degree of an equation in rectangular coordinates is not changed by transformation to other axes .... 76. Transformations between polar and rectangular systems . Chapter VIII. — The Conic Sections Equation of the Second Degree Ax' + 2 Sxy + By' + 2Gx + 2:Fy+C = 77. Recapitulation 78. The conic sections . . I. THE PARABOLA Special Equation of the Second Degree Ax^ + 2Gx + 2Fy + C = 0, or By' + 2 Gx + 2 Fs+ = 79. The parabola defined 80. First standard equation of the parabola 81. To trace the parabola y- = ipx 82. Latus rectum 83. Intrinsic property of the parabola. Second standard equation . 84. Every equation of the form Ax' + 2 Gx + 2 Fy + C = 0, or By' + 2 Gx + 2 Fy + C =0, represents a parabola whose axis is parallel to one of the coordinate axes II. THi: ELLIPSK Special Equation iif the Second Degree Ax- + U:/' + 2 (r'..- + 2Fy+ C = 85. The ellipse defined . . 8(). The first standard equation of the ellipse H7, To trace the ellipse — + ^ = 1 . a'^ li- CONTENTS 88. Intrinsic property of the ellipse. Second standard equation . 148 89. Every equation of the form Ax^ + Btp' + 2Gx + 2Fy + C = 0, in which A and B have the same sign, represents an ellipse whose axes are parallel to the coordinate axes . . , 161 III. THE HYPERBOLA Special Equation of the Second Degree Ax''-Sy^ + 2Gx + 2Fy+ C=0 90. The hyperbola defined . . 153 91. The first standard equation of the hyperbola .... 163 92. To trace the hyperbola ?! _ I!? = i 164 a^ 62 93. Intrinsic property of the hyperbola. Second standard equation 155 94. Every equation of the form Ax^ + Bx+2Gx + 2Fy+ C = 0, in which A and B have unlike signs, represents an hyperbola whose axes are parallel to the coordinate axes . . . 158 96. Summary 169 96. Polar equation of the conic 160 IV. THE GEKERAL EQUATION Ax^ + 2 Bxy + Bti^ + 2Ox + 2Fy+C=0 Beduction to the Standard Form 97. Condition that the general quadratic expres.. Exponential anil lojjarithniic curves 274 16l'>. Trigonnmelric curves 275 Hi", l-nci of eombiiied equations ■276 liiH. I'olar coiirdiiiiites 273 109. Parametric equations 280 ANALYTIC GEOMETRY PART I CHAPTER I INTRODUCTION* ALGEBRAIC AND TRIGONOMETRIC CONCEPTIONS 1. Number. When one quantity is measured by another quantity of the same kind chosen as a unit, the result is ex- pressed as a number, positive (+) or negative (— ). If the algebraic operations, addition, etc., are performed upon given numbers, the result is in each case a number. A number is imaginary if it involves in any way an indicated even root of a negative number; otherwise it is real. The absolute value of a number is its value irrespective of its sign, a .^"-'^ \ B • 2. Constants and variables. If AB and AO are two given straight lines making an angle a at the point A, and if any two points X and Y, on these lines, respectively, are joined by a straight line, then Area of triangle ^XF= \ • AX' AY' sin a, i.e., A=^ • X • y ' sin a, • This introduction is in the nature of a review, which may be omitted, or made more complete, at the discretion of tlie teacher. 1 2 INTRODUCTION [Ch. 1 where x is the length of AX, y is the length of ^ 3^ and A is the area of the triangle. If now the points X and Y are moved along the lines AB and AC in any way whatever, then A, x, and y will each pass through a series of different values, — they are variable num- bers or variables ; while ^ and sin a will remain unchanged, — they are constant numbers or constants. It is to be remarked that ^ has the same value wherever it occurs, — it is an absolute constant ; while «, though constant for this series of triangles, may have a different constant value for another series of triangles, — it is an arbitrary con- stant. Because x and y may separately take any values whatever, they are independent variables ; while A, whose value depends upon the values of x and y, is a dependent variable. The illustrations just given may serve to give a clearer con- ception of the following more formal definitions. An absolute constant is a number that has the same value wherever it occurs ; such are the numbers 2, 7, f, 6^, jr, e (where ir = 3.14159265 •••, approximately ^, the ratio of the circumference of a circle to its diameter; and e = 2.71828182 -. = 1 + J+,5 + ,|+ •". approximately ^, the base of the Naperian system of loga- rithms). An arbitrary constant is a number that retains the same value throughout the investigation of a given problem, but may have a different fixed value in another problem. An independent variable is a number that may take any value whatever within limits prescribed by the conditions of the problem under consideration. A dependent variable is a number that depends for its 2-4] ALGEBRAIC AND TRIGONOMETRIC CONCEPTIONS 8 value upon the values assumed by one or more independent variables. A variable that becomes greater than any assigned number, however great, is an infinite number ; a variable that becomes and remains smaller (numerically, not merely algebraically less) than any assigned number, however small, is an infini- tesimal number. All other numbers are finite. 3. Functions. A number so related to one or more other numbers that it depends upon these for its value, is a function of these other numbers. E.g., the circumference and the area ot a circle are functions of its radius ; the distance traveled by a railway train is a function of its time and rate; if y = 3iB' + 5a! — 8, then y is a function of x. 4. Identity, equation, and root If two functions involving the same variables are equal to each other for cdl values of those variables, they are identically equal. Such an equality is expressed by writing the sign = between the two functions, and the expression so formed is an identity. If, on the other hand, the two functions are equal to each other only for par- ticular values of the variables, the equality is expressed by writing the sign = between the two functions, and the expres- sion so formed is an equation. The particular values for which the two functions are equal, i.e., those values of the variables which satisfy the equation, are the roots of the equation. E.g., (x + yy=x' + 2xy+y'', (x+a)(x-a)+a' = 3^, Q jp2 a!4- 3 and x-\ — - — = ^— are identities; x—1 x—1 while 3x' — 10x + 2 = 2s^—ix — 6, is an equation. The roots of this equation are the numbers 2 and 4 4 INTRODUCTION [Cii. I Special attention is called to the fact that an equation always imposes a condition : an identity does not. E.g., 3p — 6x + 8 = if, and only if, a; = 2 or a; = 4. So also the equation ax + by + c = imposes the con- dition that X shall be equal to — by — c a 5. Notation. In general, absolute constants are represented by the Arabic numerals, while arbitrary constants and varia- bles are represented by letters. A few absolute constants are, however, by general consent, represented by letters ; examples of such constants are ir and e. [Art. 2.] Variables are usually represented by the last letters of the alphabet, such as u, v, ic, X, y, z; while the first letters, a, 6, c, ••• are reserved to represent constants, especially arbitrary constants. Particular fixed values from among those that a variabie may assume are sometimes in question; e.g., the values, x=2 and a; = — 1, for which the function a^ — a; — 2 vanishes ; such values may conveniently be denoted by aflSxing a sub- script to the letter representing the variable. Thus x^, Xo, x^,--- will be used to denote particular values of the variable x. Similarly, variables which enter a problem in analogous ways are usually denoted by a single letter having accents attached to it; thus x', x", x'", ••• denote variables that are similarly involved in a given problem. The word "function" is usually represented by a single letter, such as f,F, 4>j 'I') "• j t^iis y= («) means that y is a function of the independent variable x, and is read "y equals the (^function of x"; so also z = F{u, v, x) means that 2 is a function of the independent variables, n, v, and x, and it is read, " z equals the i^-function of u, v, and x." 4-0] ALUEHIiAIC' AND TUltJONOMKTIUC CUNCIil'TIONS 5 Each of the two equations, ?/ = 3 x-^ — 4 a; + 10 and y = (.c), asserts that y is a function of x ; but while the former tells precisely how y depends upon x, the latter merely asserts that there is such a dependence, without giving any information con- cerning the form of that dependence. If, however, the form of ^ in a given problem is defined by the equation 3aJ'_a;4^.5 ^(x)=: then, in the same problem 2x-\-l ^^^^^3jJ^_^^ *W = |' ^"<1 «A(0)=5. 6. The quadratic equation. Its solution. The most general equation of the second degree, in one unknown number, may be written in the form ax? + bx + c = 0, (1) where o, 6, and c are known numbers. Its roots are -h + W¥-4.ac and - _ -&- Vi^'-^ac ,^, ^ Ta ' ' 2^ ' ^-'' Their nature depends upon the number b'—iac, called tha discriminant of the quadratic equation ; there are three cases to be considered, viz. : if 6' — 4 ac > 0, then the roots are real and unequal, if i'' — 4 ac = 0, then the roots are real and equal, if 6' — 4 ac < 0, then the roots are imaijinary. (3) 6 INTRODUCTION [Ch. 1 EXERCISES 1. Show whicli of the following equalities are identities : (1) x^-4a; + 4 = 0; (4) {p + qy=p' + q' + 3pq(p+q); (2) (s + t)(s-t):=^-t^; (5) ar'+5a! + 6 = (a! + 3)(a! + 2). (3)^^=«'-«/3 + /3»; a + fS 2. Determine, without solving the equation, the nature of the roots of SaP + Sx + l^O. Solution. Since 6* — 4 oc = 64 — 12 = 52, i.e., is positive, therefore the roots are real and unequal; again since a, b, and c are all positive, therefore both roots are negative [cf. (2), Art. 6]. 3. Without solving the equation, determine the character of the roots of 8!B'-3a; + l=0. 4. Given the equation a!* — 3a — m(a; + 2a!* + 4)=5a!* + 3. Eind the roots. For what values of m are these roots equal ? 5. Determine, without solving, the character of the roots of the equations : (1) 52''-2« + 5 = 0; (2) a!» + 7 = 0; (3)3^-« = 19. 6. Determine the values of m for which the following equa- tions shall have equal roots : (1) a^-2a!(l+3m)+7(3 + 2m) = 0; (2) 7nx' + 2a?-2m=3mx~9x + 10; (3) 4ar' + (l + TO)a! + l = 0; (4) ar' + (6x4-m)» = a« 7. If in the equation 2 ax(ax 4 nc) + (nP — 2)(^ = 0, x is real, show that h is not greater, in absolute value, than 2. 6,7] ALGEBRAIC AND TRIGONOMETRIC CONCEPTIONS 7 8. If X is real in the equation — = a, show that a ar — 5x + 9 is not greater than 1, nor less than — -jJj-. 9. For what values of e will the following equations have equal roots ? (1) 3ar' + 4a! + c = 0; (2) (maj + c)' = 4 te; (3) 'Lx'+9(2x + cy = 36. 10. Solve the equations in examples 2, 3, and 5. 11. Solve the equations ; a) «^-25«.= -144; (2) ^-^ + ^ = 0. 7. Zero and infinite roots. In equations (2) of Art. 6, x^ and Xi, i.e., the roots of 0^+ bx+c = 0, were given; and it is seen that -b + V^- -4ac •"1 2a -b+Vb^ -4ac 2a and that " ,m —b—^V' — ^ao — 6— V*" — 4ac _6_V6*-4ac 2c 2a _6+V6^^^ac (2) Equations (1) and (2) show that : (1) If a and 6 remain unchanged while c grows smaller (numerically), then x^ grows smaller and x^ grows larger: the roots of as? + 6* + = are a^ = 0, atj = • a (2) If a remains unchanged while c and h grow smaller, then both Xi and x^ grow smaller : the roots of aii? + • a; + = are a:,= 0, a;a=0. 8 INTRODUCTION [Ch. I (3) If b and c remain unchanged while a grows smaller, then one root, x^, grows larger, indefinitely ; the roots of -x^ + bx H-c = are a^= — ,Xi = . b (4) If c remains unchanged while a and 6 become smaller, then both a;, and Xg grow larger, indefinitely. (5) If a and c remain unchanged while b grows smaller, then the roots approach numerical equality: the roots of aa^ + • x + c = &Texi = -J—~, 052= —-i/—-"- X a 'a 8. Properties of the quadratic equation. If the roots of a quadratic equation are not themselves needed, but only their sum or product is desired, these may be obtained directly from the given equation by inspection ; for by adding, and again by multiplying, the two roots of aa^+bx+c = 0, (1) are obtained the relations a!,+xj,= ---, XiXt = ?. (2) a a Ejg,, the half sum of the roots of the equation mV + 2(6m-2Z)a! + 6' = . Xi + Xj _ 2{bm. — 2r) _ 2l—brn, 2 2 m' ~ m" ' Again, if x^ and «, are the roots of the equation a^+px + q = 0, then x — Xi and x— x^ are the factors of its first member. Also conversely: if a quadratic function can be separated into two factors of the first degree, then the roots can be im- mediately written by inspection. E.g., the roots of (2 as — 3)(a; + 5) = are f and — 5. 7-9] ALGEBRAIC AND TRIGONOMETRIC CONCEPTIONS 9 9. The quadratic equation involving two unknowns. One equation involving two unknown numbers cannot be solved uniquely for the values of those numbers which satisfy the equation ; but if there is assigned to either of those numbers a definite value, then at least one definite and corresponding value can be found for the other, so that, this pair of values being substituted for the unknown numbers, the equation will be satisfied. In this way an infinite number of pairs of values, that will satisfy the equation, may be found. If, liowever,the equation is homogeneous in the two unknowns, i.e., of the form ax' + bxy+ cf = 0, then the ratio x : y may be regarded as a single number, and the equation has properties precisely like those discussed in Arts. 6, 7, and 8. To solve a system consisting of two or more independent simultaneous equations, involving as many unknown elements, it is necessary to combine the equations so as to eliminate all but one of the unknown elements, then to solve the resulting equation for that one, and, by means of the roots thus obtained, find the entire system of roots. EXERCISES 1. Given the equation a;" + 3 x — 4 + m(3 a^ — 4)— 2 ma? — 0, find the sum of the roots ; the product of the roots ; also the factors of the first member. 2. Factor the following expressions : (1) a:^-5a; + 4; (3) mx=-3a;4-c; (5) 3w^-94w?-64; (2) a;= + 2x-8; (4) aa^+fcxiz+cy''; (6)11-27^-183/'. 3. Without first solving the equation r^- 3 a; - m(a; + 2 ar* + 4) = 5 x^ 4- 3, 10 INTRODUCTION [Ch. I find the svun, and the product, of its roots. For what value of m are its roots equal ? For what value of m does one root become infinitely large? If all the terms are transposed to one member, what are the factors of that member ? 4. Without first solving, determine the nature of the roots of the equation (m —2) (log xy— (2 m + 3)log a;— 4 m = 0. [Re- gard log X as the unknown element.] For what values of m are the roots equal ? real ? one infinitely great ? one zero ? Find the factors of the first member of the equation. 5. Find five pairs of numbers that satisfy the equation : (1) a; + 3y-7 = 0; {S) f=lQx; (2) a:»4-3/* = 4; (4) 3x + Gxy-2,y^ + Zoi? = Q. 6. Without solving, determine the nature of the roots of the equation: 93^^12 ay + 42/^ = 0, iu^-uv + 19v^=0. 7. Solve the following pairs of simultaneous equations : (1) 3x-5«/ + 2 = 0, and 2a;+7?/ — 4 = 0; (2) 6y + 22-f-3=0, and 72/ + 4«+2 = 0; (3) y— 3x + c = 0, and y^ = 9x; (4) a? + y^ = 5, and y^ = &x\ (6) 6 V -1- a'y^ = d?b^, and y = ax + b; 8. Determine those values of b for which each of the follow- ing pairs of equations will be satisfied by two equal values of y : (1) \x'+y'=a^ y = 6x+b\; (2) \y=mx + b, f = ix\; (3) ]3y + 2x = b, 6a^ + f=12l. 9. Determine, for the pairs of equations in Ex. 8, those values of 6 which will give equal values of x. 9, 10] ALGEBKAIC AND TRIGONOMETKIC CONCEPTIONS 11 10. Directed lines. Angles. A line is said to be directed when a distinction is made between the segment from any point A of the line to another point B, and the opposite seg- ment from B to A. One of these directions is chosen as posi- tive, or -f-, and the opposite direction is then negative or — . The portion of the line on one side of a point is a half line, or ray. An angle is the figure formed by two half lines from a point. If the lines be directed, the angle is measured by the amount of rotation about their point of intersection necessary to bring the positive end of the initial side into coincidence with the positive end of the terminal side. The point in which the lines meet is called the vertex of the angle. The angle is positive, or -}-, if the rotation from the initial to the terminal side is in counter-clockmse direction ; the angle is negative, or — , if the rotation is clockwise. The angle formed by two directed straight lines in space, which do not meet, is equal to the angle between two rays which meet and are respectively parallel to the given lines. The angle from the directed line a to the directed line 6 may be denoted as A (ab). For the measurement of angles there are two absolute units : (1) The angular magnitvde about a point in a plane, i.e., a complete revolution. An angle of one fourth of a complete revolution is called a right angle; ^\y of a right angle is a degree (1°); ^^ of a degree is a minute (1'); and -^^ of a minute is a second (1"). (2) Tlie angle whose subtending circular arc is equal in length to the radius of that arc ; this angle is called a ^ radian jl<''{ ; it is independent of the length of ^^ the radius. These units have the following relation : (f ^^ «•(■•) = 180°; 12 INTRODUCTION [Ch. I Therefore, 1<'> = — = 57° 17' 44.8", approximately. A right angle is 90° or f-^ ' 11. Trigonometric ratios. If from any point P in the ter- minal side of an angle 6, at a distance r from the vertex, a perpendicular MP is drawn to the initial side meeting it in Fig. 3 a. M, and if MP be represented by y and VM by x, then, by general agi-eement, ?/ is + if MP makes a positive right angle with the initial line, and — if this right angle is nega- tive ; similarly, a; is -(- if VM extends in the positive direction of the initial line, and — if it extends in the opposite direction. The A VMP is the reference triangle for Z. 6. Its three sides form six ratios, known as the trigonometric ratios or functions of the angle 6, and are named as follows : sine = ^, r cosine = -, r tangent $ = ^, X cotangent & = -, y secant $ = cosecant 6 = These functions are not all independent, but are connected by the following relations : Functions of an angle. (1) sin 6 • CSC = 1, (2) COS « • sec e = 1, (3) tantf.cote = l, (4) tan ^ = sin 5 : cos 6, (5) cot = cos : sin 0, (6) sin='e + cos='(9 = l, (7) tan=ie+l = sec''e, (8) cot2e-f-l = csc»ft 10-12] ALGEBRAIC AND TRIGONOMETRIC CONCEPTIONS 13 Functions of related angles. Based upon the definitions of the trigonometric functions the following relations are readily established for any plane angle, 6. (9) function of (180° ± ^, or (360° ± ^ = ± same function of 6, (10) function of (90° i^), or (270°±^= ± co-function of 6. The proper sign in each case is shown in the following table: Ql-adrant I QUADBANT II Quadrant III Quadrant IV sin 6 cscO + + — — cos^ sec 6 + — — + tanS cote + — + 12. Other important formulas. If Oi and 6^ are any two plane angles, then sin (^1 ± $i) = sin Oi cos $2 ± cos flj sin $2, cos ($1 ± $2) = cos 61 cos 62 T sin O^ sin 62, 4. /a . a\ tan ^1 ± tan $2 tan (6, ± $2) = ^-= h" ^ '^ 1 T tan 61 tan «, If $ is any plane angle, then sin 2 e = 2 sin e cos 6, cos 2e = cos^e - sin=' e = 1 - 2 sin»e= 2 cos^fl- 1, 2tane tan 29 = l-tan^e 14 INTRODUCTION [Ch. I o sin - = V^(l — cos 6), cos - = V|(l 4- cos ff), 2 ^'l^-cose • costf sinfl sin 6 1 + cos 6 If a, 6, and c are the sides of a triangle, directed continuously as in the figure, from Ato'B,B to O, C to A, and lying respectively opposite the interior angles A, B, and C; and if A is the area of this triangle; then ^^^j,,;^^. a' = b'^ + c' — 2 be cos A. EXERCISES 1. Express in radians the angles : 15°; 60°; 135°; -252°; | rt. angle; 10° 10' 10" ; 88° 2'; (3t)°. 2. Express in degrees, minutes, and seconds the angles : /ttV" /'37r\<" /IN") /2\<'> 7 „ , .. 5 ^ , 3. Find the values of the other trigonometric functions, given : (1) tane = 3; (2) sec a; = - -v/2 ; (3) cos <^ = -i=; V3 (4) sin { = I ; (5) cot ^ = I ; (6) esc u = — 2. Solution of (1). Construct a right triangle ABC with the sides AB = 1 and 2?C = 3, then ABAC is an angle whose tangent is 3. Therefore 12,13] ALGEBRAIC AND TRIGONOMETRIC CONCEPTIONS 15 ^C = V3F + 5C' = VI0, and the other func- tions of the angle BAC are at once seen to be : sintf = -4=:. cose = -4=, csce = ^^, VlO VlO 3 sec 6 = VlO, and cot = ^. 4. By means of a right triangle, with appropriate acute angles, find the numerical values of the trigonometric ratios of the following angles: 30°; 45°; 60°; 90°; 135°; and - 45°. 5. Express the following functions in terms of functions of positive angles less than 90° : tan 3500°; - esc 290°; sin (-369°); -cos^i^; cot (-1215°). 6. Solve the following equations : (1) sin ^ = -cos 210°; (2) cos^ =sin2fl; (3) — £2?.^_ = V3; sinajcot^a; and (4) (sec^ x—1) (csc^ a; + 1) = |. 7. In the following identities transform the first member into the second : >.., tan tf — cot^ _ 2 ^. ^g, sec x + esc a; _ 1 + cot x tanfl + cot^~~csc^tf ' secx — esc x~l — cotaj' (3) CSC X (sec a; — 1) — cot a;(l — cos x) = tan x — sin x ; (4) (2 r sin a cos af + /^(cos" a — sin^ a)^ = r' ; (5) (cos a cos 6 + sin a sin b)' + (sin a cos b — cos a sin 6)' = 1 ; (6) (r cos (t>y + (r sin cos Oy + (?• sin <^ sin 6)' = r'. 13. Orthogonal projection. The orthogonal projection* of a point upon a line is the foot of the perpendicular from the * Hereafter, unless otherwise stated, projection will be understood to mean orthogonal projection. 16 INTRODUCTION [Ch. I point to the line. In the figure, M is the projection of P upou AB. The projection of a segment PQ of a line upon another R^ < ^ H B A , -< ■^''' Fio. 6 6. N Fia. 6 a, line AB, is that part of the second line extending from the projection of the initial point of the segment to the projec- tion of the terminal point of the segment. Thus MN is the projection of PQ upon AB, and NM is the projection of QP upon AB. J^rom the definition, M]Sr PH -;r:r = -zzt:: = COS a, PQ PQ ^ .: MN= PQ ■ cos a ; i.e., the projection of a segment of a line upon another line is equal to the product of its length by the cosine of the angle which it makes ivith that other line. A linemade up of parts PQ, QB, RS, ••• (Fig. 7 a, 7 6), which are straight lines having different directions, is a broken line; L B Fio. 76. 13] ALGEBRAIC AND TRIGONOMETRIC CONCEPTIONS 17 and the projection, of a broken line * upon any line is the alge- braic sum of the projections of its parts upon the same line. Thus the projection of PQRST upon AB is the projection of PQ+ the projection of QR + •••, upon AB; i.e., proj. PQRST upon AB = MN+ ]SrK+ KL + LH= MH; but MH is the projection of the straight line PT which joins the first initial to last terminal point of the broken line. Similarly for any broken line. The following theorems may therefore be stated : (1) Hie projection of a segment of a line upon any straight line in space equals the product of its length by the cosine of the angle between tJie tivo lines. (2) Tlie projections of two parallel segments of equal length upon any given line in space are equal in absolute value. (3) Tim projection of any broken line in space upon any straight line equals the projection, ujmn the same line, of the straight line which joins the extremities of the broken line. (4) Tlie projection of the perimeter of any closed polygon upon any given line is zero. EXERCISES 1. Two lines of lengths 3 and 7 respectively meet at an angle - ; find the projection of each upon the other. 2. The center of an equilateral triangle, of side 5, is joined by a straight line to a vertex ; find the projection of this joining line upon each side of the triangle. 3. A rectangle has its sides respectively 4 and 6 ; find their projections upon a diagonal. » In space, as well as in a plane. 18 INTRODUCTION [Ch. I 4. A given line AB makes an angle of 30° with the line MN, and BC is perpendicular to AB and of length 15 ; find the pro- jection of BC upon MN. Solve this problem if the given angle be a instead of 30°. 5. Two lines in space, of length a and 6 respectively, make an angle w with each other ; find the projection of b upon a line that is perpendicular to o. 6. Project the perimeter of a square upon one of its diagonals. CHAPTER II GEOMETRIC CONCEPTIONS. THE POINT L COORDINATE SYSTEMS 14. Coordinates of a point. Position, like magnitude, is relative, and can be given for a geometric figure only by reference to some fixed geometric figures (planes, lines, or points) which are regarded as known, just as magnitude can be given only by reference to some standard magnitudes which are taken as units of measurement. The position of the city of New York, for example, when given by its latitude and longitude, is referred to the equator and the meridian of Greenwich, — the position of these two lines being known, that of New York is also known. So also the position of Baltimore on the railroad from Washington to New York may be given by its distance from Washington ; while a particular point in a room may be located by its distances from the floor and two adjacent walls. If, as in the last illustration, a point is to be fixed in space, then three magnitudes must be known, referring to three fixed positions. If, on the other hand, the point is on a known surface, as New York oi: Baltimore on the surface of the earth, then only two magnitudes need be known, referring to two fixed positions on that surface ; while if the point is on a known line, only one magnitude, referring to one fixed posi- tion on that line, is needed to fix its position. 19 20 GEOMETRIC CONCEPTIONS. THE POINT [Ch. 11 These various magnitudes which serve to fix the position of a point, — in space, on a surface, or on a line, — are called the coordinates of the point. In Part I of this book it will be understood that the work is restricted to a given plane surface. 15. Positive and negative coordinates. If a point lies in a given directed straight line, its position with reference to a -.1 p' \0 P X fi^^'i point of that line is com- L — 3--- it — 5— J pletely determined by one coor- *■'«■ 8- dinate. E.g., let X'OX be a given directed straight line,' and let distances from toward X be regarded as positive, then distances from toward X' are negative. A point P in this line and 3 units from toward X may be designated by +3, where the sign + gives the direction of the point, and the number 3 its distance, from O. Then the point P' lying 3 units on the other side of would be designated by "3. In the same way there corresponds to every real number, positive or negative, a definite point of this directed straight line; the numbers are called the coordinates of the points; and the point 0, from which the distances are measured, is called the origin of coordinates. 16. Cartesian coordinates of points in a plane. Suppose two directed straight lines X'OX and Y'OY are given, fixed in the plane and intersecting in the ^ point 0. These two given lines /r are called the coordinate axes, / / X'OX being the »-axis, and F' or , -J-— -/^ being the y-axis ; their point of j^' q/ x intersection O is the origin of coor- / /^ dinates. Any other two lines, par- fy-' / allel respectively to these fixed fig. 9. 14-17] COORDINATE SYSTEMS 21 lines, and at known distances from them, will intersect in one and but one point P, whose position is thus definitely fixed. If these lines through P meet the axes in M and L respec- tively, then the directed distances LP and MP, measured j>arallel respectively to the axes, are the Cartesian coordinates of the point P. The distance LP, or its equal OM, is the abscissa of P, and is usually represented by x, while MP, or its equal OL, is the ordinate of P, and is usually represented by y. The point P is designated by the symbol (x, y), — often written P = (x, y}, — the abscissa always being written first. Thus the point (4, 5) is the point for which OM = 4 and MP =5; while the point ("3, 2) has 0M= "3 and MP =2. 17. Rectangular coiirdinates. The simplest and most com- mon form of Cartesian coordinate axes is that in which the angle XO Yis a positive right angle; the abscissa (a;) of a point is, in this case, its perpendicular distance from the y-axis, and its ordinate (y) is its perpendicular distance from the x- axis. This is known as the rectangu- lar system of coordinates. The axes divide the entire plane into four parts called quadrants, which are usually designated as first (I), sec- "°' ^ ' ond (II), third (III), and fourth (IV), in the order of rotation from the positive end of the x-axis toward the positive end of the y-axis, as indicated in the accompanying figure. These quadrants are distinguished by the si'grjis of the co- ordinates of the points lying within them, determined as in Art. 15, thus:* * Compare with table in Art. 11, signs of cosine and sine. Y II. L I. — ^p •■ ^, M ni. rv. 22 GEOMETRIC CONCEPTIONS. THE POINT [Ch. II Quadrant I Quadrant II Quadrant III Quadrant IV abscissa x + — + ordinate y + + — — Four points having numerically the same coordinates, but lying one in each quadrant, are symmetrical in pairs with re- gard to the origin, even though the axes are not at right angles; if, however, the axes are rectangular, then these points are symmetrical in pairs, not merely with regard to the origin as before, but also with regard to the axes, and they are severally equidistant from the origin. Because of this greater symmetry rectangular coordinates have many advantages over an oblique system. Ill the folloii'ing pages rectangular coordinates will always he understood unless the contrary is expressly stated. EXERCISES 1. Plot accurately the points : ("l, -7) *, (+4, +5), (0, -3), and (+3, 0). 2. Plot accurately, as vertices of a triangle, the points : (-1, -3), (-2, -7), and (+4, +4). Find by measurement the lengths of the sides, and the coordinates of the middle point of each side. 3. Construct the two lines passing through the points (a, -6) and ("a, 6), and (a, b) and (~a, "6), respectively. What is their point of intersection? Find the coordinates of the middle point of each line. * These minus sifjns are written high merely to indicate that they are signs of quality and not of operation. 17, 18] COORDINATE SYSTEMS 23 4. If the ordinate of a point is 0, where is the point ? if its abscissa is ? if its abscissa is equal to its ordinate ? if its abscissa and ordinate are numerically equal but of opposite signs ? 5. Express each of the conditions of Ex. 4 by means of an equation. 6. The base of an equilateral triangle, whose side is 5 inches, coincides with the a;-axis; its middle point is at the origin ; what are the coordinates of the vertices ? If the axes are chosen so as to coincide with two sides of this triangle, respectively, what are the coordinates of the vertices ? 7. A square whose side is 10 inches has its diagonals lying upon the coordinate axes ; find the coordinates of its vertices. If a diagonal and an adjacent side are chosen as axes, what are the coordinates of the vertices ? of the middle points of the sides ? of the center ? 8. Find, by similar triangles, the coordinates of the point which bisects the line joining the points (~2, ~7) and (5, 6). 9. Show that the distance from the origin to the point (o, b) is Va' + b\ How far from the origin is the point (a, ~b) ? (-a, b) ? (-a, -6) ? 10. Prove, by similar triangles, that the points (4, 0), (0, 3), and ("4, 6) lie on the same straight line. 11. Solve exercises 1 to 4, and 10, if the coordinate axes make an angle of 30°. Also if this angle is 45°. 18. Notation. In the following pages, in accordance with Art. 6, a variable point will be designated by P, and its coordi- nates by {x, y) ; so that P = (x, y). Several variable points under consideration at the same time will be called P' = (x', y'), P" = (x", y"), P" = (x'", y'") ; etc. Fixed points similarly will be designated as Pj = (cci, ?/i), P, = (x.,, y,), etc. 24 GEOMETKIC CONCEPTIONS. THE POINT [Ch. II II. ELEMENTARY APPLICATIONS 19. Distance between two points. Let Pj = (xj, y,) and Pj = (x2, Pi) be two given points with rectangular axes OX and OY. To find distance PiP^ = d in terms of a;,, o^, y^, y^. L. L, P. y Q X 31, i U r Q L, P, p. L, k M, X Fia. 11 a. Fig. 11 6. Construction : Extend the abscissa iiPi of point P„ to meet the ordinate JfjPj of point Piin Q: then in A PiQP.^, P[^+QP^=P^,'; but PiQ = sc2 — Xjand QP2 = 2/2 — yi; hence «f = VCXg - «,)« + (j/ij - jr,)«. [1]« Since either of the two points may be named P^, this formula may be expressed in words thus : with rectangular coordinates, the square of the distance between two points is equal to the square of the difference between their abscissas plus the square of the dif- ference between their ordinates. 20 Direction from one point to a second. The direction of a line is indicated by the angle that it makes with the 9!-axis, * The demonstration applies to each figure. By making and examin- ing other possible constructions the student should assure himself that the formula is entirely general. The more important formulas a^-e printed in bold-faced type. They should be committed to memory by the learner. 19-21] ELEMENTARY APPLICATIONS 25 measured as in Art. 10. This angle is called the inclination of the line ; it is usually found by means of its tangent, which is called the slope * of the line. From this definition it follows that with rectangular axes the slope of the line from the point Pi = (o^, ^i) to the point P-i = {Xi, 2/2), and with an inclination , is m = tan <^ = ^; that is, m = — - • [2] Fig. 12. 21. Parallel lines. Perpen- dicular lines. If two straight lines are parallel, evidently their inclinations and therefore their slopes, are equal. If the lines are perpendicular to each other, with inclinations, 6 and 6' respectively, so that d'=zd + 90°, then 1 tan e' = - cot e = — tan 6' i.e., each slope is the negative re. ciprocal of the other. In brief, if rrii and m, are the slopes of two lines, then if the lines are parallel, »»2 = '"^i [3] if the lines are perpendicular, wtj = — - [4] EXERCISES 1. rind the distances between the points (2, 6), (4, 14), and (~8, ~8), taken in pairs. 2. Find the distahces for the points of Ex. 1, if the axes are oblique with u = 30° ; if 3 + yi)iXi-Xi)l. 28 GEOMETRIC CONCEPTIONS. THE POINT [Ch. II By expanding and rearranging the second member, this latter equation may also be written in the form ^ = l\^i(.y2- ys> + ^'a (vs - y\^ + "'s (vi -yi>\- [s] The symmetry * in formula [5] should be carefully noted. It may be seen from the figure that with the vertices as there named, with the triangle upon the left as it is traced from Pi to Pj to P3, the value of A is positive ; while if the vertices be named in the reverse order, the sign of each term of eq. [5] will be reversed, and the area A will be negative. 23. One great advantage of the analytic method of solving problems lies in the fact that the analytic results which are obtained from the simplest arrangement of the geometric figure with reference to the coordinate axes are, from the very nature of the method, equally true for all other arrangements. Thus formulas [1], [2], and [5] can be most readily obtained if the points are all taken in quadrant I, i.e., with their coordinates all positive ; but because of the convention adopted concerning the signs as essential parts of the coordinates, these formulas remain true for all possible positions of P^ and P^. By draw- ing the figures and making the proofs when Pi and P^ are taken in various other positions, the student should assure himself of the generality of formulas [1], [2], and [5] in articles 19, 20, and 22, and of the underlying importance in applying them of a careful interpretation oftJie negative number. •This kind of symmetry is known as cyclic (or circular) symmetry. If the numbers 1, 2, and 3 are arranged thus UL , then the subscripts in 2 the first term begin with 1 and follow the arrowheads around the circle (i.e., their order is 1, 2, 3), those of the second terra begin with 2 and follow the arrowheads (their order is 2, 3, 1), and those of the third term begin with 3 and follow the arrowheads. 22 24] ELEMENTAUr APPLICATIONS 29 EXERCISES 1. Find the areas of the following triangles : (1) vertices at the points (3, 5), (4, 2), and (1, 3) ; (2) vertices at the points (7, 3), ("4, 6), and (3, -2) ; Solve without using the formula, and then verify by substi- tuting in the formula. 2. Prove that the area of the triangle whose vertices are at the points (1, 9), (11, ~1), and (6, 5) is zero, and hence that these points all lie on the same straight line. 3. Are the points (2, 3), (1, "3), and (3, 9) coUinear ? 4. Prove that the area of the triangle formed by the points (0, 0), (a, b), and (mc, md) is m times the triangle formed by the points (0, 0), (a, b), and (c, d). 5. Derive formula [6] from the relation A = ^ base x alti- tude. 6. Derive formula [5] from the relation A = ^ 6c sin A. 24. To find the coordinates of the point which divides in a given ratio the straight line from one given point to another. Let FiQ. 15. Pi = (a;,, y,) and Pj = (^21 1/2) be the two given points, P3 = (a;,, ^3) the required point, and let the ratio of the parts into which P3 divides P1P2 be j/ii : wij ; i.e., let P1P3 : P3P2 = wii : wij. Draw the ordinates JfiPi, M-iP^, M^P^, and through P, and P3 draw lines 30 GEOMETRIC CONCEPTIONS. THE POINT [Ch. II parallel to OX, meeting M3P3 and JfjPj in It and Q respectively. To find OM^ = Xj and MsP^ = y^ in terms of Xj, x^, y^ y^, TOi, and wij. The triangles PjRPs and P3QP2 are similar ; 'P^P2 therefore But PyPs m, P3P, m,' and PlR = X3—Xi, P3Q = Xi — X3, RPi = yi—y\, QPi = y2—y3- [In Fig. 15 6, a^, y„ .y^, and 3/3 are negative.] Therefore a^-gg ^ ya-yi ^m, Kj — ^3 ^2 — .'/3 Wis Whence »^i^2 + m,^^ ^^^ j,, ^ ^M^i + i?^ . ^g. The above reasoning applies equally well whatever the value of u> (the angle made by the coordinate axes), hence formulas [6] hold whether the axes be rectangular or oblique. If the dividing point P3 is not between Pj and P^, but upon P1P2 extended, then segments P1P3 and P3P2 have opposite signs, and the ratio mi: wij is negative. In such a case point P, is said to divide P1P2 externally in the ratio m^ : m^. Corollary. If Pg is the middle point of P1P2, then jjij = wij, and formulas [6] become a^a- 2 1 yg- 2—. [7] i.e., the abscissa of the middle point of the line joining two given points is half the sum of the abscissas of those points, and the or- dinate is half the sum of their ordinates. 24] ELEMENTARY APPLICATIONS 31 EXERCISES 1. By means of an appropriate figure, derive formulas [7] independently of [6]. 2. The point P^ = (2, 3) is one third of the distance from the point P, = (~1, 4) to the point P^s (xj, y^) ; to find the coordi- nates of Pj. Solution. Here Pj and P3 are given, with a^ = — 1, yi = 4, X3 = 2, ^3 = 3, also wii = 1, and m.2 = 2; there- fore, from [6], », + 2(-l) ^^3^y.+ 2(4) l-h2 ' 1-1-2 ' which give aij = 8 and yj = 1 ; therefore the required point Pj is (8, 1). 3. Find the points of trisection of the liue joining (2, 3) to (4, -5). 4. Find the point which divides the line from (1, 3) to (~2, 4) externally into segments whose numerical ratio is 3 : 4. Solution. Here a;, = 1, 2/1 = 3, 0:2 = — 2, y2 = i, «ii = 3, and m2 = 4, but the point of division being an external one, the two segments are oppositely directed ; therefore one of the numbers 3 or 4, say 4, must have the minus sign prefixed to it. Substituting these values in [6], 3(-2)-4(l)^^ ^^ 3(4)-4(3)^ ' 3-4 3-4 ' the required point is, therefore, Pj = (10, 0). 5. Solve Ex. 4 directly from a figure, without using [6]. 6. Find the points which divide the line from (2, 10) to (4, 14) internally and externally into segments that are in the ratio 2 : 3. 32 GEOMETRIC CONCEPTIONS. THE POINT [Cii. U 7. A line AB is produced to C, so that BC=^AB; if the points A and B have the coordinates (5, 6) and (7, 2), respec- tively, what are the coordinates of G ? 8. Prove, by means of Art. 24, that the median lines of a tri- angle meet in a point, which is for each median the point of trisection nearest the side of the triangle. 25. Analytic Geometry is the study of geometric figures by algebraic methods. It may conveniently be divided into Plane Analytic Geometry, which concerns only figures in a given plane surface, and is treated in Part I of this book ; and Solid Analytic Geometry, which concerns space figures, and is taken up in Part II. The fundamental basis of analytic geometry is the represen- tation of a geometric point by means of algebraic numbers, which has been the subject of the present chapter. The next important step is to find the relation that exists between an algebraic equation and a geometric curve, or locus. This will be taken up in the next chapter. EXAMPLES ON CHAPTER II 1. Find the area of the quadrilateral whose vertices are the points (0, 0), (5, 0), (2, 3), and (4, 6). Draw the figure. 2. Find the lengths of the sides and the altitude of the isos- celes triangle (0, 5), (5, 0), (9, 9). Find the area by two different methods, so that the results will each be a check on the other. 3. Find the coordinates of the point that divides the line from (2, 3) to (~1, ~6) in the ratio 6:8; in the ratio 2 : "3 ; in the ratio 3 : ~2. Draw each figure. 4. One extremity of a straight line is at the point (-3, 4), and the line is divided by the point (1, 6) in the ratio 4 : 6 ; find the other extremity of the line. 24,25] ELEMENTARY APPLICATIONS 33 5. The line from ("6, "2) to (3, "1) is divided in the ratio 5:4; find the distance of the point of division from the point (-4, 6). 6. Show analytically that the figure formed by joining the middle points of the sides of any quadrilateral is a parallelogram. 7. Show that the points (3, "1), (V3, V7), and (V6, 2) are equidistant from the origin. 8. Show that the points (1, 1), ("1, -1), and (a, -a) form an isosceles triangle. Find the slopes of its sides. 9. Prove analytically that the diagonals of a rectangle are equal. 10. Show that the points (0, ~a), (2 a, a), (0, 3 a), and ("2 o, a) are the vertices of a square. 11. Express by an equation that the point Qi, k) is equidistant from (-1, 1) and (1, 2) ; from (1, 2) and (1, -2). Then show that the point (f, 0) is equidistant from ("1, 1), (1, 2), and (1, -2). 12. Prove analytically that the middle point of the hypote- nuse of a right triangle is equidistant from the three vertices. 13. Three vertices of a parallelogram are (1,2), (~6, ~3), and (7, ~C) ; what is the fourth vertex ? 14. The center of gravity of a triangle is at the point in which the medians intersect. Find the center of gravity of the triangle whose vertices are (2, 3), (4, ~5), and (3, ~6). [Cf. Ex. 8, p. 32J. 15. The line from (x„ y^ to (a;^, y,) is divided into four equal parts ; find the points of division. 16. Prove analytically that the two straight lines which join the middle points of the opposite sides of a quadrilateral mutu- ally bisect each other. 34 GEOMETRIC CONCEPTIONS. THE POINT [Ch. II 17. Prove that (2, 10) is on the line joining the points (0, 4) and (4, 16), and is equidistant from them. 18. If the angle between the axes is 60°, find the perimeter of the triangle whose vertices are (2, 2), (~7, "1), and (~1, 5). Plot the figure. 19. Show analytically that the line joining the middle points of two sides of a triangle is half the length of the third side. 20. A point is 7 units distant from the origin and is equidis- tant from the points (4, 2) and (~4, ~2) ; find its coordinates. 21. Prove that the points (a, 6 + c), (6, c + a), and (c, a + b) lie on the same straight line. [Cf. Ex. 2, p. 29]. 22. What is the ordinate of a point whose abscissa is c, which lies on the line from the origin to the point (a, 6) ? 23. Find five pairs of values of x and y that satisfy each of the following equations, and plot the corresponding points : (1) 3x + 2y + 6 = 0. (3) 2a; + 7 = 3y. (2) «= + s'» = 9. (4) xy = i. CHAPTER III THE LOCUS AND THE EQUATION 26. The locus of an equation. A pair of numbers (as, y) is represented geometrically by a point in a plane. If these two numbers (x, y) are variables, but connected by an equa- tion, then this equation can, in general, be satisfied by an infinite number of pairs of values of x and y, and each pair may be represented by a point. These points will not, how- ever, be scattered indiscriminately over the plane, but will all lie in a definite curve, whose form depends only upon the nature of the equation under consideration ; and this curve will contain no points except those whose coordinates are pairs of values which, when substituted for x and y, satisfy the given equation. This curve is called the locus or graph of the equation ; and the first fundamental problem of analytic geom- etry is to find, for a given equation, its graph or locus. 27. Illustrative examples : rectangular coordinates. (1) Given the equation x+5 = Q, to find its locus. This equation is satis- fied by the value — 5 for x, whatever the value given to y, Now such points as A=(-5,2), P,= (-5,3), P3 = (-5, -3),etc., all lie on the line MN, parallel to the jr-axis, and at the dis- tance 5 on the negative side of it, — this line extending indefi- nitely in both directions. Moreover, each point of M2f has for 36 THE LOCUS AND THE EQUATION [Ch. hi A P. M N its abscissa — 5, hence the coordinates of each of its points satisfy the equation a;+5=0. lu the chosen system of coordi- nates, the line MN is the locus of the equation a; + 5 = 0. Similarly, the equation a;— 5 = is satisfied by any pair of values of which X is 5, such as (5, 2), (5, 3), (5, 4), etc. ; all the corresponding points lie on a straight line M'N', lif Fio. le parallel to the j/-axis, at the distance 5 from it, and on its positive side; i.e., M'2f is the locus of the equation a; - 5 = 0. (2) Griven the equations y ±3 = 0, to find their loci. By the same reasoning as in (1) it may be shown that the locus of the equation y+3=0 is the straight line A B, parallel to the x-axis, situated at the distance 3 from it, and on its negative side. Also that the locus of the equation y — 3 = is CD, a line parallel to the x-axis, at the distance 3 from it, and on its positive side. More generally it is evident that in Cartesian coordinates (rectangular or oblique), an equation of the first degree, and co)i- taining but one variable, represents a straight line parallel to one of the coordinate axes. (3) Given the equation Sx — 2y + 12 = 0, to find its locus. In this equation both the variables appear. By assigning any definite value to either one of the variables, and solving the equation for the other, a pair of values that will satisfy the equation is obtained. Thus the following pairs of values are found: 27, 28] THE LOCUS AND THE EQUATION 37 aH = 0, yi = 6 x-. = l, y2=7| Xj := z, ^3 = y 3^1 = 3, 3/4=10^ 9:5= -1, ^5 = 41 ^6 = - 2, 2/^ = 3 x,= -3, 2/7 = 1^ a;8=-4, ^8 = x=+(x>, y=+ca «=— 00, y=— 00 Plotting the corresponding points where P, = (x,, y{) = (0, 6), A = (xi, y^) = (1, 7|), etc., Fio. 17. they are all found to lie on the straight line EF, which is the locus of the equation 3a; — 2y + 12 = 0. In Chap. V, it will be shown that, in Cartesian coordinates, an equation of the first degree in two variables always repre- sents a straight line. EXERCISES Plot the loci of the following equations : 1. x = 0. 2. y = 0. 3. mx = 0. 4. 7a; = 3. 5. 22/ — 5=0. & x-\-y = 0. 7. 3x + 2 = y. 8. 2x^ + 2f = i. 9. x + y = 8. 10. x — y = 0. 11. a^-y''=16. 12. a^ + 2y^ = 'i. 13. v = 32t. 14. x = 3y + 2. 15. «'' + «» = 25. 16. u^ — v=0. 17. s=16«='. 18. ?4-^ = l. 3 2 19. «!/ = 4. 20. x* = 4y. 21. 2/ = — «'• 28. The locus of an equation. By the process illustrated above, of constructing a curve from its equation, the first con- ception of a locus is obtained, viz. : 38 THE LOCUS AND THE EQUATION [Ch. Ill (1) The locus of an equation containing two variables is the line, or set of lines, which contains all the points whose coordinates satisfy the given equation, and lohich contains no other points. It is the place ivhere all the points, and only those points, are found whose coordinates satisfy the given equation. A second conception of the locus of an equation comes directly from, this one, for the line or set of lines may be regarded as the pr/f/i traced by a point which moves along it. The path of the moving point is determined by the condition that its coordinates for every position through which it passes must satisfy the given equation. Thus the line EF [the locus of eq. (3), Art. 27] may be regarded as the path traced by the point P, which moves so that its coordinates (a;, y) always sat- isfy the equation 3a; — 2y + 12 = 0. Thus arises a second conception of a locus, viz. : (2) The locus of an equation is tlie path traced by a point which mooes so that its coordinates always satisfy the given equation. In either conception of a locus, the essential condition that a point shall lie on the locus of a given equation is, that the coordinates of the point when substituted respectively for the varia- bles of the equation, shall satisfy the equation ; and in order that a curve may be the locus of an equation, it is necessary that there be no other points than those of this curve whose coordinates satisfy the equation. 29. Classification of locL The form of a locus depends upon the nature of its equation ; the curve may therefore be classi- fied according to its equation, an algebraic curve being one whose equation is algebraic, and a transcendental curve one whose equation is transcendental. In particular, the degree of an algebraic curve is defined to be the same as the degree of its equation. The following pages are concerned chiefly with algebraic curves of the first and second degrees. 28-30] THE LOCUS AND THE EQUATION 39 30. Construction of loci. Discussion of equations. The process of constructing a locus by plotting separate points, and then connecting them by a smooth curve, is only approxi- mate, and is long and tedious. It may often be shortened by a consideration of the peculiarities of the given equation, such as symmetry, the limiting values of the variables for which both are real, etc. Such considerations will often show the general form and limitations of the curve ; and, taken together, they constitute a discussion of the equation. The points where a locus crosses the coordinate axes are almost always useful ; they are given by their distances from tiie origin along the respective axes, called the intercepts of the curve. The following examples may serve to illustrate these con- ceptions. (1) Discussion of the equation 3a! — 2y + 12 = 0. Intercepts: if a; = 0, then «/ = 6; hence the ?/-intercept is 6 (see Fig. 17); if y=0, then x— —4; hence the ic-intercept is —4. The equation may be written : « = l2/-4, which shows that as y increases continuously from to oo , a; in- creases continuously from — 4 to co ; therefore the locus passes from the point Pg through the point P^, and then recedes indefi- nitely from both axes in the first quadrant. Written as above, the equation also shows that as y decreases from to — X , X also decreases from — 4 to — oo ; therefore the locus passes from Pg into the third quadrant, receding again indefinitely from both axes. Since for every value of y, x takes but one value (i.e., each value of y corresponds to but one point on the curve), therefore the locus consists of a single branch. The proof that the locus of any first-degree equation, in two variables, ^3 a straight line is given in Cliap. IV. 40 THE LOCUS AND THE EQUAITON [Ch. Ill (2) Disciission of the equation a? + y- = a^. Intercepts: if x=0, then y= ±a, and if y=0, then x=±a; lience for each axis there are tivo intercepts, each of length a, and on opposite sides of the origin ; i.e., four positions of the tracing point are: A=(a,0), A'={-a,0), B = {0, a), and B' = (0, "a). This equation may also be written Fio. 18. y = ± Va^ — x', which shows that every value of x gives two corresponding values of y which are numerically equal, but of opposite sign ; the locus is, there- fore, symmetrical with regard to the a>axis. It shows also that, corresponding to any value of x numeri- cally greater than a, y is imaginary; the tracing point, there- fore, does not move farther from the 2/-axis than ± a, i.e., farther than the points A and A'. Moreover, as x increases from to ■ a, y remains real and changes gradually from -)- a to 0, or from — a to ; i.e., the tracing point moves continuously from B to A, or from B' to A. Again, if x decreases from to — a, y remains real and changes continuously from -f- a to 0, or from — a to ; i.e., the tracing point moves continuously from B to A' or from B to A'. Similarly, the equation may be written x = ± Va'^ — y-, which shows that the curve is symmetrical with regard to the j/-axis, and that the tracing point does not move farther than ± a from the a>axis. From these facts it follows that this locus is a closed curve of only one branch. It is a circle of radius a, with its center at the origin ; this curve will be studied in detail in Chap. V. 30] THE LOCUS AND THE EQUATION 41 (3) Given the equation i/' = i x, to find its locus. This equa^ tion may be solved by the forms y= ± VJ^, and X--, from which it is clear that for each value given to a; there are two values of y, numerically equal but of opposite sign ; hence the locus is symmetrical with respect to the x-axis ; while for each value given to y corresponds only one value of x. Again, for negative values of x there are no corresponding real values for y. Hence the locus lies wholly on the positive side of the 2/-axis. The intercepts are both zero. The fol- lowing points are points of the locus : P,= (0,0),P,= (1,2), P3S(1, -2),P, = (2,2.8...), P, = (2, -2.8...), Po = (4, 4), andP„=(+oo, ±Qc). All these points are found to lie on the curve as plotted in Fig. 19. This curve is called a parabola, and will be studied in a later chapter. The parabola is one of the curves obtained by the intersec- tion of a circular cone and a plane. It will be shown in Chap. VIII that in Cartesian coordinates the locus of any algebraic equation in two variables and of the second degree is a " conic section." (4) Discussion of the equation y' = (x — 2)(x — 3){x — 4). Intercepts : if x = 0, then y is imaginary ; if t/ = 0, then a; = 2, 3, or 4 ; hence the locus crosses the a>axis at the three points A = {2, 0), B = {3, 0), and C=(4, 0), and it does not Fig ly. 42 THE LOCUS AND THE EQUATION [Ch. in cut the y-axis at all. Moreover, since y is imaginary if x is negative, the locus lies wholly on the positive side of the j/-axis. This locus is symmetri- cal -with regard to the a:-axis ; it has no point nearer to the y-axis than A; between A and B it consists of a closed branch ; and it has no real points between B and C, but is again real beyond C. The entire locus con- sists, then, of a closed oval, and of an open branch which recedes in- definitely from both axes. (See Fig. 20.) In discussing an equation it is well to consider (1) intercepts; (2) symmetry, with respect to each axis; (3) limiting values, if any, of each variable; (4) whether the locus is a closed or open curve. EXERCISES Construct and discuss the loci of the following equations : ^- f-I=^- ^- f + 1 = ^- *• ^+^' =«• 2. 3i?-f = ^a\ 4. a?-f = Q. 6. ar'-42/^ = 0. 31. The equation of a locus. The second fundamental prob- lem of analytic geometry is the reverse of the first [cf. Art. 26], and is usually more difBcult. It is to find, for a given geo- Fio. 20. 30-32] THE LOCUS AND THE EQUATION 43 metric figure, or locus, the corresponding equation, i.e., the equation which shall be satisfied by the coordinates of every point of the given locus, and which shall not be satisfied by the coordinates of any other point. The geometric figure may be given in two ways, viz. : (1) As a figure with certain known properties ; and (2) As the path of a point which moves under known con- ditions. In the latter case the path is usually unknown, and the com- plete problem is, first to find the equation of the path, and then from this equation to find the properties of the curve. The two ways by which a locus may be " given " correspond to the two conceptions of a locus mentioned in Art. 28, and they lead to somewhat different methods of obtaining the equation. The first method may be exemplified clearly by considering the familiar cases of the straight line and the circle. 32. Equation of a locus with known properties. LetP,= (3, 2) and P2= (12, 5) be two given points; and let P = (x, y) be any other point in the straight line through Pj and P,. Y >^ P^_ Jr^ i. ^ ;B iR. ^ : X M, M Fio. 21 a. M, The slope of the line is the same, whatever two points are taken to determine it ; hence , _ ya - ?/i _ y-?/i [Art. 20] 44 THE LOCUS AND THK EQUATION [Oil. Ill On substituting the respective values for the coordinates, 5- 12- _ o = ?'— X — 2 .3' -X- -3 = 0. which reduces to 3 ?/ — x — 3 = 0. (1 ) This is the required equation of the straight line through P, and Pj, because it fulfills both the requirements of the definition [cf. Art. 28 (1)] ; i.e., it is satisfied by the coordi- nates of any (i.e., of every) point of this line, because x, y are the coordinates of any such point ; and it is not satisfied by the coordinates of any point which is not on this line ; for let P,^(x„y,) be any point not on the line through P, and Pj, the ordinate M^P^ will T7.„ on meet PiPj m some point Pt=(xt, 2/i), for which x^^x^ but y^^^ys- Since P, is on the line PiPi, its coordinates satisfy eq. (1), therefore -3 32/,- ■3 = 0, and y, = Xi- j/3^^4^, sothat3y3-a;3-3^0; i.e., the coordinates of P3 do not satisfy the equation 3 jf-a;-3 = 0. NoTB. If the coordinates of any point on the straight line P1P2 are suhstituted for x and y in eq. (1), the first member will be equal to zero. Substituting the coordinates of a point on one side of the line, the first member will be negative ; of a point on the other side, it will be positive. The line is the boundary which separates the part of the plane for which 3 3/ — a; — 3 is negative, the negative side of the line, from the part for which this function is positive, the positive, side. To determine the posi- tive side of the line, the x term is usually taken positive. 32, 33] TIIK LOCUS AND THE EQUATION 45 Again, let P={x, y) be any point on a circle of radius 5 about the origin as center. Then, by the well-known prop- erty of right triangles, «.e. 0^ + 2/^=25; and this equation is not satisfied by the coordinates of any point not on the circle. Hence this is the equation of the circle. Fia. 23. EXERCISES 1. Find the equation of the straight line through the two points (~1, ~7) and (11, 6) ; through the points (2, S) and (8, 3). 2. Find the equation of the straight line through the two points (4, 6) and (~4, ~6). Through what other point does this line pass ? Does the equation show this fact ? 3. Find the equation of the straight line through the point (10, ~14), and making an angle of 4i5° with the a;-axis ; making the angle — 45° with the aj-axis. On which side is the origin ? 4. Find the equation of the line through the point (6, 2), and making the angle 150° with the a^axis. 5. With rectangular coordinates, find the equation of the circle of radius 5, which passes through the origin, and has its center on the y-axis. Is its positive side outside or inside ? 33. Equation of a locus traced by a moving point. In the exer- cises given above, the geometric figure in each case was com- pletely known; and, in obtaining its equation, use was made of the known properties of straight lines, similar triangles, right triangles, and trigonometric functions. In only a few eases. 46 THE LOCUS AND THE EQUATION [Cii. Ill however, is the curve so completely known ; in a large class of important problems, the curve is known merely as the path traced by a point which moves under given conditions or laws. Such a curve, for instance, is the path of a cannon ball, or other projectile, moving under the influence of a known initial force and the force of gravity. Another such curve is that in which iron filings arrange themselves when acted upon by known magnetic forces. The orbits of the planets and other astro- nomical bodies, acting under the influence of certain centers of force, are important examples of this class of " given loci." In such problems as these, the method used in Art. 32 can- not, in general, be applied. A method that can often be em- ployed, after the construction of an appropriate figure, is : (1) From the figure, express the known law, under which the point moves, by means of an equation involving geometric magni- tudes ; this may be called the "geometric equation." (2) Find for each geometric magnitude used an equivalent alge- braic value, expressed in terms of the coordinates of the moving point and given constants. (3) Replace each geometric magn itude by its equ ivalent algebraic value, thus obtaining the " algebraic equation." (4) Simplify the algebraic equation. The result is "the" equation of the locus. 34. Equation of a circle : second method. To illustrate this second method of finding the equation of a locus, consider the circle as the path traced by a point which moves so that it is always at a given constant distance from a fixed point. From this definition, find its equation. C Let C = (3, 2) be the given fixed ^s^^ point, and let P=(x, y) be a point ^ that moves so as to be always at the Fig. 2i. distance 2^ from C. 33-35] THE LOCUS AND THE EQUATION 47 Then CP=^, [geometric equation] but CP = V(a; - 3y+{y- 2)\ [algebraic value] V(»— 3)*+ (3/ — 2/= I; [algebraic equation] f.&, (a. _ 3)'+ (2, -2)' = 1^5 hence 4a;» + 4y'' — 24a? — 16y+27 = 0, whicli is the required equation of the locus. EXERCISES 1. Find the equation of the path traced by a point which moves so that it is always at the distance 5 from the point (4, 0). Trace the locus. 2. Find the equation of the path traced by a point which moves so that it is always equidistant from, the points (~2, 3) and (7, 5). On which side is the origin ? 3. A line is 3 units long ; one end is at the point (2, -3). Find the locus of the other end. 4. A point moves so as to be always equidistant from the y-axis and from the point (8, 0). Find the equation of its path, and then trace and discuss the locus from its equation. 5. A point moves so that the sum of its distances from the two points (0, Vo), (0, ~V5) is always equal to 8. Find the equation of the locus traced by this moving point. 6. A point moves so that the difference of its distances from the two points (0, V5), (0, ~V5) is always equal to 2. Find the equation of the locus traced by this moving point. 35. General Theorems. A few general theorems about loci and their equations are of constant use. The locus of an equation remains unchanged : (a) by any trans- position of the terms of the equation ; and (P) by multiplying both members of the equation by any finite constant. 48 THE LOCUS AND THE EQUATION [Ch. Ill (a) If in any equation the terms are transposed from one member to the other in any way whatever, the locus of the equation is not changed thereby ; for the coordinates of all the points which satisfied the equation in its original form, and only those coordinates, satisfy it after the transpositions are made. [See Art. 28 (1)]. (j8) If both members of an equation are multiplied by any finite constant k, its locus is not changed thereby. For if the terms of the equation, after the multiplication has been per- formed, are all transposed to the first member, that member may be written as the product of the constant k and a factor containing the variables. This product will vanish if, and only if, its second factor vanishes ; but this factor will vanish if, and only if, the variables which it contains are the coordinates of points on the locus of the original equation. Hence the co- ordinates of all points on the locus of the original equation, and only those coordinates, satisfy the equation after it has been multiplied by k; hence the locus remains unchanged if its equation is multiplied by a finite constant. E.g., (a) 3!c' + 2f-2x + 3 = and 3a^ + 2y^=2x-B have the same locus. (6) 5a;-7y-f6 = and 15a;-21 2/-f 18 = have the same locus. Note. The theorem of this article is of great service in studying the properties of curves represented by equations. Certain standard equations are studied thoroughly, and the properties of the corresponding curves found. Other equations shown to differ from the standard equations by a constant factor only, are then also completely known. 36. Points of intersection of two loci. Since the points of intersection of two loci are points on eadi locus, therefore the coordinates of these points must satisfy each of the two equa- tions; moreover, the coordinates of no other points can satisfy both equations. Hence, to find the coordiuates of the points of 35, 30] THE LOCUS AND THE EQUATION 49 intersection of two curves, regard their equations as simultaneous and solve. E.g., Find the coordinates of the points of intersec- tion, Pi and Pji of the loci of x —2y = 0, and y' = x. The point of intersection Pi = (xi, y,) is on both curves, •■• a;i-2yi=0, and yi= xi.* Solving these two eqiia- tions X, = 0, or 4, and 2/i = 0, or 2; i.e., ^. = (4,2) and P,= (0,0) Fig. 25. are two points, the coordinates of which satisfy each of the two given equations ; therefore they are the points of intersection of the loci of these equations. EXERCISES Find the points of intersection of the following pairs of curves : 3. j3x-4?/ + 16 = 0, 3x-y-7 = 0. 2_ ix + y = i, 1 a; — V = 4. \y = 3x+2, I a,-2 + r = 4. (2y-5x = 0, g la? + f = 9, [aP + 6xy + y^ = 0. 6. < \.2xy = 9. * II X and y are regarded as the coordinates of the point of intersec- tion, the subscripts may be omitted here. 50 THE LOCUS AND THE EQUATION [Ch. Ill 7. J2/- = 4a;, i y — a; = 0. fa;= + 2/^ = 16, -( x" = 4 y, 2/- -a; = 8. 37. Product of two or more equations. Given two or more equations ivith their second members zero ; * the product of their first members, equated to zero, has for its locus the combined loci of the given equations. This follows at once from the fundamental relation between an equation and its locus [see Art. 28 (1)], by reasoning as illustrated in the following example. Let x + y = Q (1) and x~y = Q (2) be two given equations, whose loci are respectively the straight lines CD and AB, bisecting the angles between the axes. To show that the equation ix-lry){x-y)=0 i.e., a?-f = (3) has for its locus both these lines. Proof. If Pi = (a^ y,) is any point on CD, then a^ + ^1 = 0, ^■°-26- hence (x,+yi}{x,-y^) = 0; therefore P, is a point on the locus of equation (3). * 1i equations whose second members are not zero are multiplied to- gether, member by member, the resulting equation is not In general satis- fied by any points of the loci of the given equations except those in which they intersect each other ; the new equation therefore represents a locus through the points of intersection of the loci of the given equations. 36-38] THE LOCUS AND THE EQUATION 51 But Pj is any point on CD, hence every point on CD is a point of the locus of equation (3). In the same way it is shown that AB belongs to the locus of equation (3). Moreover, if P^ — (x^, y^ be any point not on AB nor on CD, then X3 + 2/3 ^i: 0, and x^ — y^^f^ 0, hence (a;3 + 2/3)(a'3-2/3)=?s=0; i.e., Pg does not belong to the locus of equation (3). Hence the locus of equation (3) contains the loci of equation (1) and (2), but contains no other points. The above theorem may be stated briefly thus : if «, v, w, etc., be any functions of two variables, then the equation uvw • ■•■ =0 has for its locus the combined loci of the equa- tions M = 0, v = 0, w = 0, etc. ; and conversely. Note. When possible, factoring the first member of an equation, whose second member is zero, simplifies the work of finding the locus of the given equation. EXERCISES What loci axe represented by the following equations ? 1. ^ = 0. 2. f-f = 0. 3. 3x' + 2xy-7x=0. 4. 2xy''-5x'y =0. 5. ar' + 4x-|-4 = 0. 6. (ar" + 2/^ - 10)(/ - 10 ar^ = C. 38. Locus represented by the sum of two equations. Given the equations ^^_^y_^^Q ^^^ x + 2y-7=0. representing respectively the loci AB and CD. Required to find the locus of their sum, i.e., of 4a;-2y-8 = 0. 52 THE LOCUS AND THE EQUATION [Cii. Ill Plotting, the locus LM is found to pass through the point of intersection P of the two given loci (Fig. 27). Again, given equations 2 2/ — a; = and (1) 2/=-a; = (2) representing respectively the loci AB and CP^P.D which intersect in P, and Ps (Fig. 25). Then the locus of the equation f-x+{2y-x)=Q, Fig. 27. i.e., y"-+2y—2x=Q, (3) must represent some curve through Pj and P^. For, let Pi=(x„ y^ be one of the common points; then 2^1 — !ri = and ^i* — 351 = 0, since P| is on each curve. yi-Xi-\-2yi-Xi — 0, i.e., Pi satisfies equation (3), and therefore the locus of (3) passes through P,. Similar reasoning would show that the locus of equation (3) passes through every other point in which the loci of equations (1) and (2) intersect each other. In pre- cisely the same way it may be proved generally that the locus of the Slim of two equations passes through all the points in which the loci of the two given equations intersect each other. If either of the given equations (1) or (2) had been multi- plied by any constant factor before adding, the above reason- ing would still have led to the same conclusion ; in fact, this theorem may be briefly, and more generally, stated thus : if u and V are any functions of the two variables x and y, anO, k is any constant, then the locus of u + Tcv = 38, 39] THE LOCUS ASD THE EQUATION 53 passes through every point of intersection of the loci of ti = and v = 0. Note, This theorem is of great service in finding the points of inter- section of the loci of two equations, as in Art. 36. E.g., the points of intersection of the loci of equations (1) and (2) above, are also on the locus of i/i-x-{2y-x)=0, i.e.,ot y^-2y = 0, i.e., y(y-2) = 0, hence are on the lines y = 2 and y = 0. Therefore, the points are Pi =(4, 2) and Pa =(0, 0). Geometrically, this means that instead of fixing the points by the given parabola and oblique straight line, they are fixed by the straight lines parallel to the axes. [Cf. Art. 144]. 39. Fundamental problems of analytic geometry. The elemen- tary applications already considered have indicated how alge- bra may be applied to the solution of geometric problems. Points in a plane have been identified with pairs of numbers, — the coordinates of those points, — and it has been seen that definite relations between such points correspond to definite re- lations between their coordinates. It has been found also that the relation between points, which consists in their lying on a definite curve, corresponds to the relation between their coordinates, which consists in their satisfying a definite equation. From this fact have arisen the two fundamental problems of analytic geometry : I. Given an equation, to find the corresponding geometric curve, or locus. II. Given a geometric curve, to find the corresponding equation. From this relation between a curve and its equation a third problem arises: III. To find the properties of the curve from, those of its equation. 54 THE LOCUS AND THE EQUATION [Cu. Ill The remaining chapters of Part I will be concerned chiefly with the third problem. In this application of analytic methods, however, only algebraic equations of the first and second degrees will for the most part be considered. EXERCISES 1. Verify Art. 38 by first finding the coordinates of the points of intersection of the loci of equations (1) and (2), and then substituting these coordinates in equation (3). 2. Find the equation of a curve that passes through all the points in which the following pairs of curves intersect ; ^ ^\a^ + 2x + y' = 0. j ^^ ly=cosa!. ] ^'^ [y' = 4:X. EXAMPLES ON CHAPTER III 1. Are the points (~3, 17), (4, 6), and (5, 5) on the locus of 3x + 2y = 25? 2. Is the point f% 5 j on the locus of 9 a^ + 4 y» = 2 a« ? 3. The ordinate of a certain point on the locus of aP+y^ =25 is 3 ; what is its abscissa? What is the ordinate if the abscissa is a ? Find by the method of Art. 36 where the following loci cut the axes of x and y: 4. y={x+2)(x + 3). 5. 16a!«-9/ = 144. 6. ar' + 6a; + y» = 4y + 3. 7. The two loci ^=1. -r+^=i intersect in four 4 9 4 9 points ; find the lengths of the sides and of the diagonals of the quadrilateral formed by these points. 8. A triangle is formed by the points of intersection of the loci of x+y=a, x—2y=ia, and y—x+7a=0. Find its area. 39] THE LOCUS AND THE EQUATION 55 9. Find the distance between the points of intersection of the curves 3x — 2y + 6 = 0, and ar" + y' = 9. 10. Does the locus of y-=:5x intersect the locus of 9 x — 6 .v + 5 = 0? 11. Does the locus of a^ — 4^ + 4 = cut the locus of :x? + f=l? 12. For what values of m will the curves ar + j/^ = 9 and y = 6x+7n not intersect ? [Cf. Art. 6]. Trace these curves. 13. For what value of 6 will the curves j/' = 4 » and y = X + 6 intersect in two distinct points ? in two coincident points ? in two imaginary points (i.e., not intersect) ? 14. Find those two values of c for which the points of inter- section of the curves y = 3x + c and a^ + y' = 3(i are coincident. 15. Find the equation of a curve that passes through all the points of intersection of x' + y' = 25 and j/" = 4 x. Test the correctness of the result by finding the coordinates of the points of intersection and substituting them in the equation just found. 16. Write an equation that shall represent the combined loci of {1), (2), and (3) of Art. 30. Discuss and construct the loci of the equations : 17. (x' — y^(y — smx} = 0. 20. y = 4:a^. 18. a? — f — 0. 21. f = a^. 19. a!' — y* = 0. 22. ^ = 10". 23. Show that the following pairs of curves intersect each other in two coincident points ; i.e., are tangent to each other ; (a) f-8x-6y-63 = 0, 2y-x = 23. (j3) 9ar'-42/^ + 54x-162/ + 29 = 0, • " 15 X - 8 2/ + 11 = 0. 56 THE LOCUS AND THE EQUATION [Cii. Ill 24. Find the points of intersection of the curves ^ + 2^ = 1 and t-^^1. 9 25 9 25 25. Find the intersection of y = a* and a; = 0. Plot. 26. Find the equations of the sides of the triangle whose vertices are the points (2, 3), (4, -5), (3, "6). [Cf. Art. 32]. Test the resulting equations by substitution of the given coordinates. 27. Find the equations of the sides of the square whose vertices are (~1, 0), (1, 2), (3, 0), (1, ~2). Compare the equa- tions of the parallel sides ; of perpendicular sides. 28. Find the coordinates of the center of the square in Ex. 27. Then find the radius of the circumscribed circle, and (Art. 34) the equation of that circle. Test the result by find- ing the coordinates of the points of intersection of one of the sides with the circle. [Art. 36]. 29. Find the equation of the path traced by a point which is always equidistant from the points (a) (4, 0) and (0, -4); (fi) (6, 4) and (3, 3); (y) (« + 6> a — b) and (a — b, a+ 6). 30. A point moves so that its abscissa always exceeds -f of its ordinate by 6. Find the equation of its locus, and trace the curve. Which side is positive ? [Note, p. 44.] 31. A point moves so that the square of its abscissa is always 4 times its ordinate. Find the equation of its locus and trace the curve. 32. Find the equation of the locus of a point which moves 80 that the sura of its distances from the points (~1, 3) and (7, 3) is always 3 0. Trace and discuss the curve. 39] THE LOCUS AND THE EQUATION 57 33. Find the equation of the locus of the point in Ex. 32, if the difference of its distances from the fixed points is always 2. 34. Express by a single equation the fact that a point moves so that its distance from the y-axis is always numeri- cally 3 times its distance from the o^axis. 35. A point moves so that the square of its distance from the point (a, 0) is 4 times its ordinate. Eind the equation of its locus, and trace the curve. 36. A point moves so that its distance from the ^-axis is \ of its distance from the origin. Find the equation of its locus, and trace the curve. Which side is positive ? 37. A point moves so that the difference of the squares of its distances from the points (1, 3) and (4, 2) is 5. Find the equation of its locus and trace the curve. 38. Solve Ex. 37 if the word "sum" is substituted for " difference." 39. Let A = {a, 0), B = {b, 0), and A' = {-a, 0) be three fixed points; find the equation of the locus of the point P = {x, y) which moves so that FB^ + p2^ = 2 PZ'. 40. A point moves so that \ of its ordinate exceeds ^ of its abscissa by 1. Find the equation of its locus and trace the curve. 41. Find the equation of the locus of a point that is always equidistant from the points (3, "4) and (5, 3) ; from the points (3, -4) and (2, 0). By means of these two equations find the coordinates of the point that is equidistant from the three given points. 68 THE LOCUS AND THE EQUATION [Ch. Ill 42. 'LetA = (-l, 3), B = (-3, "3), C = (l, 2), Z>=(2, 2) be four fixed points, and let P={x, y) be a point that moves sub- ject to the condition that the triangles PAB and PCD are always equal in area ; find the equation of the locus of P. 43. If the area of a triangle is 25 and two of its vertices are (5, "6) and (~3, 4), fiud the equation of the locus of the third vertex. 44. Find the equation of the straight line through the points (1, 7) and (6, 11), using the fact that three points of a straight line determine a triangle whose area is zero. CHAPTER IV THE STRAIGHT LINE. EQUATION OF FIRST DEGREE Ax +By+C=0 40. In Chapter III it was shown that to every equation between two variables there corresponds a definite geometric locus, and that if the geometric locus be given, its equation may be found. It will now be helpful to consider some of the more elementary loci and their equations, and to apply ana- lytic methods to the study of the properties of these curves. Since the straight line is a simple locus, and one whose prop- erties are already well understood by the student, its equa- tion will be examined first. In studying the straight line, as well as the circle and other second degree curves, to be taken up in later chapters, it will be found best first to obtain the simplest equation which represents the locus, and to study the properties of the curve from that simple or standard equation ; then to find methods for reducing to this standard form any other equation that represents the same locus. 41. Equation of straight line through two given points. The method of the present article is precisely the same as that illustrated by a numerical example in Art. 32. Let two given fixed points be Pi = (x^, yi), and P., = (x^, i/j), and let P = (x, y) be any other point on the straight line through Pi and P^ 69 60 THE STRAIGHT LINE [Ch. IV By definition of 'straight' line, the slope of the line remains constant; hence yzL2i = yi:zl., [Art. 20] SO ~~ iCl Xq iCl which may be written y-Vi x-ac-i Vi ~ 2/1 ^2 — •*'! [8] Since P = (x, y) is any point on the line through P-^ and P2, therefore equation [8] is satisfied by the coordinates of every O M, M M, Fio. 28 a. Fig. 286 point on this line. That equation [8] is not satisfied by the coordinates of any point except such as are on the line PyP^ may be proved as was done in Art. 32. Equation [8] then fulfills both requirements of the definition in (1) of Art. 28 and is therefore the equation of the straight line through the two points (wj, y-^ and (kj, y^. This equation will be frequently needed and will be referred to as a standard form; it should be committed to memory. Note. Since eq. [8] states a property of similar triangles which is mot dependent upon the angle XOY, this equation is true for oblique 42. Equation of straight line in terms of the intercepts which it makes on the coordinate axes. If the two given points in Art. 41 41,42] EQUATION OF FIRST DEGREE 61 are those in which the liue cuts the axes of coordinates, i.e., A = (a, 0) and B = (0, 6) (Fig. 29 o), then equation [8] becomes y — _ X — a _ 6_0"~0-a' that is, - +— = 1 [9] where a and b are the intercepts which the line cuts from the axes (where it is assumed a^O, b^O). r Xf- V, B ^^ x^x- Fig. 29 a. M Fig. 29 6 This is another standard form of the equation of the straight line; it is known as the symmetrical or the intercept form. Note. In equation [9] both x and y are variables, but are not inde- pendent ; eacli is an implicit function of the other. For any particular line a and 6 are constants, but they may represent other constants in the equation of another line, i.e. , they are arbitrary constants, and are often called parameters of the line. EXERCISES 1. Show that equation [9] is not satisfied by the coordinates of any point except those lying on MN. 2. Write down the equations of the lines through the follow- ing pairs of points : («) (3, 4) and (4, 2); (y) ("6, 1) and ("2, "5); (P) (3, 4) and (4, -2) ; (S) ("15, "3) and (^|, y')- 62 THE STRAIGHT LINE [Ch. IV 3. Write the equations of the lines which make the follow- ing intercepts on the x- and jz-axes respectively. 1 4 (a) 7and4; (y) -- and + -; 03) -6 and 10; (8) -3a and -^. 4. What do equations [8] and [9] become if one of the given points is the origin ? 5. By drawing, in Fig. 29 b, a perpendicular PQ from P to the X-axis, derive equation [9] from the similar triangles QAP and OAB. 6. Is equation [9] true if P is on MN but not between A and £ in Fig. 296? 7. Are equations [8] and [9] true if the coordinate axes are not at right angles to each other ? Show by similar triangles. 8. Is the point (3, 3) on the line through the points (2, 3) and (5, 7) ? On which side of this line is it ? Which is the negative side of this line ? 9. What intercepts does the line through the points (~6, 1) and (5, ~3) make on the axes ? 10. The vertices of a triangle are : (8, "10), (4, 6), and (6, -12). Find the equations of the sides ; also of the three medians ; then find the coordinates of the point of intersection of two of these medians, and show that these coordinates satisfy the equation of the other median. What proposition of plane geometry is thus proved ? 11. Find the tangent of the angle [the " slope," of. Art. 20] ■which the line in Ex. 9 makes with the x-axis. 12. Draw the line whose equation is - + ^ = 1, and then find the equations of the two lines which pass through the origin and trisect that portion of this line which lies in the first quadrant. 42,43] EQUATION OF FIRST DEGREE 63 43. Equation of straight line through a given point and in a given direction. Let P, = (xj, y^ be the given point, and let the direction of the line be given y by the angle XAP = d which the line makes with the aj-axis; also let P= (x, y) be any point on ^ ^,*^ the given line and denote ^'"^ the slope, i.e., tan 6, by m. Using again the fact that the slope of the line is constant, m = ^LlJh., [Art. 20] that is, V-yi = m{x — x{), [lO] which is the desired equation. Corollary. If the given point be 5 = (0, 6), t.e., the point in which the line meets the ^-axis, then equation [10] becomes y = mx + b. [ll] Equation [11] is usually spoken of as the slope form of the equation of the straight line. EXERCISES 1. What do the constants m and 6 in equation [11] mean ? Draw the line for which m = 3 and 6 = 4, also that for which m = — 1 and 6 = — |. 2. What is the effect on the line represented by eq. [11] if 6 is changed while m remains the same ? What if m be changed and h left unchanged? 3. Describe the effect on the line [10] of changing m while Xi and ^1 remain the same; also the effect resulting from a change in a-, while m and y^ remain the same. 64 THE STRAIGHT LINE [Ch. IV 4. Write the equation of a line through the point ("3, 7), and making with the aj-axis an angle of 30°; of ~60°; of r) Ct)"'-'(-t)"' 5. Write the equations of the following lines : (a) slope 4, j^-intereept 4 ; (fi) slope \, ^-intercept "3 ; (y) slope ~2, a!-intercept ~|. 6. A line has the slope ~6; what is its y-intercept if it passes through the point (7, 1) ? 7. What must be tlie slope of a line whose y-intercept is +3, in order that it may pass through the point ("5, 5) ? 8. Is the point (1, ^) on the line passing through the point (0, ~3), and making an angle tan"' ^- with the-avaxis? 9. How do the lines y = 3x—\, y = Sx + 7, and ky — 3 kx + 15 = differ from each other ? What have they in common ? Draw these lines. 10. What is common to the lines y=3x — l, 2y=5x —2, and 7 a; -33/ = 3? 11. What is the slope of line [8], p. 60 ? of line [9], p. 61 ? 12. Derive equation [11 j, p. 63, independently of equation [10], p. 63. 44. Equation of straight line in terms of the perpendicular from the origin upon it and the angle which that perpendicular makes Fig. 31a. ^^" Fio. 31 6. with the X-axis. Let KII be the line whose equation is sought. and let the perpendicular (ON=i}) from O upon this line. 43,44] EQUATION OF FIRST DEGREE 65 and the angle (a) which this perpendicular makes with the X-axis, be given. Also let P = {x, y) be any point on KH ; then by projection upon ON [Art. 13], OM cos (a) + MP oos(j-v^= ON, i.e., a;cosa-(-ysino=j», [l2] which is the required equation. Equation [12] is known as the normal form of the equation of the straight line. In the following pages a will, unless otherwise stated,* be regarded as positive, and less than 180°. Corollary. If ^ is the angle to p from the y-axis, equation [12] takes the symmetrical form a; cos «+ y cos fi=p. Simi- larly, if a and /3 are respectively the angles which the line makes with the x and y axes, equation [10] takes the symmet- rical form x-Xj ^ y-yi cos a cos /3 [Compare Arts. 148 and 161]. EXERCISES 1. The perpendicular from the origin upon a certain line is 6 ; this perpendicular makes an angle of ^ with the x-axis ; what is the equation of the line ? 2. If in equation [12] p is increased while a remains the same, what is the effect upon the line ? If a be changed while p remains the same, what is the effect ? 3. A certain line is 4 units distant from the origin, and makes an angle of 120° with the a;-axis; what is its equation? *Cf. Art. 51. 66 THE STRAIGHT LINE [Ch. IV 4. Given a = 45°, what must be the length of p in order that the line HK (see Fig. 31 a) shall pass through the point (2, 7) ? 5. A line passes through the point (~4, ~3), and a perpen- dicular upon it from the origin makes an angle of 225° with the a!-axis. What is the equation of this line ? 6. A line passes through the origin with the slope m = 1. What is the value of a ? of j) ? 7. If two parallel lines are on opposite sides of the origin do they have the same angle a ? Explain. 8. State the parameters [cf. note, Art. 42] for each of the standard equations of the straight line. 9. Show by projection that the distance d of Pi = (x^, yi) from the line Fio. 32. IS X cos a + y sin a =p d = a^ cos a + yi sin a — p. 45. Summary. The results of Arts. 40 to 44 may be briefly summarized as follows : The position of a straight line is determined by I. Two points through which it passes. Eq. [8], p. 60 Special case : cue point on the o^axis, the other upon the ^-axis. Eq. [9], p. 61 II. One point and its direction. Eq. [10], p. 63 Special cases : (1) the given point on the y-axis, Eq. [11], p. 63 (2) the point given by its distance and direction from the origin, and the direction of the line being given by the per- pendicular to it from the origin. Ea. ri21. n. fi.*; 44-46] EQUATION OF FIRST DEGREE 67 Of the five standard forms of the equation of the straight line, [8] and [9] are independent of this angle between the coordinate axes,* while [10], [11], and [12] (to, a, and p retain- ing their present meanings) are true only when the axes are rectangular. It may also be pointed out that, from the nature of its derivation, equation [8] is inapplicable when the line is parallel to either axis ; equation [9] is inapplicable when the line passes through the origin ; and equations [10] and [11] are not applicable when the line is parallel to the ^-axis. 46. Reduction of the general equation Ax + By + C = O to the standard forms. Determination of a, b, m, p, and a in terms of A, B, and C. The most general equation of the first degree in two vari. ables may be written in the form where A, B, and C are constants, with A positive, and neither A nor B is zero.f (1) Reduction to the standard form - + 2 = 1 (^symmetric or . . a intercept form). The equation Ax + By + C=0 (1) represents some locus which is not changed by multiplication by a constant, and transposition [Art. 35] ; therefore -^ + -f^=U (2) A B * See Art. 41, Note. t If either A or B, say A, is zero, then the equation may be written in the form : j^ = — — , which is the equation of a straight line parallel to the ^ C K-axis, and at the distance from it [of. Art. 27, (2)]. B X Where C ^fc 0. See Art. 42. 68 THE STBAIGHT LINE [Cb. IV represents the same locus. But equation [2] is in the stand- ard form for a straight line [Art. 42], and the intercepts are : C J r G a= , and o= — — . A' B These intercepts are in practice readily found by substitut- ing y = 0, and X = 0, respectively, in the given equation. Since this demonstration does not depend upon the angle between the axes, therefore it applies whether the axes are oblique or rectangular ; hence the theorem : Every equation of the first degree between two variables, when interpreted m Cartesian coordinates, represents a straight line. Because of this fact, such an equation is often spoken of as a linear equation. Note. In the equation Ax + Sy + C = 0, there are apparently three constants; in reality, there are but two independent constants, viz. the ratios of the coefficients. [Cf. Art. 35]. This corresponds to the fact that a straight line is determined geometrically by two conditions. The equa- tion may be written in the form Ax-{- By + 1 =0, wherein A and B may be fractions. (2) Reduction to the standard form y = mx + b (slope form). The equation Ax -\-By+C=0 has the same locus as has the [see Art. 35]; hence, by comparison with equation (11) ^ Ah G ni = --,and & = -— ; and eauation (3) is of the required form. * If B = 0, the line represented by equation [1] is parallel to the y-axis, and the slope form of the equation is inapplicable [Art. 45]; in that case, this reduction also fails. 40, 47] EQUATION OF FIRST DEGREE 69 (3) Reduction to the standard form x cos a-\-y sin a =p (normal form). If any two numbers A and B be taken as the lengths of the legs of a right triangle, then the hypotenuse makes with the leg A an angle a such that and sin a = - V^^+W \'A' + ^ Now, divide equation [1] through by ■y/A' + B'; and trans- pose the constant term : :X + B y=- -0 ^A^ + B^ -yjA^ + B? V^'-l-^ (4) This is therefore of the form required, -C ... A and p = -s/A^ + B , cos « = \/A' + B' , sm a = ^A' + B' Since a is to be taken positive, and less than 180° [Art. 44], the sign of -VA' + E^ is to be taken the same as that of B in the given equation. 47. To trace the locus of an equation of the first degree. Since the locus of an equation of the first degree in two variables is a straight line, and a straight line is determined by two of its points, therefore to trace the locus it is necessary to find only the two points most easily determined, viz. those points where the line cuts the axes, respectively. If the line is parallel to an axis only one point is needed. F M 70 THE STKAIGHT LINE [Ch. IV E.g. The locus of the equation 2a; — 3y + 12 = passes through the two points (0, 4) and (~6, 0) ; therefore LM is the locus. EXERCISES 1. Reduce the following equations to the intercept (sym- metric) form, and draw the lines which they represent: (a) 2x-32^ + 12 = 0; (fi) 6 x-4y + 1 = 10 x + 3; (y) 2y = 15-y + 5x; (8) x-2y + l ^^ S + 7y 2. Beduce to the slope form, and then trace the loci : (a) 7a;-5j^ + 6(y-3x)=-10a; + 4; (fi) 3a: + 2y + 6=0; (y) a,-5 = 3-3y. Which is the positive side of the line (fi)? 3. Reduce to the normal form, and then trace the loci : (a) 4a;-|-3y = 15; (j3) 4 a; -32/-l-15 = 0; ^y) x-3y = 5 + 6x; (8) |a; = y-5. 4. Show that the lines 9 a; + 5 = 3 y and 3 a; — y = 81 are parallel. 5. What is the slope of the line between the two points (3, ~1) and (2, 2) ? What is its distance from the origin ? 6. A line passes through the point (6, 5) and has its inter- cepts on the axes equal and both positive. Find its equation and its distance from the origin. 7. A straight line passes through the point (~2, 1) and is such that the portion of it between the axes is bisected by that point. What is the slope of the line ? 47] EQUATION OF FIRST DEGREE 71 8. What are the intercepts which the line through the points (3, -1) and (7, 6) makes on the axes? Through the points ("a, "2 a) and (~6, "2 b) ? 9. What systejn of lines obtained by varying the parameter b is represented by the equation y = 8 .r + 6 V 10. What system of lines obtained by varying the parameter m is represented by the equation y = mx + 10 ? 11. What family (system) of lines obtained by varying the parameter a is represented by the equation x cos a + y sin a = 10 ? To what curve is each line of the family tangent ? 12. Find cos a and sin a for the lines («) y=mx + bi W f-f=l; (y) - = -; (8) 7a;-52/+10 = O. X y 13. Find by means of cos a and sin a what quadrant is crossed by each of the lines : (a) 2a;+2 = 3y; (j8) 3 a;+ 5 2/+15 = 0; (y) « + V3?/-10=0. 14. What must be the slope of the line + fco; — 4 y = — 17 in order that it shall pass through the point (1, 3) ? Has Tc a finite value for which this line will pass through the origin ? 15. Determine the values A, B, C in order that the line Ax + By + C=0 shall pass through the points (0, 3) and (— 12, 0). [Art. 46, Note]. 16. Derive equation [8] by supposing (.r„ .Vi) a^^id (^ij Vi) to be two points on the line 2/ = ma; + 6; and thence finding values for m and 6. 17. Find the slopes of thB lines 5y — 3x = 9 and 3y + 5x — 10 = ; and thence show that these lines are perpendicular to each other. 72 THE STRAIGHT LINE [Ch. IV 18. Find cos « for each of the lines 9x +y — 9 = and x — 9y + 2=0, and then show that the two lines are perpen- dicular to each other. 19. Show by means of : (1) the slopes ; (2) the angles ; that the lines 2y — 39;=7, iy-6x + 5 = 0, lOy- 15a;-|-c = are all parallel. 20- Eednce the equation Ax + Bi/+C=0 to the normal form, i.e., to the form x cos a + y sin a =p. Suggestion : the two equations, as representing the same line, make the same intercept on the axes. 21. Prove that the lo- cus of Ax+By + C=0 is a straight line, (a) by using similar triangles PiHPi and F2KP3, thus showing that the sum of the angles at Pj is equal to 2 rt. A. (6) by using Article 24. Fig. 35. 48. To find the angle made by one straight line with another. Let the equations of the lines be y = mjx + bi, (1) and y = m^ + 62, (2) where mi = tan tfj, wij = tan 02, and 61, 63 are the angles which these lines make, respectively, with the ie-axis. It is required to find the angle , measured from line (1) to line (2). 47-49] EQUATION OF FIRST DEGREE 73 Since <^ = ^2 — ^i, tan,^ = /^"^^~^^"\ , [Art. 12] ^ 1 + tan fl, ■ tan flj' "■ -■ i.e., tan + = ~f !-■ [13-1 If the angle were measured from line (2) to line (1), it would be the negative, or else the supplement, of <^; in either case its tangent would be the negative of that given by formula [13]. If the equations of the lines be given in the form : Aix + B,y+Ci = 0, (3) and Aix + B^+C2 = 0, (4) then formula [13] becomes AiA^ + BiB^ EXERCISES Find the tangent of the angle from the first line to the second in each of the following cases, and draw the figures : 1. 2x—ly-7 = 0, 3a; — 4y-3 = 0; 2. a;-2y+l = 0, 5 a; + 122/ + 6 = 0; 3. 4:X=6y + 9, 3y = 2x + 2; 4. ^-1 = 1, ^ + 1=]; a a 5. xcos a + y sintt=p, -4--=l. a 49. Condition that two lines are parallel or perpendicular. From formula [13] can be seen at once the relations that were shown in Art. 21 to hold for parallel and perpendicular lines. 74 THE STRAIGHT LINE [Ch. IV For if lines (1) and (2), Art. 48, are parallel, then ^ = 0, tan <^ = 0, and ^^ ^ ^^ . If lines (1) and (2) are perpendicular to each other, then ^ = ^ , tan <^ = oo , and nii = . Similarly, if the lines be given by equations in the forms (3) and (4), Art. 48, then from equation 14, if the lines are parallel, Bj Bg if the lines are perpendicular, Bi A^' The conditions just found enable one to write down readily the equations of lines which are parallel or perpendicular to given lines, and which also pass through given points. E.g., a line parallel to the line y = 3x + 7 (1) must have the same slope, m = 3, and hence must have its equation of the form y = 3x + b. (2) Now 6 may be determined by a second condition, e.g., by the requirement that the line shall pass through the point (1, 5). Then the coordinates must satisfy equation (2) ; i.e., 5 = 3-1 + 6, .-.6 = 2, and y=3a; + 2, (3) • It must not be forgotten that this conclusion is drawn only for lines that are not perpendicular to the x-axis ; because if the lines are perpen- dicular to the z-axis then equations (1) and (2) are inapplicable. [Cf, Art. 45]. 49] EQUATION OF FIRST DEGREE 75 represents a line parallel to line (1) and through the point (1, 5). Similarly, y = —^x + b (4) is the equation of a line perpendicular to line (1), for all values of b. And again, for all values of K, the lines 3a; + 6y-15 = and 3a! + 6y + £'=0 are parallel ; while the lines 3a; + 5y-15 = and 5x—3y + E' = Q are perpendicular. Here also the values of K and K' may be so determined that the respective lines shall pass through any given point. This condition for parallelism and for perpendicularity of two lines may also be stated thus : tvjo lines are parallel if their equations differ (or may be made to differ) only in their constant terms; two lines are perpendicular if the coefficients of x and y in the one are equal (or can be made equal), re^ectively, to the coefficients of — y and x in the other, EXERCISES 1. Write down the equations of the set of lines parallel to thelines: (a) y = 6x-2; (fi) 3x-7 y = 3; (y) a!cos30° + ysin30'' = 8; (8) |-f =1- 2. Explain why it is that the constant term in the answers to Ex. 1 is left undetermined or arbitrary. 3. Find the tangent of the angle between the lines (a) and (j8) in Ex. 1; also for the lines (fi) and (8), and (a) and (8) of Ex. 1. 76 THE STRAIGHT LINE [Cu. IV 4. Write the equations of lines perpendicular to those given in Ex. 1. 5. By the method of Art. 49 find the equation of the line that passes through the point ("9, 1), and is parallel to the line y = 6x-2. 6. Solve Ex. 6 by means of equation [10], Art. 43. 7. Find the equation of the line that is parallel to the line ,4a! + By+C = and that passes through the point (ajj, y,) ; make two solutions, one by the method of Ex. 6, and the other by Ex. 5. Find the equation of the straight line : 8. Through the point (2, ~5) and parallel to the line 2y=x-10. 9. Through the point (~1, ~1) and perpendicular to 2 y = 2 a; — 10 ; solve by two methods. 10. Through the point (1, 1) and parallel to the line 3 7 x—y+1 2 5* 9 11. Perpendicular to the line 7y + 2x — 1 = 0, and passing through the point midway between the two points in which this line meets the coordinate axes. 12. Find the foot of the perpendicular from the origin to the line 7x — 5y=10. 50. Line which makes a given angle with a given line. The formula a. a ^ a ^ tanA^^ [Art. 48] 1 + tan ^1 tan 0^ states the relation existing between the tangents of the angles ^1, flj, and (see Fig. 36), hence if any two of these angles are known, this equation determines the value of the third. Thus 49,50] EQUATION OF FIRST DEGREE 77 this formula may be employed to determine the slope of a line that shall make a given angle with a given line. E.g., to find the equation of a line that shall make the angle 60° with the line 3y — 5x + 7 = 0. Here <^=60°, .".tan <^ = V3, and m, = |. Since in equation [13] <^ is measured from line (1), substi- tuting in equation [13], 5+3V3 V3 = tan 6, 1 + f tan flj. _L whence tan 6, = ' 3-5V3 and 3-5V3 is the equation of a line fulfilling the required con- ditions ; k may be so determined that this line shall also pass through any given point. It is to be remarked that through any given point there may be drawn two lines, each of which shall make, with a given line, an angle of any desired magnitude. «>^<" u Z^x^^"^ ^ L^ ^^ '/^ ^ / /A, Fio. 37. Thus, through P, = (ar,, y^ the lines (1) and (2) may be so drawn that each shall make an angle <^ with the given line LM: but line (1) makes the angle -|- <^, while line (2) makes the angle — <^ ; which must be remembered in applying formula [13]. 78 THE STRAIGHT LINE [Cu. IV EXERCISES 1. Fiud the equations of the two lines which pass through the point (8, 5), and each of which makes an angle of 45° with the line 3x —2y= 6. 2. Show that the equations of the two straight lines pass- ing through the point (3, 5) and inclined at 45° to the line 2x — 3y-7 = are 6a;-2^-10 = and x+5y-28 = 0. Find the equation of the straight line : 3. Making an angle of + 7 with the line 4 x — 3 3/ = 7 : con- 4 struct the figure. Why is there an undetermined constant in the resulting equation ? 4. Making an angle of —60° with the line 5a; + 12y + l = ; construct the figure. 5. Making an angle of + 30° with the line x — 2y+l = 0, and passing through the point (1, 3) ; making an angle of — 30°, and passing through the same point. 6. Making an angle of ± 135° with the line x — y = 2, and passing through the origin. 7. Making the angle tan-'| + - ] with the line - — ^ = 1, \ aj a b and passing through the point (-, — ^j* 8. Find the equation of aline through the point (4, 5) form- ing with the lines 2 x—y + 3 =0 and 3 .?/-!- 6 .1; = 7 a right-angled triangle. Find the vertices of the triangle (two solutions). 9. Show that the triangle Avhose vertices are the points (4, 2), (6, -4), (-8, -2) is a rieht trianele. 50] EQUATION OF FIRST DEGREE 79 10- Prove analytically that the perpendiculars erected at the middle points of the sides of the triangle, the equations of whose sides are a; — 1 = 0, ^ — 1 = 0, and x + y = 10, meet in a point which is equidistant from the vertices. 11. Find the equations of the lines through the vertices and perpendicular to the opposite sides of the triangle in exer- cise 10. Prove that these lines also meet in a common point. 12. A line passes through the point (~2, 3) and is parallel to the line through the two points (4, 7) and (~1, ~9) ; find its equation. 13. Find the equation of the line which passes through the point of intersection of the two lines 10 a; + 5 .y + 11 = 0, and x + 2y + 14. = 0, and which is perpendicular to the line x + 7y + l = 0. This problem may be solved by first finding the point of intersection (J^, — i^-) of the two given lines, and then, by formula [10], p. 63 [see also Art. 49], writing the equation of the required line, viz. : which reduces to 7 x — y = 31. Or, it may also be solved somewhat more briefly, and much more elegantly, by employing the theorem of Art. 38. By this theorem the equation of the required line is of the form 10a;-f 5i/ + ll + fc(a; + 2?/ + 14) = 0, i.e., (10 + K)x+(r, + 2k)y + n + Uk = 0. It only remains to determine the constant A', so that this line shall be perpendicular to x + 7 »/ + 1 =0. 80 THE STRAIGHT LINE [Ch. IV By Art. 49 its slope must be r~^' ^^iice _ 10 + fc ^ 7 whence A; = -3. 5 + 2fc ' Substituting this value of k above, the required equation becomes 1 x — y =. 31, as before. 14. By the second method of exercise 13 find the equation of the line which passes through the point of intersection of the two lines 2 x + y = 5 and a; = 3 y — 8, and which is : (1) parallel to the line iy=6x + l; (2) perpendicular to this line ; (3) inclined at an angle of 60° to this line ; (4) passes through the point (~1, 3). 15. Solve exercise 10 by the method of exercise 14. 16. Do the lines 4*— 2y = 3, 3a; — y+^ = 0, and 5x — 2y — l = meet in a common point? What are the angles they make with each other? 17. Find the angles of the triangle of exercise 10. 18. When are the lines x+{a + b)y + c = and a(x + ay) + b(x—by) +d = parallel? when perpendicular? 19. Find the value of p for each of the two parallel lines y = 4 a; + 14 and y = 4 a; — 10 ; and hence find the distance between these lines. [Cf. Art. 46 (3) and (4)]. 20. What is the distance between the two parallel lines 3x~Ry + G = and 10y-Gx = 7? 21. Find the cosine of the angle between the lines 4:y — x + 8 = pnrl fi7i_T-LQ — n 60, 51] EQUATION OF FIRST DEGREE 81 22. What relation exists between the two lines y = 4x + 8 and y = — 4:X — 6? 23. Find the angle between the two straight lines 3x = iy + 7 and 5 y = 12 a; + 6 ; and also the equations of the two straight lines which pass through the point (4, 5) and make equal angles with the two given lines. 24. Find the angle between the two lines x + 3y + 10 = and 2x + y — 10 = 0. Find also the coordinates of their point of intersection, and the equations of the lines drawn perpendicular to them from the point (-2, 3). 51. The distance of a given point from a given line. This prob- lem is easily solved for any particular case, thus : Find the equation of the line which passes through the given point and which is parallel to the given line [Art. 49], then find the distances pi and p^ from the origin to the given line and the parallel line, respectively [Art. 46, (3) and (4)], and finally subtract the former distance from the other ; the result is the distance /rom the given line to the given point.* E.g., find the distance d of the point Pi = (2, f) from the l"ie 3a; + 4y-7 = 0. (1) The parallel line Sx + iy + k = contains the point (2, f) : hence i=-6-6=-12, The equation becomes 3x + 4:y-12 = 0. (2) FlO. 38. * Cf. Art. 44, Ex. 9. 82 THE STRAIGHT LINE [Ch. IV Now for lines (1) and (2) the perpendiculars from the origin are respectively, by Art. 46 (3), P, ±1?^=+^; and p, = \. + V4=+3=' 5 6 Hence, d=p2—P\=+'^. Similarly, in general, to find the distance of any given point Pi = (xi, ^i) from any given line, Ax-irBy+C=Q. (1) The equation of the parallel line through Pi is Ax + By- {Ax^ + By^) = 0. (2) Therefore jo, = ^^ ' + -^y i . p^ = ^-^ — wherein the sign of the radical is to be chosen in accord with Art. 46 (3) ; , , Ax. + Byi + C . ^ hence d = ' ' — ■ [15] If the equation of a given line is so written that its second member is zero, this formula may be translated into words thus : To get the distance of a given point from a given line, vorite the first member of the equation alone, substitute for the variables therein the coordinates of the given point, and divide the result by the square root of the sum of the squares of the coefficients of x and y in the equation, — the sign of this square root being chosen the same as that of the number represented by B. In formula [15] the numerator is positive or negative according as P, is on the positive or negative side of the given line [see Art. 32, Note]; i.e., for points on opposite sides of the given line the corresponding distances are opposite in sig^n. The origin is at a negative distance from the given line if B and C have opposite signs. 51] EQUATION OF FIKST DEGREE 83 EXERCISES 1. Find the distance of the point (2, "7) from the line 3x-6y + l=0. By formula [15], d = 3-2-6(-7) + l ^ _ J9_ . -V3^+6^ 3V5 The origin is on the positive side of the line 3x— 6y + 1 =0, but at a negative distance; hence the point (2, ~7), also, is on the positive side of the line. 2. Find the distance of the point (5, 4) from the line 5y + 4lx=20. On which side of the line is the point ? 3. Find the distance of the point (2, 7) from the line 6 y - 4 a; = 17. 4. Find the distance of the point (~a, ~b) from the line a 5. Find the distance of the intersection of the two lines, y + 4: = 3x and 5x=y — 2, from the line 2y — 7 = 9x. On which side of the latter line is the point ? 6. Find the distance of the point of intersection of the lines 2 a; — 6 y = 11 and 4 a; = 3 y + 15 from the line ^x + ^"~ = 6. On which side of the latter line is the point ? Plot the figure. 7. How far is the point (+6, +1) from 3y = 7a; + 8? On which side ? 8. By the method of Art. 51, find the distance of the origin from the line 2x — 5y — 7; also from the line Ax + By+ (7=0. Check the results by Art. 46 (3). 9. Find the distance of the point (+4, +5) from the line joining the two points (3, ~1) and (~4, 2). On which side is it ? 10. Find the distance of the point (a:i, y^) from tlie line y = mx + b. 84 THE STRAIGHT LINE ICh. IV 11. Find the altitudes of the triangle formed by the lines whose equations are 3x + y + 4 = 0, 3x — 5y + 34: = 0, and 3x— 2y + l = 0. Check the result by finding the area of the triangle in two ways. 12. Show analytically that the locus of a point which moves so that the sum of its distances from two given straight lines is constant is itself a straight line. 13. Express by an equation that the point P, = {xi, yi) is equally distant from the two lines x—2 y = 11 and 3x = 4y+5. (Give two answers.) Should P, move in such a way as to be always equidistant from these two lines, what would be the equation of its locus ? 14. Find, by the method of exercise 13, the equations of the bisectors of the angle formed by the lines 4:X+3y = 12 and 3x + 4y=24. 52. Bisectors of the angles between two given lines. The bisector of an angle is the locus of a point which moves so that it is always equally distant (numerically) from the sides of the angle. From this property its equation may easily be found, as in Art. 51, exercises 13 and 14. The fact that in two opposite quarters formed by the intersecting lines the distances of the moving point from the given lines are alike in sign, while in the other two quarters the distances differ in sign [Art. 51] leads to the two equations for the two bisectors. If Q,P,= + R,Pi then QjP, = - R^P^. The resulting equations will be found to show the well-known fact that the two bisectors are FiQ. 39. neroendicular to each other. Jl5 p «SsA/ ""^ / \ \ / X / \ N^ m \{4) 61, 62] EQUATION OF FIKST DKGREE 85 E.g., the bisectors of the angles, Fig. 39, between the lines 3x + 4:y — l = and 12a;+5y + 6 = are given by the equations + 5 "^ +13 ' 3a! + 4y— 1 _ 12a! + 5.v+6 . + 5 +13 ' i.e., by 21 X — 27 y + 43 = and 99x + 77y + lT = 0. The slopes are m = -J and m' = — -^j hence the lines are perpendicular to each other. EXERCISES 1. Find the equations of the bisectors of the angles between the two lines a; — y + 6 = and — ^^— = 5y — 7. 2. Show that the line x + y + 10 = bisects one of the angles between the two lines 4a;— 3y+17 = 0, and 3x — 4y + 7 = 0. Which angle is it ? Find the equation of the bisector of the other angle. 3. Show analytically that the bisectors of the interior angles of the triangle whose vertices are the points (0, 0), (0, 3), and (4, 0) meet in a common point. 4. Show analytically, for the triangle of Ex. 3, that the bisectors of one interior and the two opposite exterior angles meet in a common point. 5. Find the angle from the line 3a; + 2/ + 12 = Oto the line ax + by +1 = and also the angle from the line ax + by + 1=0 to the line x + 2y — l = 0. 86 THE STRAIGHT LIXR [Ch. IV By imposing upon a and b the two conditions : (1) that the angles just found are equal, and (2) that the line ax + by + 1 = passes through the intersection of the other two lines, determine a and b so that this line shall be a bisector of one of the angles made by the other two given lines. 53. The equation of two lines. By the reasoning given in Art. 37 it is shown that if two straight lines are represented by the equations ^ ^ ^ « ,< < and A^ + B^ + Cj = 0, (2) then both these lines are represented by the equation {AyX + Bi!i+C^){,A^JrBa-\-Ci) = 0; (3) i.e., two straight lines are here represented by an equation of the second degree. Conversely, if an equation of the second degree, whose second member is zero, can have its first member separated into two first degree factors, with real coefBcients, as in equa- tion (3), then its locus consists of two straight lines. Thus the equation may be written in the form (2a5-3 2/ + 7)(a:-fy + l) = 0, which shows that it is satisfied when 2a; — 3y + 7 = 0, and also when x + j/ + 1 = 0. Its locus is therefore composed of the two lines whose equations are : 2a; — 32/ + 7 = 0, and x + y + l = 0. The test to be satisfied in order that an equation of the second degree in two variables may be factored is given in Chapter 52, 53] EQUATION OF FIRST DEGREE 87 VIIT. A method of factoring, token factoring is possible, is shown in the following example : [See also Art. 97] 2x'-xy — 3f + 9x + iy + 7. Take out 2 as a factor, complete the square of the a>-terms, and collect the other terms : then the expression takes the form of the difference between two squares : = (x + y+l){2x-3y+7). EXERCISES The following equations represent pairs of straight lines. Find in each case the equations of the lines, their common point, and the angle between them: 1. 6y'-xy-x' + 30y + S6 = 0. 2. T'—2xy-3y^ + 2x-2y + l = 0. 3. a;" — 2 a;y sec a + 2/^ = 0. 4. ar' + 6a:y + 9y= + 4a; + 123/-5 = 0. 5. The equations of the opposite sides of a parallelogram are x'-J x + 6 = and y^-Uy + 4:0=0. Find the equations of the diagonals, and their common point. 6. Show that 6 x' + Say— 6y'' = is the equation of the bisectors of the angles made by the lines 2x' + 12xy + 7 y'=0. 88 THE STRAIGHT LINE [Ch. IV EXAMPLES ON CHAPTER IV 1. The points ("2, 4) and (6, ~4) are the extremities of tlie base of an equilateral triangle. Find the equations of the sides, and the coordinates of the third vertex. (Two solutions). 2. Three of the vertices of a parallelogram are at the points (2, 2), (6, 8), and (10, "4). Find the fourth vertex. (Three solutions). Find also the area of the parallelogram. 3. Find the equations of the two lines drawn through the point (3, 0), such that the perpendiculars let fall from the point (~6, ~6) upon them are each of length 3. 4. Perpendiculars are let fall from the point (5, 0) upon the sides of the triangle whose vertices are at the points (4, 3), (~4, 3), and (0, ~5). Show that the feet of these three perpen- diculars lie on a straight line. Find the equation of the straight line: 5. Through the point (1, 2) and the point of intersection of the lines x — y = and x-\-y — 2 = 0. Prove that it is a bisector of the angle formed by the two given lines. 6. Through the intersection of the lines 2x~3y + 2 = and 3x — 4y — 2 = and parallel to 5 x — 2 y + 3 = 0. 7. Through the point (1, 2), and intersecting the line a! + y = 4 at a distance ^ V6 from this point. 8. Find the equation of a straight line through the point (5, 0) and making equal angles with the lines y — 2x + 6 = and y = — 2x + 16. 9. Prove analytically that the diagonals of a square are of equal length, bisect each other, and are at right angles. 10. Prove analytically that the line joining the middle points of two sides of a triangle is parallel to the third side and equal to half its length. 63] EQUATION OF FIRST DEGREE 89 11. Find the locus of the vertex of a triangle whose base is 4 a and the difference of the squares of whose sides is 16 axis and the origin on the circumference? when the circle touches each axis and has its center in quad- rant II ? 8. In what respect is equation [16] more general than equation [17] ? 94 THE CIRCLE [Ch. V 56. Iq rectangular coordinates every equation of the form Aae^ + Ay^ + 2Gac + 2Fy+ C = represents a circle. The equations of the circles already obtained (equations [16] and [17], as well as the answers to examples 1 to 6 and 7) are all oftheform a-' +f + 2Ch: + 2 F)j + 0=0. (1) It will now be shown that, for all values of G, F, and G, for which the locus of equation (1) is real, this equation represents a circle. To prove this, it is only necessary to complete the square in the a5-terms and in the y-terms, and then transpose C to the second member. Equation (1) may then be written in the **""™ (x+Gf+iy + Fy=GP+F^-G = {-VG'' + F'-Cy, (2) which is (cf. equation [16]) the equation of a circle whose center is the point (— G, — F), and whose radius is ^G' + F'-C. It follows that every equation of the more general form Aa?-[-Af + 2Gx + 2Fy+ C = represents a circle; for, by Art. 35, this equation has the same locus as has a? + y' + 2—x -\- 2— y -\ — = 0, and this last A A A equation is of the form of equation (1), which, as shown above, represents a circle. Hence, interpreted in rectangular coordinates, every equation oftJw second degree from lohich the term in xy is absent, and t)i which the coefficient of i? equals thai of y^, represents a cirde. 57. Equation of a circle through three given points. By means of equation [16], or of the equation :fi_ + ,f + 2Gx+2Fy+C=0, (X) 56,57] SPECIAL EQUATION OF THE SECOND DEGREE 95 which has been shown in Art. 56 to be its equivalent, the prob- lem of finding the equation of a circle which shall pass through any three given points not lying on a straight line can be solved ; i.e., the constants h, k, and r, or G, F, and C, may be so deter- mined that the circle shall pass through the three given points. To illustrate : let the given points be (1, 1), (2, ~1), and (3, 2), and let a? + y^ + 2 Gx + 2Fy+C=0 be the equation of the circle that passes through these points ; to find the values of the constants G, F, and G. Since the point (1, 1) is on this circle, therefore [cf. Art. 28], i + i+2G + 2F+C = 0; similarly, 4 + 1+i G -2F+ C=0, and 94-4 + 6G'-f4F-f-C7=0. These equations give : G= ~-^, F= —i, and 0=4. Substituting, the equation of the required circle be- comes , » „ , „ ar + y''—5x — y-i-4: = 0; its center is at the point (f, ^), while its radius is Note. The algebraic fact that the most general equation of the circle contains three parameters (h, k, and r, or G, F, and C, above) corresponds to the geometric property that a circle is uniquely determined by three of its points. EXERCISES Find the radii, and the coordinates of the centers, of the fol- lowing circles ; also, draw the circles. 1. x' + y^-3a;-'lij—ll=0. 4. 2(a?+y'^=7y. 2. 3x- + 3 y- — 5:v — 7y + l= 0. 5. ax' + ay- = —bx — cy. 3. r'-|-2/= = 3(a;-t-3). g. {x+yf+{x-yf=ia\ 96 THE CIRCLE [Ch. V 7. What loci are represented by the equations {x+hy+{y+ky=o, and a^ + t/''-2a; + 63^+38 = 0? Find the equation of the circle through the points : 8. (1,2), (3, -4), and (5, -6); 9. (0, 0), {a, b), and (6, a) ; 10. (10,9), (4, -5), (0,5); 11. (-6, -2), (-2, 2) and having the radius 2^. 12. Find the equation of the circle which has the line join- ing the points (1, 2) and (5, "~1) for a diameter. 13. Find the equation of the circle which touches each axis, and passes through the point (8, 4). 14. A circle has its center on the line 3y + ix = 7, and touches the two lines x+y =3 and y — x = 3; find its equation, radius, and center ; also draw the circle. 15. Show that equation (1) represents a real, imaginary, or " point " circle (i.e., only one real point), according as G" + F' — O is respectively greater than, less than, or equal to, zero. SECANTS, TANGENTS, AND NORMALS 58. Definitions of secants, tangents, and normals. A straight line will, in general, intersect any given curve in two or more distinct points ; it is then called a secant line to the curve. Let P, and Po be two successive points of intersection of a secant line PjPiQ with a given curve LPjP., ■•■ K; if this secant line be rotated about the point P, so that Pj approaches P, along the curve, the limiting Fig. 41. position PiT which the secant 58-59] SECANTS, TANGENTS, AND NOUMALS 97 approaches, as Pj approaches coincidence with Pj, is called a tangent to the curve at that point. This conception of the tan- gent leads to a method of extensive application for deriving its equation, — the so-called " secant method." Since the points of intersection of a line and a curve are found [Art. 36] by considering their equations as simultaneous, and solving for x and y, it follows that, if the line is tangent to the curve, the abscissas of two points of intersection, and also their ordinates, are equal. Therefore, if the line is a tan- gent, the equation obtained by eliminating x or y between the equation of the line and that of the curve must have a pair of equal roots. A straight line drawn perpendicular to a tangent and through the point of tangency is called a normal line to the curve at that point. Thus, in Fig. 41, PiP^, P1P3 are secants, Pi 7" is a tangent, and P^N a normal to the curve at Pi. 59. Tangents : Illustrative examples. (1) Find the equation of that tangent to the circle 0:^-1-^^=5, which makes an angle of 45° with the avaxis. Since the line makes an angle of 45° with the a;-axis, its equa- tion is y = x-^b, where 6 is to be determined so that this line shall touch the circle. Clearly, from the figure, there are two values of 6 {OBi and OB^ for which this line will be tan- gent to the circle. According to Art. 68, these values of 6 are those which make the two points of intersection of the line and the circle become coincident. Considering the equations a?+y-=5 and y=x->rb assimulta- 98 THE CIRCLE [Ch. V neous, and eliminating y, the resulting equation in x is oi?-\.{x + hf = : i.e., 2a!" + 26»+ 6^-5 = 0. The roots of this equation will become equal, i.e., the abscissas of the points of intersection will be equal [Art. 6], if 6!_2(6"-5) = 0, i.e., if6 = ±VIO. The equations of the two required tangent lines are, there- fore, . — — y = x-\- V 10, and y = x — v 10. (2) Find the equations of those tangents to the circle a? -\-'!^:=&y that are parallel to the line a; -(- 2 y -(- 11 = 0. The equation of a line parallel to a:-(-2y-|-ll=0 is x-{-2y 4-^ = 0, where k is an arbitrary constant [Art. 49], and this line will become tangent to the circle, if the value of the con- stant A; be so chosen that the two points in which the line meets the circle shall become coincident. Considering the equations a^-\-y'=z6y and K-f-2y-|-Jfc = as simultaneous, and eliminating x, the resulting equation in y is {-k-2yy + y' = 6y, i.e., 5y'+(4.k-6)y + K'==0. The two values of y will be equal if [Art. 6] (4A;_6)'-20&"=0, I.e., if &"-|-12&-9 = 0, i.e., if k= —6± 3 V5, and the two required tangent lines are : x+2y~6-j-3V5 = 0, and a;-H2 y- 6-3V6=0. 69. 60] SECANTS, TANGENTS, AM) NORMALS 99 EXERCISES Find the equations of the tangents : 1. To the circle ar'+j^'=4, parallel to the line x+2y+a= 0. 2. To the circle 3{x' + f)=4:y, perpendicular to the line x + y = b. 3. To the circle oe' + y'- + 10x-6y -2 = 0, parallel to the line y = 2x + 7. 4. To the circle x' + y'' = r^, and forming with the axes a triangle whose area is r^- 5. Show that the line y = x + cV2 is, for all values of c, tangent to circle ar' + y*= c^; and find, in terms of c, the point of contact. 6. Prove that the circle a^ + y' — 2x — 2y + l = touches both coordinate axes ; and find the points of contact. 7. For what values of c will the line 3a! + 4y + c = touch the circle x' + f — 8x + 12y — -i:4:=0? 8. For what value of r will the circle a^ + ^'^ = »•* touch the line y = 3x + 5? 9. Prove that the line bx = a(y — a) touches the circle x(x — b)+ y{y — a) = ; and find the poi nt of contact. 10. Three tangents are drawn to the circle 3^+y^=25 ; one of them is parallel to the a>-axis, and together they form an equilat- eral triangle. Find their equations, and the area of the triangle. 60. Equation of tangent to the circle a;^ + y^ = r^ in terms of its slope- The equation of the tangent to a given circle, in terras of its slope, is found in precisely the same way as that followed in solving (1) of Art. 59. Let m be the given slope of the tangent, then the equation of the tangent is of the form y = m,x + b; (1) 100 THE CIRCLE [Ch. V and b is found to be b= ± 7-Vl + 1 Therefore, y = mac ± rVl + »»*, [18] is, for all values of m, tangent to the circle (2). Equation [18] enables one to write at once the equation of the tangent of given slope, and is also of service in getting the equation of a tangent through an outside point; but only, of course, to a circle vihose center is at the origin. E.g., to find the equation of the tangent whose slope m = tan 45° = 1, to the circle ar* + y" = 5 ; substitute 1 for m and V5 for r in equation [18]. This gives y = x± VlO. Again, to find the equation of the tangent to the circle aP + 1/^ = 25, from the outside point P= ("""T, ~1). Now the coordinates of P satisfy equation [18], hence, y = mx ± SViT+ot? ; i.e., ~1 = '*'7 m± 5Vm'+ 1. Hence, 12m' + 7m — 12=0 and mi = |, m2 = — |; i.e., there are two such tangent lines, and their equations are y = I a; — -2^ and i/= — ^x + ^. EXERCISES Find the equations of the lines which are tangent : 1. To the circle a? + y^ = 25, and whose slope is 4 ; 2. To the circle 3? + %^ = A:, and which are parallel to the line x + 3y + 3=0; 3. To the circle ar" + v^ = 9, and which make an angle of 30° with the avaxis ; with the 2/-axis ; 4. To the circle a? + y^ = i^, a,t the point (xi, y^ ; (!0, 61] SECANTS, TANGENTS, AND NORMALS 101 5. To the circle a;^ + y- = 25, and perpendicular to the line joining the points (~3, 7) and (7, 6). 6. To the circle aP + y' = 2x + 2y — l, and whose slope is +1. 7. To the circle aP + f + 2Gx + 2Fy + C=0, with slope m. 61. Equation of tangent to the circle in terms of the coordi- nates of the point of contact : the secant method. (a) Center of the circle at the origin. Let Pj = (x^ y^ be the point of tangency, on the given circle a;' + 2^'' = r-". (1) Through jP, draw a secant line LM, and let Pj = {x^, y^) be its other point of intersection with the circle. If tlie point Pa moves along the circle until it conies into coincidence with Pi, the limiting position of the secant LM is the tangent P, T. [Art. 58]. The equation of the line LM is of the form y-yi = m.{x-x^, (2) in which the special value of the slope aso — «i (3) Fig. 43. is to be determined by the fact that the fixed point Pj and the moving point Pj are always on the circle (1) ; therefore ^ + yi='>^; (4) and xi + yi = 7^. (5) Hence, subtracting equation (4) from equation (5), x^ — x^^ + yi — y^ = 0; 102 THE CIRCLE [Ch. V that is, (2/2 — yi) (y^ + ^i) = — (xj — a,) (xi + Xi) : whence, ^izzll = _ fdz^ .« (6; Xi — xj^ yi + yi Substituting this value of m in equation (2) gives y-y, = -^^^(x-x,), (7) which is the equation of the secant line LM of the given circle (1). Now let Pj move along the circle until it coincides with P^, i.e., until x^ = x^, and ^2 = Vu then equation (6) becomes i.e., y-yi = --(.x-x;), which, by clearing of fractions and transposing, may be written in the form ^^ ^ ^^y ^ ^2 ^. y. . i.e., Xyx + viv = r*, [19] which is the required equation of the tangent to the circle aj* + 3/2 _ ^^ j^ a,nd y, being the coordinates of the point of tangency. NoTK. If Ps approaches Pi so that finally x^ = Xi, y^ = yi, in equation (2), then the indeterminate form y — yi = - (x — X\) is obtained. This indeterminateness arises because account has not been taken of the con- dition that Pi and P2 are on the circle, and therefore there is no definite- ness to the path by which P2 approaches Pi. * The difference between equations (3) and (6) consists in this : no matter where the points (zi, y{) and (X2, ^2) niay be, equation (8) rep- resents the slope of the straight line passing through thera ; but equation (6) gives the slope of the line through these points only when they are on the circle x^ 4- 1/' = r^. CI, 02] SECANTS, TANGENTS, AND NORMALS 103 (6) Center of circle not ai origin. By a similar process, if the equation of the given circle is 3?-iry^ + 2Gx + 2Fy+C=(i, (8) then the equation of a secant line through the two points Pj and Pj on the circle will be found to be which for the tangent at the point P, becomes y-yv ^J±|(x-rrO;* (10) and this may be reduced to the form xix+viu + Gix + x{)+F(^y-]-y{) + C = 0, [20] which is the required equation of the tangent to the circle (8), Xi and 2/i being the coordinates of the point of contact.* Note. Equation [20] may be easily remembered if it is observed that it differs from the equation of the circle (equation [7] ) only in having III, yiy, x + Xi, and y + Vi in place of x:\ y', 2 x, and 2 1^, respectively. It will be found later that any equation of the second degree (from wUich the zj/-term is absent) bears tliis same relation to the equation of a tangent to its locus, Xi and yi being the coordinates of the point of contact. Com- pare, also, equation [10] with equation (1). It must be carefully kept in mind that equations [19] and [20] rep- resent tangents only if {xi, yi) i3 a point on the circle. It may be shown that these equations represent other lines if (xi, ^i) is not on the circle, called polars of the given point. 62. Equation of a normal to a given circle. By definition [Art. 58] the normal at a given point, P, = (.v^. y,), on any curve is the line through Pj, and perpendicular to the tangent at Pi. Hence, to get the equation of the normal at any given point, it • Students sliould carefully work out all the details here. 104 THK CIRCLE [Ch. V is only necessary to write the equation of the tangent at this point [Art. 61], and tlien the equation of a line perpendicular to this tangent [Arts. 43, 49] and passing through the given point. Thus the equation of the normal to the circle x' + y^ + 2Gx-\-2Fy+C=0, (1) at the point Pi s {x^, y^, is j'-^>=!4|(— ^)- (2) Since the coordinates — G and — F ot the center of the given circle (1) satisfy equation (2), hence, every normal to a circle passes through the center of the circle. If the center of the circle is at the origin, then = 0, F= 0, and C = ~r', and the equation (2) of the normal becomes y-2/i = -(9' — ^i), (3) ■which reduces to x^y — xy^ = 0, an equation which could have been derived for the circle a^ + y' = r^ in precisely the same way that equation (2) was derived from equation (1). EXERCISES 1. Derive, by the secant method, the equation of the tan- gent to the circle aP + i^ — 2rx = 0, the point of contact being 2. Write the equation of the tangent to the circle : (a) x' + y^ = 41, the point of contact being (4, 5). (fi) a?+y'—3 x+l(iy=15, the point of contact being (4, ~11). (y) (a; + 2y+(y + 3y=10, the point of contact being ("5, "4). (8) 3 a^+3 f+2 y-f-4 a;=0, the point of contact being (0, 0). 3. Find the equation of the normal to each of the circles of Ex. 2, through the given point. 62, 63] SECANTS, TANGENTS, AND NORMALS 105 4. A tangent to a circle is perpendicular to the radius drawn to its point of contact. By means of this fact, derive the equation of the tangent to the circle (x — a)' + (jr — 6)- = r' at the point (x^, y^). (Of. equation [20]). 5. From the fact that a normal to a circle passes through its center, find the equation of the normal to the circle aP + y'—Sx + 6 y + 21 = at the point (4, "1). 6. Find the equations of the two tangents, drawn through the external point (11, 3) to the circle 3? + y^ — 40. 7. What is the equation of the circle whose center is at the point (5, 3), and which touches the line Sx + 2y + 10 = 07 8. If the line — t- - = 1 touches the circle a,-^ + «" = r* find m 11 the equation connecting m, n, and r. 9. Find the equation of a circle inscribed in a triangle whose sides are the lines a;=0, y=0, and - + " = 1. (a and b being .... a b positive). 10. Solve Ex. 6 by assuming a^ and i/i as the coordinates of the point of contact, and then finding their numerical values from the two equations which they satisfy. 11. Prove that from an outside point two tangents, and only two, can be drawn to a circle. 63. Lengths of tangents and normals. Subtangents and sub- normals. The tangent and normal lines of any curve extend in- definitely in both directions ; it is, however, convenient to consider as the length of the tangent the length TP-i, meas- ured from the point of inter- section (T) of the tangent with the av^xis to the point of tangency (P,) ; and simi- 106 THE CIRCLE [Ch. V larly to consider as the length of the normal the length NPi, measured from the point of intersection (N) of the normal with the a;-axis to Pj. The subtangent is the length TM, where M is the foot of the ordinate of the point of tangency Pi; and the subnormal is the corresponding length 2f^M. As thus taken, the subtangent and the subnormal are of the opposite sign ; ordinarily, however, one is concerned merely with their absolute values, irrespective of the algebraic sign. The subtangent is the projection of the tangent length on the a>axis, and the subnormal is the like projection of the normal length. 84. Tangent and normal lengths, subtangent and subnormal, for the circle. The definitions given in the preceding article furnish a direct method for finding the tangent and normal lengths, as well as the subtangent and subnormal, for a circle. E.g., to find these values for the circle x'+y'=25, and corresponding to the point ^ of contact (3, 4), proceed thus : The equation of the tangent y ^ PiT is [Art. 61] 3x+iy = 25; hence, 0T=^- and the subtangent is TM=OM- 0T = 3- Fig. 45. Similarly, the tangent length is -H- TPi = Vmt^+MP} = V(J^)24- 4* = ( Again, the equation of the normal at (3, 4) is 4a; — 3y = 0; hence, the subnormal is O.V= +3, and the normal is OPi = 5. 63-05] SECANTS, TANGENTS, AND NORMALS 107 EXERCISES Find the lengths of the tangent, subtangent, normal, and subnormal : 1. For the point ("4, +11) on the circle a? + y' + 3 x — 10 y = 15; 2. For the point (~1, +3) on the circle cr" + j/' + 10 a; = 0. 3. For the point whose abscissa is 3 on the circle x'+y'= 25. 4. The subtangent for a certain point on a circle, whose center is at the origin, is 5^, and its siibnormal is 3. Find the equation of the circle, and the point of tangency. 5. Show that 7W = ^, and NM= — yitn, where m is the slope of the tangent. 6. Show that the square of the length of the tangent from the outside point (xj, ^i) to the circle 3?-\-y^= ir is x^-\- y^ — r*. 65. Circles through the intersections of two given circles. Given two circles whose equations are a?+f + 2 0,x + 2F^ + C,=Q, (1) and 3?+f-is-2Gfi + 2Fiy + C^ = Q. (2) These circles intersect, in general, in two finite points Pi = (a^, ^i) and P^ = (arj, y^, and [Art. 38] the equation a?+f + 2Q^x+2F^y+0^ + k{x' + y' + 2G^ + 2F^+G,) = Q, (3) where Tc is any constant, represents a curve which passes through these same common points. The locus of equation (3) is, moreover, a circle [Art. 56] ; hence, a series of different values being assigned to the param- eter k, equation (3) represents what is called a " family " of 108 THE cieclp: [Ch. V circles ; each one of these circles passing through the two points /', and Pj in which the given circles (1) and (2) inter- sect each other. 66. Common chord of two circles. If in equation (3), Art. 65, the parameter k be given the particular value — 1, the equation reduces to 2{G,-G,)x + 2(F,-F,)y+C,-C, = 0, (4) which is of the first degree, and therefore represents a straight line; but this locus belongs to the family represented by equation (3) of Art. 65, hence it passes through the two points Pi and P^ in which the circles (1) and (2) intersect. This line (4) is, therefore, the common chord of these circles. 67. Radical axis ; radical center. The angle formed by two inter- secting curves. The line whose equation is obtained by elimi- nating the 3? and y^ terms between the equations of two given circles, as in Art. 66, whether the circles intersect in real points or not, is called the radical axis of the two circles. If the two given circles intersect each other in real points, then this line is called also their common chord; that is, the common chord of two circles is a special case of the radical axis of two circles. Three circles, taken two and two, have three radical axes. It is easily shown that these three radical axes pass through a common point ; this point is called the radical center of the three circles. By the angle between two intersecting curves is meant the angle formed by the two tangents, one to each curve, drawn through the point of intersection. Hence to find the angle at which two curves intersect, it is necessary to find the point of intersection, then to find the equBr tions of the tangents at this point, one to each curve, and finally to find the angle formed by these tangents. 0O-67] SECANTS, TANGENTS, AND NORMALS 109 EXERCISES 1. Find the equation of the common chord of the circles a^ + y^-Sy-'ix-U^O, x' + y'' + 8y-8x-2 = 0. 2. Find the points of intersection of the circles in exercise 1, and the length of their common chord. 3. Find the radical axis, and also the length of the common chord, for the circles of -\-i^ + ax + by + c = 0, x' + i/' + bx + ay + c = 0. 4. Find the radical center of the three circles x' + y' + ix+l^O, 2(x' + y^+3x + 5y + 9=0, x' + y^ + y = 0. 5. Show that tangents from the radical center, in exercise 4, to the three circles, respectively, are equal in length. 6. Prove analytically that the tangents to two circles from any point on their radical axis are equal. 7. Find the angle at which the circle a? + y^ = 25 cuts the circle (x-Tf + y^ + 2 y + 2i = 0. 8. At what angle does the circle (a; — 4)' + y' — 2 y = 15 meet the line a; + 2 y = 4 ? 9. Show that if in Art. 65, equation (3), the radius be- comes infinite, the common chord is obtained, as in Art. 66, equation (4). 10. For the circles of Ex. 1, show that the radical axis is perpendicular to the line of centers. 11. The radical axis is perpendicular to the line joining the centers of two circles. 110 THE CIRCLE [Ch. V EXAMPLES ON CHAPTER V 1. Find the equation of the circle circumscribing the tri- angle whose vertices are at the points (10, 9), (4, ~5), sind (0, 5). What is its center ? its radius ? 2. Determine the center of the circle (x-ay + (y-by=a' + b*. What family of circles is represented by this equation, if a and b vary under the one restriction that a* + 6* is to remain constant? 3. What must be the relations among the coefficients in order that the circles a^ + y^-2 Gix-2F^ + Ci = 0, and aP + y^—2O^ — 2F0+C2 = O, shall be concentric? that they shall have equal areas? 4. Under what limitations upon the coefficients is the circle x' + y^ + Dx+Ey + F=0 tangent to each of the axes ? 5. Find the equation of the circle which has its center on the as-axis, and which passes through the origin and also through the point ("2, "3). 6. Find the points of intersection of the two circles x' + f-ix — 2y-31 = 0a.nd.aP + y'-4x + 2y + l = Q. 7. Circles are drawn having their centers at the vertices of the triangle (7, 2), (~1, ~4), and (3, 3), respectively, and each passing through the center of a fourth circle which circumscribes this triangle ; find their equations, their common chords, and their radical center. 8. Circles having the sides of the triangle (7, 2), (~1, 4), (3, 3) as diameters are drawn ; find their equations, their radi- cal axes, and their radical center. 67] SECANTS, TANGENTS, AND NORMALS 111 9. Find the equation of the circle passing through the origin and the point (a?j, y{), and having its center on the j/-axis. 10. The point (~1, ~1) bisects a chord of the circle v? -\- y' = 25 ; find the equation of that chord. 11. A circle touches the line 4^ + 3a; + 3=0 at the point (3, ~3) and passes through the point (5, 9) ; find its equation. 12. A circle whose center coincides with the origin touches the line y — 2x+Z = 0; find its equation. 13. At the points in which the circle a? + jf •{■ ax + hy = cuts the axes, tangents are drawn ; find the equations of these tangents. 14. A circle, whose radius is V74, touches the line 5x = 7 y — 1 at the point (11, 8) ; find the equation of this circle. 15. A circle is inscribed in the triangle (0, 0), (~ 8 a, 0), (0, 6 a) ; find its equation. 16. Through a fixed point (a^, y^) a secant line is drawn to the circle x' + y^=r^; find the locus of the middle point of the chord which the circle cuts from this secant line, as the secant revolves about the given fixed point (xi, 2/1). 17. Prove analytically that an angle inscribed in a semicircle is a right angle. 18. Prove analytically that a radius drawn perpendicular to a chord of a circle bisects that chord. 19. Angles in the same segment of a circle are equal. 20. Two straight lines touch the circle x'+y' — Sx — Sy + 6 = 0, one at the point (I, 1) and the other at the point (2, 3) ; find the point of intersection {h, Jc) of these tangents. 21. Find the condition among the coefficients that must be satisfied if the circles 112 THE CIRCLE [Ch. V ar' + / + 2 (?ix + 2 Fiy = and ar'+ 2/' + 2 Gjs; + 2 J";?/ = Q shall touch each oth'er at the origin. 22. Determine F and C so that the circle shall cut each of the circles a;2 + y2_4a;-22/ + 4 = and a!2 + / + 4a; + 2y = l at right angles. [Cf. Art. 67]. 23. Given the two circles a!' + /-4a;-2!/ + 4 = and ar' + / + 4a; + 22/-4 = 0; find the equation of their common tangents. 24. Find the radical axis of the circles in example 23 ; show that it is perpendicular to the line joining the centers of the given circles, and find the ratio of the lengths of the segments into which the radical axis divides the line joining the centers. 25. Given the three circles : a:! + y»_16a;+60 = 0, 3ar' + 32/''-36a; + 81 = 0, and ar' + y'-16a;-12y + 84 = 0; find the point from which tangents drawn to these three circles are of equal length, also find that length. How is this point related ia position to the radical center of the given circles ? Prove that this relation is the same for any three circles. 26. Find the locus of a point which moves so that the length of the tangent, drawn from it to a fixed circle, is in a constant ratio to the distance of the moving point from a given fixed point. 27. Find the length of the common chord of the two circles (a;-a)»+(i/-6y = r» and {x-hf + (jii-af = r^. From this find the condition that these circles shall touch each other. G7] SECANTS, TANGENTS, AND NORMALS 113 28. For what point on the circle a:? + 3/" = 9 are the subtan- gent and subnormal of equal length? the tangent and normal ? the tangent and subtangent ? 29. An equilateral tr'.angle is inscribed in the circle 3?-\-y- = 25 with its base parallel to the jz-axis ; through its vertices tangents to the circle are drawn, thus forming a circumscribed triangle ; find the equations, and the lengths, of the sides of each triangle. 30. Find the equation of a circle through the intersection of the circles 3?-\-y- — A. = (i, a^ + y^ — 2x — 4y + 5 = 0, and tan- gent to the line a; + y — 3 = 0. 31. Find the locus of the vertex of a triangle having given the base = 4 a, and the sum of the squares of its sides = 4 c^. 32. Find the locus of the middle points of chords drawn through the positive end of the horizontal diameter of the circle o^ + ^ = a'. 33. Through the external point Pi = (a\, y^, a line is drawn meeting the circle a?-\-y^ = a? in Qand K; find the locus of middle point of PjQ as this line revolves about Pj. 34. A point moves so that its distance from the point (1, 3) is to its distance from the point (~4, 1) in the ratio 2 : 3. Find the equation of its locus. 35. Do the circles ^^f-2x-4.y-20 = and a:» + ?/» - 14 a; - 16 y + 100 = intersect? Show in two ways. 36. Find the equation of a circle of radius V85 which passes through the points (1, 2) and (4, ~3). 37. Find the equations of the tangents through (2, 3) to the circle 9{x' + f) +^x-12y + 4: = Q. 114 THE CIRCLE [Oh. V 38. What are the equations of the tangent and the normal to the circle 3? + y^ = 13, these lines passing through the point (3, 2)? through the point (6, 0)? 39. At what angle do the circles a^ + y^ — 6y4-2a! + 5 = and ar + ^ + 4j/ + 2a! — 5 = intersect each other ? 40. A diameter of the circle 4ar'4-4y^ + 8a; — 12 2/ + 1=0 passes through the point (1, ~1). Find its equation, and the equation of the chords which it bisects. 41. Find the locus of a point such that tangents from it to two concentric circles are inversely proportional to the radii of the circles. 42. A and B are two fixed points, and P a point such that AP= mBP, where m is a constant ; show that the locus of P is a circle, except when m = 1. 43. A point moves so that the square of its distance from the base of an isosceles triangle is equal to the product of its distances from the other two sides. Show that the locus is a circle. 44. If m is arbitrary, the circles represented by the equation (m + l){!i? + y^ = x + my have a common chord. CHAPTER VI POLAR COORDINATES 68. Of the many kinds of coordinates that may be used to fix the position of a point [Art. 14] thus far only the Cartesian has been studied, and especially the rectangular form, because of its greater symmetry [Art. 17]. But another kind, called polar coordinates, has special advantages for many problems, and is often used. It is implicitly used in the statement ' town A is 50 miles northeast from B ' ; i.e., the polar coordinates of a point are its direction and distance from a fixed point. 69. Polar coordinates. Given a fixed point O in a fixed di- rected line OR, and any point P in the plane, -with the distance 0P= p and the angle ROP= 6; then the position of P in the plane is fully determined by p and 6. -«.. o Fig. 46 a. The fixed line OR is called the initial line or polar axis, the fixed point the pole of the system, and the polar coordinates 115 116 POLAR COORDINATES [Ch. VI of the point P are the radius vector p and the directional or vectorial angle Q. The usual rule of signs applies to the vec- torial angle 6, and the radius vector is positive if measured from along the terminal side of the angle 6. The point P is designated by the symbol (p, 6). It is clear that one pair of polar coordinates (i.e., one value of p and one of 6) serve to determine one, and but one, point of the plane. On the other hand, if is restricted to values lying between and 2 v, then any given point may be desig- nated by four different pairs of coordinates. In this respect polar coordinates differ from Cartesian coordinates. •P. +240° ,.' -^ Vy^ R Fig. 47 o. 7 Fig. 47 6. / E.g., the polar coordinates (3, 60°) determine one point only, P^; but Pj may also be given by the three other pairs of coordinates : ("3, 240°), (3, -300°\ and (-3, -120°). -300^- R — >■ '-120° ^'°-47<=' / FIG. 47 d. 09, 70] POLAR COORDINATES 117 EXERCISES 1. Plot accurately the following points : (2, 20°), (2, ^\ (-7, ly (^'^'t)' ^^' ^^"°)' ^'^' ~^^^°^' ^^' '^^"^ ^"^' ^^^''^' (^'t) (^'i) (0.^) (6,0"), and (-6,0°). 2. Construct the triangle whose yertices are : ( 4, ^ ], [ 6, — p ), I 2, — ^ ) ; find by measurement the lengths of the sides and the coordinates of their middle points. 3. The base of an equilateral triangle, whose side is 5 inches, is taken as the polar axis, with the vertex as pole ; find the coordinates of the other two vertices. 4. Write three other pairs of coordinates for each of the points (2,^; (-3, 75°); (5, 0°); (0, 60°). 5. Where is the point whose radius vector is 7? whose Radius vector is ~1 ? whose vectorial angle is 25° ? whose vec- torial angle is 0'''' ? whose vectorial angle is ~180° ? 6. Express each of the conditions of Ex. 5 by means of an equation. 7. What is the direction of the line through the points (a,|)and(a,^y 70. Applications. The elementary problems given in Chap- ters II, III, and IV may often be solved to advantage by the use of polar coordinates. Some elementary formulas are ob- tained below. (a) Distance between two points. 118 POLAR COORDINATES [Ch. VI Let OR be the initial line,* the pole, and let Pi = (pi, d,) and 1\ = (pa ^2) be the two given fixed points. To find the dis- Fig. 48 a. Fig. 48 5 tance PjP, = d in terms of the given constants pi, p^, 61, and 63. In the triangle OP1P2 [cf. Art. 12] P^ = OP^ + OPi - 2 • OP, OPi • cos P^OP^ I.e., cP = pi" + pi — 2 p,pj cos (^2 - e,), hence ^ = "^Pi* + P2* - 2 p,P2 cos («2 - 'i) [31] (j8) .4rea 0/ a tHwiigle. r, Let the vertices of the triangle be Pi = (pi, 61), P,= (p„ e,), and P,= (fi3, 0$) ; to find its area A in terms of pi, pj, pj, 6j, 62, and ^3. Now, A = OP2P3 f"""-*"- +OP3P, + OP,P2;t but OP2P3 = \p2p3 sin (*8 — ^2). OP3P1 = i P3(Ji sin (61 — 6^, and OP1P2 = \ piP2 sin (63 — ^i) . .-. A = ^|pip2 sin (^2 - 61) + P2P3 sin (63 - e^) + p^,sm{0i-0,)l. [22] * The demonstration applies to each figure. t Area OPaPs and OP3P1 are positive ; but OP1P2 is negative, for the included angle is negative, — from OPi to OP2. 70] POLAR COORDINATES 119 (y) Ijoci. The study of loci by polar coordinates may be seen from an example : Given the equation p = 4 cos 6, to find its locus. This equation is satisfied by the following pairs of values, found as in Art. 27 (3) : ^2 = 30° ^3 = 60° 04 = 45° Os = 90° ^,= -30'' e, = - 60° 08 =-45° 0,= -9O° etc. p, = 4 P2 = 2 V3 = 3.46+ p3=2 P4=2V2 = 2.8+ P5 = pe = 3.46+ p, = 2 Pa = 2.8+ p, = etc. The corresponding points are : P, = (4, 0°) ; P, = (3.46+, 30°) ; P, = (2, 60°) ; P, = (2.8+, 45°) ; P, = P5 = the poleO=(0, ±90°); P„=(3.46+, -30°); P,= (2, -60°); etc. All these points are found to lie on a circle whose radius is 2, the pole being on the circumference, and the polar axis being a diameter. This circle is the locus of the equation p = 4 cos e. [Cf. Art. 71 (/3), equation 3]. EXERCISES 1. Find the distances between the points (2, 30°), f 3, -y^l, and ( 1, — ^ ], taken in pairs. 2. Prove that the points (a, ^J, (a, -^\ and (0, 0) form an equilateral triangle. 120 POLAK COORDINATES [Ch. VI 3. Do the points (7, 30°), (0, 0°), and ("11, 210°) lie on one straight line ? Solve this by showing that the area of the tri- angle is zero, and then verify by plotting the figure. 4. Find the area of the triangle (t, ^ ], [2ir, ^ V and 5. Find the area and also the perimeter of the triangle whose vertices are the points (3, 60°), (5, 120°), and (8, 30°). Find by the method of Ai-t. 36 where the following loci cut the polar axis (or initial line) : 6. p = ism''0. 7. p2 = a'cos2e. 8. A point moves so that its distance from the pole is nu- merically equal to the tangent of the angle which the straight line joining it to the origin makes with the initial line. Find the polar equation of its locus and plot the figure. Find the points of intersection of the curves, fp = 9cos0, Jp = 9cos(45°-0), ®' |pcostf = 4. ^°- ]pcoa(^-i\ = l. Discuss and construct the loci of the equations : 11. p'^ a" cos 2 0. 12. p = 3 0. 13. p = osin2tf. 14. Find the equation of a curve passing through the points of intersection of the curves p=2 cos 0, p cos = \. [Cf. Art. 38]. 71. Standard equations of the straight line and circle. (a) The straight line. (1) Line throiigh two given points. If point P be any point on the straight line through Pj and Pj, then the area of the triangle PPiPi is zero : hence, (equation [22]), for evei^/ point of the line PiPj the equation is [Fig. 51] 70, 71] POLAR COORDINATES ■ 121 PPi sin (9 — Oi) + pip2 sin (^i - O^) + p^ sin {6^ -6) = (i. [23] (2) Equation of the z, line in terms of the perpendicular upon it from the pole, and the angle which this perpendicular makes with the initial line. Fig. 51. Fio. fi2 Let OR be the initial line, O the pole, and LK the line whose equation is sought. Also, let N= (p, a) be the foot of the perpendicular from O upon LK, and let P= (j>, ff) be any other point on LK. Draw 0.?7"and OP; then ^= cos NOP, OP ' i.e., pcos(e-o)=i», [24] which is the required equation. (/8) The circle. Let OR be the initial line, O the pole, C= (fii, 0i) the center of the circle, r its radius, and P=(p, 6) any point on the circle. Draw OC, OP, and CP; then, by trigonometry, >-^ = p^ + p,= - 2 pp, cos (]9 - «,), i.e., p^ — 2 pip cos (0 — tfj) + p.' -7^ = 0, [26] which is the equation of the given circle. By changing the relative positions of the polar axis, the pole, and the center of the circle, equa- tion [25] takes several special forms, which follow. These the student should verify, and plot. Fio. 53. 122 POLAR COORDINATES [Ch. VI (1) If the center is on the polar axis ; p^ — 2 pi/j cos 6 + pi^ — 1^ = 0; 2) If the pole is on the circle ; p — 2rcos(e — ei) = 0; (3) If the pole is on the circle and the polar axis a diameter; p-2»'cose = 0; [25 «] (4) If the center is at the pole ; P = r. [25 ft] EXERCISES 1. Construct the lines : (a) p cos (6 + 30°) = 5 ; (c) pcos(e + ^=9; (6) p sin e = 4 ; (d) p cos (tf — w) = 10. 2. Find the polar equations of straight lines at a distance 3 from the pole, and : (1) parallel to the initial line ; (2) per- pendicular to the initial line. 3. A straight line passes through the points (10, 60°) and /10V3^ _ 3Qo\ . gjj^ .^g p^jg^j. equation. 4. Find the polar equation of a line passing through a given point (pi, ^i) and cutting the initial line at a given angle <(> = tan~' k. 5. Find the polar coordinates of the point of intersection of the lines p cos fe - ^= 10, p cos fe -f) = 5. 6. Derive equation (j3) (3) directly from a diagram. 7. Derive equation p (2) directly from a diagram. 8. Construct the circle whose equation is p = 4 cos 6. 71] POLAR COORDINATES 123 9. Find the polar equation of the circle whose center is at the point (4, ^ j and whose radius is 5; determine also the points of its intersection with the initial line. 10. Find the polar equation of a circle whose center is at the point (10, ^ J and whose radius is 2. Find also the equations of the tangents to the circle from the pole. 11. A circle of radius 4 is tangent to the two radii vectores which make the angles 60° and 120° with the initial line : find its polar equation, and the distance of the center from the origin. 12. Find the points of intersection of the loci : (a) p cos ($—^=a and /ocosf 6— - j = a; (fi) pcos [9-'^ = — and p^aaind. Construct the curves: 13. p = 2 e. IS. p = sec' e. 17. p = , 14. 2p= d. 16. p = sin 2 tf. 1 — cos fl CHAPTER VII TRANSFORMATION OF COORDINATES 72. That the coordinates of a point which remains fixed in a plane are changed by changing the axes to which this fixed point is referred, is an immediate consequence of the definition of coordinates. It is evident also that the different kinds of coordinates of any- given point (Cartesian and polar, for example) are connected by definite relations if the elements of reference (the axes) are related in position. E.g., the point Q, when referred to the polar axis OX and the pole O, has the coordinates ^ (5, 30°), but when it is referred to the ^^ rectangular axes OX and Y the co- j^ \30° ordinates of this same point are ^ ' "" (fV3,|). Again : while a curve remains fixed Fig. 54. jjj 3^ plane, its equation may often be greatly simplified by a judicious change of the axes to which it is referred. E.g., the line L^L, Fig. 65, when referred to the axes OX and OT, has the equation y = tan 6- x + b, but when referred to the axes O'X and O'V, the former of which is parallel to the given line, its equa- tion is y = c. 124 72, 73] CARTESIAN COOKDINATES ONLY 125 For these and other reasons, in the study of curves and sur- faces by the methods of an- alytic geometry, it will often be found advantageous to trans- form the equations from one set of axes to another. Formulas for the simpler changes of axes are derived in the next few articles.* Y L o ,,'• Fig. 56. I. CARTESIAN COORDINATES ONLY 73. Change of origin, new axes parallel respectively to the original axes. Let OX and OF be the original axes, O'X' and O'Y' the new axes, respectively parallel to the old, and let the coordinates of the new origin when referred to the original axes be h and k, i.e., O' = (h, k), where h = OA and k=AO'. Also let P, any point of the plane, be (x, y) when referred to the first axes OX and OY, and (x', y') when referred to O'X' and O'T. Draw MM'P parallel to the ?/-axis ; then OM=OA+ AM= OA + O'M', Fig. 56. as = 05' + ft, 1 y = y' + k ;i [26] I.e., and similarly, which are equations for translating the axes to the new origin. These formulas are, of course, independent of the angle between the axes. * For a more complete treatment of this subject see Tanner and Allen, Analytic Geometry, unabridged edition. Chapter VI, 126 TRANSFORMATION OF COORDINATES [Ch. VII As an illustration of the usefulness of such a change of axes, suppose there is given the equation x'-2hx + y^-2kyz=a^-h^-ie, (1) referred to the axes OX and Y. Now let P= {x, y) be any point on the locus L^L of this equation, and let (a;', y') be the same point P referred to the axes O'X' and 0' F', so that x = x' + h and y =y' -^-k. Substituting these values in the given equation gives: (a! + Kf-2h{x' + h) + {y' + Kf-21c(]i' + lc) = a^-h?-l2. 13. />*cos^ = fci. EXAMPLES ON CHAPTER VII 1. Find the equation of the locus of 2 ay — y + 6a; — 11 = referred to parallel axes through the point (-J-, ~3). 2. Transform the equation ar* — 4a^+43/^ — 6a; + 12.v = to new rectangular axes making an angle tan"' \ with the given axes. 3. Transform n^— x — xy + 2y — 2 = to parallel axes through the point (2, 3). Draw an appropriate figure. 4. Transform the equation of Ex. 3 to axes bisecting the angles between the old axes. Trace the locus. 5. To what point must the origin be moved (the new axes being parallel to the old) in order that the new equation of the locus 2a!»-5a!y — St^ — 2a! + 13y — 12 = shall have no terms of first degree ? Solution. Let the new origin be (h, k); then x = x' -{-h, y = y'-{-k, and the new equation is 2(x' + hy-5{x' + h)(ji,' + k)-3(y' + ky-2ix< + h) + 13(.v' + fc)-12 = 0, i.e., 2x"'-5x'y'-3y'<' + (4:h-5k-2)x'-{5h + 6k -13)y' + 2h*-5hk-3k'-2h + 13k-12 = 0; 76] POLAR COORDINATES 131 but it is required that the coefficients of x' and y' shall be ; i.e., h and fc are to be determined so that 4A-5fc- 2 = 0, and 5A + 6A;-13 = 0; hence li-=^ and fc = ^. Therefore the new origin must be at the point (JjL, ^), and the new equation is 2a;"'-5xy-32/'2-8 = 0. 6. The new axes being parallel to the old, determine the new origin so that the new equation of the locus ar' + 2an/-/ + 8as + 43/-8 = shall have no terms of first degree. 7. Transform the equations x + y — 4 = and Zx — 2y + 5 = to parallel axes having the point of intersection of these lines as origin. 8. Transform the equation - — 2 = 1 to new rectangular axes through the point (3, ~2), and making the angle tan~' (|) with the old axes. 9. Through what angle must the axes be turned that the new equation of the line Sx + y — 10 = shall have no y-term? Show this geometrically, from a figure. 10. Through what angle must the axes be turned in order that the new equation of the line 3x-\-2y = 12 shall have no a>term ? Show analytically. Solution. Let 6 be the required angle ; then the equations of transformation are X = x' cos d — y' sin 6 and y = x' sin 6 + y' cos 6 ; and the new equation is (6 cos e + 4 sin e)x' — (G sin fl - 4 cos ffjy' = 24 ; 132 TRANSFORMATION OF COORDINATES [Ch. VII but it is required that the coefficient of x be 0, 6cos6 + 4 sin^ = 0, i.e., tan6 = — |; whence = tan"' ( — |), and the equation becomes (6 sin fl - 4 cos e)y' + 24 = 0, which reduces to y' + 24 = 0, Vl3 t.e., to A/i3y + 12=0. 11. Through what angle must the axes be turned to remove the as-term from the equation of the locus ^ + By + C=0? to remove the y-term ? 12. Show that to remove the xy-teTia from the equation of the locus, 23p — 5xy — 3y' = S [cf. Ex. 6], the axes must be turned through the angle 6 = 67° 30', i.e., so that tan 2 = — l. What is the new equation ? 13. Through what angle must a pair of rectangular axes be turned that the new x-axis may pass through the point (2,5)? 14. What point must be the new origin, the direction of axes being unchanged, in order that the new equation of the line Ax + By + C=0 shall have no constant term ? 15. To what point, as origin of a pair of parallel axes, must a transformation of axes be made in order that the new equa- tion of the locus, a^ — 2xy—2x + iy = 0, shall have no terms of first degree ? Construct the locus. 16. Find the new origin, the direction of axes remaining unchanged, so that the new equation of the locus, x' + xy — x — 3^+4 = 0, shall have no constant term. Construct the figure. 76] POLAR COORDINATES 133 17. Transform the equation 5!i^ + 2xy + 5y' = 12 to new- rectangular axes making an angle of — 45° with the given axes, — origin unchanged. 18. Transform y^ = 4 a; to new rectangular axes having the point (9, 6) as origin, and making an angle tan"' ^ with the old. 19. Transform to rectangular coordinates, the pole and initial line being coincident with the origin and a>-axis, respectively : (a) p' = a'' cos 2 6, ()3) / cos 2 ^ = a', (y) p = k sin 2 6. Transform to polar coordinates, the aj-axis and initial line being coincident : 20. (a^ + 2r)' = Jc\a? — y^, the pole being at the point (0, 0). 21. a^— 2/^—4 X — 62/— 54 = 0, pole being at the point (2, ~3). 22. 3^+y'=S ax, the pole being at the point (4 a, 0). 23. Transform the equation y' + iay cot 30° — 4:xy = to an oblique system of coordinates, with the same origin and avaxis, but the new y-axis at an angle of 30° with the old avaxis. 24. Prove the formula for the distance in polar coordinates [21], p. 118, by transformation of the corresponding formula [1], p. 24, in rectangular coordinates. 25. Transform the equation xcosa4-y sin a=p to polar coordinates. 26. Given oblique axes making the angle X'OY' = u, and a second set, of rectangular axes, XO Y, with the same origin and ZXOX' = 0, ZX0Y'=4,, show by projection that a; = a;' cos 6 + y' cos <^ y = a^ sin ^ +y' sin <^. CHAPTER VIII THE CONIC SECTIONS EQUATION OF THE SECOND DEGREE Aa^ + 2 Mxy + Bj/*+ 2 Gx + 2 J'y + C = 0.» 77. Recapitulation. The position of a point may be repre- sented by numbers — its coordinates [Chapter II] — and there- fore a geometric curve may be represented by an algebraic equation [Chapter III], and the properties of the curve studied by means of the equation [Chapters III, IV, V]. The funda- mental relation is that the coordinates of every point of the curve, and of no other point, shall satisfy the equation. For a given point, the values of the coordinates will depend upon the system chosen — rectangular, or polar, etc. — and also upon the particular origin and axes chosen [Chapters VI, VII]. Hence, for a given curve, the corresponding equation will depend upon the system of coordinates chosen, and also upon the particular origin and axes. But [Art. 75] the degree of the equation in a chosen system does not depend upon the particular axes taken for reference, but upon the nature of the curve. For example, in rectangular coordinates the equa- tion of the first degree represents a straight line [Chapter IV]. In taking up the general study of loci, it is natural to classify * See Art. 153, general equation of second degree with three varialjles, to compare coefficients with those used here. 134 77, 78] EQUATION OF THE SECOND DEGREE 135 the loci according to the degree of their equations. Equations of the first degree having been studied, the various equations of the second degree will now be taken. It has, however, been found convenient with the straight line and circle to find, first, simple standard equations of the loci [equations 8-12, 16, and 17], to study the curves by means of these, and then to reduce more complicated equations to these familiar forms. This method of work is, in fact, what gives analytic geometry much of its usefulness. It can be used for equations of the second degree, since, happily, each represents some conic section, i.e., one of the curves of intersection of a plane and a right circular cone. These loci are the ellipse, the parabola, and the hyperbola (with the circle and straight line as special cases) ; they are of primary importance in astronomy, where it is found that the orbits of the earth, the comets, and other heavenly bodies are curves of this kind. 78. The conic sections. The conic sections, or more briefly, conies, may be defined thus : A conic is a plane curve traced by a point ivhich moves so that its distance from a fixed point hears always a constant ratio to its distance from a fixed straight line. The general equation of this locus will first be derived, then from it the various special standard forms will be obtained. (a) Tlie equation of the locus. Let F be the fixed point, — the focus of the curve ; D'D the fixed line, — the directrix of the curve ; and e the given ratio, — the eccentricity of the O curve. L ■ Let D'D be the y-axis, and the per- pendicular to it through F, i.e., the line OFX, be the avaxis. Let P = {x, y) be any position of the generating point, -,-JP F Fig. 60. 136 THE COMC SECTIONS [Ch. VIII and let 0F= k be the fixed distance of the focus from the directrix ; then F = (fc, 0). Draw FP, also LP ± D'D. Then FP : LP = e, [geom. eq.] but FP=\'{x-ky+i/, and LP = x, hence ■\/(x — ky- + y' = ex; [alg. eq.J I.e., {l-^x' + y'-2kx + le' = 0, (1) which is the equation of the given locus. This equation is of the second degree ; later it will be shown that every equation of the second degree between two variables represents a conic section. (6) Discussion of equation (1). Equation (1) shows that for every value of x, the two corresponding values of y are numerically equal but of opposite signs, hence the conic is symmetrical with regard to the x-axis as here chosen. For this reason the line drawn through the focus of a conic and perpendicular to the directrix is called the prin- cipal axis of the conic. If x = 0, then y= ±A;V— 1, which shows that this curve does not intersect the y-axis as here chosen ; i.e., a conic does not intersect its directrix. If y = 0, then(l-e=^ar'-2fea! + »r' = 0, k k whence « = - , or x = - , (2) 1+e 1 — e ^ ' hence, ife=^l, the conic meets the principal axis (the x-axis as here chosen) in two points. These points are called the vertices of the conic. If e = 1, tlie conic meets the principal axis in only one point at a finite distance from the directrix. This point is then spoken of as the vertex. The form of the locus of equation (1) depends upon the value of the eccentricity, e ; if e = 1, the conic is called a parabola ; if 78, 79] THE PARABOLA 137 e < 1, an ellipse; and if e > 1, an hyperbola. Each of these cases will now be sepai'ately considered. I. THE PARABOLA Special Equation of Second Degree Ax^ + 2Gx + 2Fy + C = 0, or By^ +2Gx + 2Fy + C = 79. The parabola defined. A parabola is the locus of a point which moves so that its distance from a fixed point is equal to its distance from a fixed line. It is the conic section with ec- centricity e = 1. The equation of a parabola, with any given focus and direc- trix, can be obtained directly from this definition. Example. To find the equation of the parabola whose directrix is the line x — 2y—l = 0, and whose focus is the point (2, "3). Let P = {x, y) be any point on the parabola ; then ^ — is the distance of P from the directrix -^^ [Art. 51], and V(a; — 2)'-|- {y -\- SY is the distance of P from the focus [Art. 193; , X — 2v — 1 , ^"""^ _^g =V(a;-2)^-Ky-h3y, by definition ; that is, 4a^ + 4a:y-|-?/=-18x-|-26y-f-64 = 0; which is the required equation. The equation obtained in this way is not, however, in the most suitable form from which to study the properties of the curve, but can be simplified by a proper choice of axes. If the line of symmetry is taken as tlie a>axis, the equation will have no y-ievn\ of first degree [cf. Art. 78, eq. (1)]; 138 THE CONIC SECTIONS [Ch. VIII ■while if the vertex of the curve be taken as origin, the equation will have no constant term, since the point (0, 0) must satisfy the equation. With this choice of axes, the equation of the parabola will reduce to a simple form, the first standard equa- tion of the pat-abola. 80. First standard form of the equation of the parabola. Let D'D be the directrix of the parabola, and F its focus ; also let ZFX, perpendicular to the directrix, be the a!-axis; denote the fixed distance ZF by 2 J), and let 0, its middle point, be the origin ; then Y, perpendicular to OX, is the y-axis. Let P = {x, y) be any point on the curve, and draw LQP perpendicular to OY; also draw the ordinate MP, and the line FP. The line FP is called the focal radius of P. To apply the method of Art. 79, the equation of the directrix and the coordinates of the focus are needed. Since ZO=OF=p, the equation of the directrix is x+p = 0, (1) a,T\d the fociig is the point (jJ, 0). (2) Now, from the definition of the parabola, FP = LP; [geom. eq.] but FP:=^/(x-py + y% and LP= ZO + OM=p + x; D L S T ^ P^ M X Z 6 s' \ hence V(a; —pf + y^= {x +p), [alg. eq.] whence V^ = 4c px, [32] which is the desired equation. This first standard form [32] is the simplest equation of the parabola, and the one which will be most used in the sub- sequent study of the curve. It will be seen later [Arts. 84, 98, T9-82] THE PARABOLA 139 102] that any equation which represents a parabola can be reduced to this form. 81. To trace the parabola y^ =4Lpx. From equation [32] it follows that (Art. 30) ; (1) The parabola passes through the point 0, halfway from the directrix to the focus, the vertex of the curve. (2) The parabola is symmetrical with regard to the avaxis ; i.e., with regard to the line through the focus perpendicular to the directrix, the axis • of the curve. (3) The abscissa x has always the same sign as the constant p, i.e., that the entire curve and its focus lie on the same side of a line parallel to the directrix, and midway between the directrix and the focus. (4) The abscissa x may vary in magnitude from to oo ; and when X increases, so also does y (numerically); hence the parabola is an open curve, receding indefinitely from its directrix and its axis. The parabola is, then, an open curve of one branch which lies on the same side of the directrix as does the focus; when constructed it has the form shown in Fig. 61. 82. Latus rectum. The chord through the focus of a conic, parallel to the directrix, is called its latus rectum. In Fig. 61 this chord is .B'B. Now B'R = 2FIi=z2SR = 2ZF=ip. Hence the length of the latus rectum of the parabola is 4 p; that is, it is equal to the coefficient of x in the first standard equation. • The axis of a curve should be carefully distinguished from an axis of coordinates ; though they olten are coincident lines in the figures to be studied. 140 THE CONIC SECTIONS [Oil. vin 83. Intrinsic property of the parabola. Second standard equa- tion. Equation [32j may be interpreted as stating a geometric property of the parabola, — a property which belongs to every point of the parabola, whatever coordinate axes be chosen. For (see Fig. 61) the equation y'^=.^:px gives the geometric relation mP = 4: of- 0M= E'E ' OM, or, expressed in words. If from any point on the parabola, a perpendicttlar is drawn to the axis of the curve, the square on this perpendicular is equivalent to the rectangle formed by the latus rectum and the line from the vertex to the foot of the perpendicular. This geometric property enables one to write down immedi- ately the equation of the parabola, whenever the axis of the curve is parallel to one of the coordinate axes. E.g., if the vertex of the parabola is the point A = (ft, fc), and its axis is parallel to the a^axis, as in the figure, F being the focus and P = {x, y) any point on the parabola. Draw MP perpendicular to the axis AK. Then MP' = 4.AF.AM, i.e, {v-hf^4.p{x-h), [33] which is the equivalent algebraic equation. This may be taken as a second standard form of the equation ; it represents the parabola whose vertex is at the point (h,Tc), whose axis is parallel to the as-axis, and, if p is positive, which lies whollj on the positive side of the line x = h. 83] THE PARABOLA 141 Equation [33] evidently may be reduced to equation [32] by a transformation of coordinates to parallel axes through the vertex as the new origin. Again, if the vertex is A = (h, k) and the axis is parallel to the y-axis, then the equation is Cx-hf=4:p(y-k), [34] which is another form for the second standard equation of the parabola. (In Fig. 63, P is negative.) Fio. 63. EXERCISES Construct and find the equations of parabolas : 1. With the focus at the point (~3, "3) and with the line x + y = 2 as directrix. 2. With the focus at the origin, and with the line y — 2x + 3=0 as directrix ; 3. With the vertex at the origin, and the focus at the point (4,0); 4. With the vertex at the origin, and the focus at the point (0, -a); 5. With the vertex at the point (~2, 5), and the focus at the point (-2, 1) ; 6. With the vertex at the point ('*^2, +4), and the focus at the point (-1, +4); 7. With the focus at the point (0, 2 p) and the line y = as directrix. 142 THE CONIC SECTIONS [Ch. VIII 8. What is the latus rectum of each of the parabolas of exer- cises 3 to 6 ? 9. Describe the effect produced on the form of a parabola by increasing or decreasing the length of its latus rectum. 84. Every equation of the form Ax^ + 2 Gx + 2 Fy + C = O, or By- + 3 Gx + 2 Fy + C = O, represents a parabola whose axis is parallel to one of the coordinate axes. Equations [32], [33], and [34] are of the form By^ + 2Gx + 2Fy+C=0, or AJ + 2Gx + 2Fy+C=0; that is, each has one and only one term containing the square of a variable, and no term containing the product of the two variables. Conversely, it may be shown that an equation of either of these forms represents a parabola whose axis is parallel to one of the coordinate axes. A numerical example will indicate the method of proof [of. Art. 56] ; it is analogous to " completing the square " in the solution of quadratic equations. ExiMPLE. Given the equation 2o/-30 2/-50.'(; + 89 = 0, to show that it represents a parabola ; and to find its vertex, focus, and directrix. Divide both members of the equation by 25, and complete the square of the 2/-terms ; the equation may then be written that is, (y-if = 2(x-l), whence ()/ — 1)2 = 4 • | • (a; — f ). Now this equation is in the second standard form (cf. equation [33]), and therefore every jjoiut on its locus has tlie geometric property given in Art. 83; 83, 84] THE PARABOLA 143 and the locus is a parabola. The vertex is at the point (I, f ) ; the axis is parallel to the x-axis, extend- ing in the positive direction ; and, since p = -^^ the focus is at the point (f^, f), and the directrix is the line a; = -j-j. Now by the same method the more general equation, viz., Ax' + 2Gx + 2Fy+C=0, (1) may be reduced to the form which is of the standard form [34] of a parabola. Note. The transformation just given fails if ^ = or if J?" = 0, for in that case some of the terms in the last equation are infinite. If, however, ^ = 0, the given equation becomes 2 Gx -i- 2 Fy + C = 0; and, this being of the first degree, represents a straight line. If, on the other hand, F=0, the given equation reduces to Ax- + 2 Gx + C = 0, and represents two straight lines each parallel to the y-axis ; they are real and distinct, real and coincident, or imaginary, depending upon the value of G^ — AC. These li^es may be regarded as limiting forms of the parabola. EXERCISES Determine the vertex, focus, latus rectum, equation of the directrix and of the axis for each of the following parabolas ; also sketch each of the figures : 1. y^ — Ay = — 4:X + 8; 4. Ay- — Say -x + ia' = b; 2. 3a^ + 6a;-y + 7=0; 5. Ax- + 2 Gx +2 Fy + C=0; 3. 5y-l=3y^ + ix; 6. By' + 2 Gx + 2 Fy + C=0. 7. Derive equation [32] as a special case of equation (1), Art. 78. 8. Reduce the equation 4 a--^ 4- 4 xy + y^ — 18 x- -f- 26 y + G4 = [cf. Art. 79] to form [32] by translating axes to the point (2, -3) as origin, then rotating the axes through the angle 6 = tan~' ^. 144 THE CONIC sections' [Ch. VIII II. THE ELLIPSE Special Equation of the Second Degree Ax^ + By^ + 2 Gas + 2 J'y + C = O. 85. The ellipse defined. An ellipse is the locus of a point which moves so that the ratio of its distance from a fixed point, to its distance from a fixed line, is constant and less than unity. This curve is the conic section with eccentricity e < 1. The equation of an ellipse with any given focus, directrix, and eccentricity may be readily obtained from this definition. Example. An ellipse of eccentricity | has its focus at (2, ~]), and has the line x-\-2y = 5 for directrix: To find its equation. Let P= (x, y) be any point on the curve, F the focus, and PQ the perpendicular from. P to the directrix. Then FP=|QP; but FP = V(a!-2)2+(2/ + l)», + Vo hence (x-2y+(l/ + i)' = M^ + 2y-By; that is, 4l!f^-16a!!/+292/'-140a;+1702/4-125=0; which is the equation of the given ellipse. To obtain the simplest form of the equation, more suitable for studying the properties of the curve, the line of symmetry should be chosen as the ar-axis, with the origin halfway between the two vertices. [Art. 78 (?>)]. The resulting equation is the Jirst standard form of the equation of the ellipse. It is derived in the next article. 86. The first standard equation of the ellipse. Let F be the focus, D'D the directrix, and ZFX the perpendicular to D'D 85, 80] THE ELLIPSE 145 through F, cutting the curve in the two points A' and A. Denote by 2 o the length of AA', and let be its middle point, making AO = OA' = a-, and let ZX be the X-axis, the ori- gin, and OT, per- pendicular to OX, the y-axis. Now the equar tion of the direc- trix and the coor- dinates of the focus are needed, so that the method of Art. 85 may be applied. For the directrix : ZO = ZAJrAO; but 2{ZA + A0)=ZA+ {ZA -|- AA!) =ZA + ZA', and, by definition of the ellipse, AF = eZA and FA' = eZA', _AF+F'A_AA' 2a L Z D Y B \ \ X a[ ]^ a-e D' ^ M jA' B' Fig. 64. hence 2{ZA + A0) = Therefore, e and the. equation of the directrix is x + - = 0. e For the focus: F0 = AO-AF=^FA' - FO-AF. 2F0 = FA' -AF=e (ZA' - ZA), i.e., = eAA' = 2 ax, therefore FO = tie, and the fonts F is the point (~ae, 0). (1) (2) 146 THE CONIC SECTIONS [Ch. VIII For the equation of the ellipse: Let Ps{x, y) be any point on the curve, and draw LP ±. U'D; then FP=eLP; [geom. eq.] a but FP=^\^(x + aey + y', LP=- + x; hence (ae + xy + f = e'fx + -\, [alg. eq . ] that is, (l-e^j^ + y' = a'(l-e^, that is, -.+ ,,f ., = !• (3) From equation (3), the intercepts of the curve on the y-axis are ± aVl — e^. Both intercepts are real, since e < 1 ; hence the ellipse outs the y-axis in two real points B and B', on opposite sides of the origin 0, and equidistant from it. If OB is denoted by b, so that 62 = a=(l-e»), (4) equation (3) takes the form This is the simplest equation of the ellipse. As will be seen later, every equation representing an ellipse can be reduced to this form. a a 87. To trace the ellipse ^ + ^ = 1. From equation [35] it follows that : (1) The ellipse is symmetrical with regard to the ai-axis ; i.e., with regard to the line through the focus and perpendicular to the directrix, the principal axis of the curve ; If a = 6 (i.e., if e = 0) this equation represents a circle. The ellipse, then, includes tlio circle as a special case. In other words : a circle is an ellipse whose eccentricity is zero. 86, 87] THE ELLIPSE 147 (2) The ellipse is symmetrical with regard to the y-smis also ; i.e., with regard to a line parallel to the directrix and halfway between the vertices (Fig. 65) ; (3) For every value of x from — a to + a, the two cor- responding values of y are real, equal numerically, bat opposite in sign ; and for every value of y from —b to b, the two values of X are real and equal numerically, but opposite in sign ; and that neither x nor y can have real values beyond these limits. The ellipse is, therefore, a closed curve, of one branch, which lies wholly on the same side of the directrix as the focus ; and the curve has the form represented in Fig. 65. D r B D, L /i « --' ,,--- ""^vf L' Z ^-, ^-ft:^-"- " \\f' \ Z' X A r* a-e- • L 1 V 1 V ( M h' D' F^ D[ Fig. 65. The segment AJ! (Fig. 65) of the principal axis intercepted by the curve is called its major or transverse axis; the cor- responding segment B'B is its minor or conjugate axis. From the symmetry of the curve with respect to these axes it follows that it is also symmetrical with respect to their intersection O, the center of the ellipse. From this symmetry about the !/-axis it follows also that the ellipse has a second focus, F* = (ae, 0) (Fig. 65) and a second directrix D\Di on the positive side of the minor axis, whose equation is X = 0. 148 THE CONIC SECTIONS [Ch. vm The latus rectum of the ellipse, i.e., the focal chord pavalle] to the directrix [Art. 82], is evidently twice the ordinate of the point whose abscissa is ae. But if Xi = ae, yi = 6Vl— c^; or, jj since & = aVl — e', 2/i=-- a Hence, the lotus rectum is — • a 88. Intrinsic property of the ellipse. Second standard equation. Equation [35] states a geometric property which belongs to every point of the ellipse, whatever the coordinate axes chosen, and to no other point : viz., if P be any point of the ellipse (Fig. 65), then i.e., stated in words : If from any point on the ellipse a perpendicular be drawn to the transverse axis; then the square of the distance from the center of the ellipse to the foot of this pei-pendicular, divided by the square of the semi-transverse axis, plits the square of the perpen- dicular divided by the square of the semi-conjugate axis, equals unity. Y Fig. 66. This geometric or physical property belongs to no point not on the curve, and therefore completely determines the ellipse. It enables one to write immediately the equation of any ellipse whose axes are parallel to the coordinate axes. 87, 88] THE ELLIPSE 149 For example : if, as in Fig. 66, the major axis of an ellipse is parallel to the a^axis, and the center is at the point G = (h, k), let P= (aj, y) be any point on the curve, and a, b be the semi- axes, then CM' id2L + ^=l, that is .2 a' GW 6« ^' [36 a] which is the equation of the given ellipse. Or again, if, as in Fig. 67, the major axis is parallel to the y-axis; then, [36 6] (y - fc)g , (x-h-)\ ^ which is the equation of the given ellipse. Equations [36 a and 6] may be considered second standard forms of the equation of the ellipse ; by a change of coordinates to a set of par- allel axes through the center C = (A, k), as the new origin, either can be reduced to the first standard form. By Art. 86 the distance from the center of an ellipse to its focus is ae ; but since 6^ = o^ (1 — e^ * [Art. 86, equation (4)], therefore ae= Va^— 6"; hence, in Figs. 66 and 67, yf A \ Y I J k \ f. ^ ' B ] J-b-. ' B' i! G i W ^ / O X K T F ■IQ. 67. * The student should observe that 6 is the semi-minor axis and not necessarily the denominator of y^ in the standard forms of the equation of the ellipse — [35] and [36 a] and [30 6] ; he should also observe that the foci are always on the major axis. 150 THE CONIC SECTIONS [Ch. VIIJ Again, the equation of an ellipse, in either standard form, gives the semi-axes as well as the center of the curve, there- fore the positions of the foci are readily determined from either standard form of the equation. EXERCISES Construct the following ellipses, and find their equations : 1. Given the focus at the point (3, 3), the equation of the directrix a; + y — 3 = 0, and the eccentricity ^; 2. Given the focus at the origin, the equation of the direc- trix a; = 9, and the eccentricity ^ ; 3. Given the focus at the point (0, 1), the equation of the directrix y — 8 = 0, and the eccentricity \ ; 4. Given the center at the origin, and the semi-axes V3) VT. Find also the latus rectum. Find the equation of an ellipse whose center is at the origin, whose axes are the coordinate axes : 5. "Which passes through the two points (~2, ~2) and (-3, "^l) ? How many such ellipses are there ? 6. Whose foci are the points (3, 0), ("3, 0), and eccentric- ityi; 7. Whose foci are the points (0, 6), (0, ~6), and eccentric- ity |; 8. Whose latus rectum is 5, and eccentricity | ; 9. Whose latus rectum is 8, and minor axis 10 ; 10. Whose major axis is 10, and which passes through the point (4, f). 88,80] THE ELLIPSE 151 Draw the following ellipses, locate their foci, and find their equations : 11. Given the center at the point (3, ~2), the semi-axes 4 and 3, and the major axis parallel to the avaxis ; 12. Given the center at the point (1, 8), the semi-axes 4 and 9, and the major axis parallel to the 2/-axis; 13. Given the center at the point (0, 7), the origin at a vertex, and (2, 3) a point on the curve ; 14. Given the circumscribing rectangle, whose sides are the lines x — a = 0, y — c = 0, x + b =0, y + d = 0; the axes of the curve being parallel to the coordinate axes. 15. If 6 becomes more and more nearly equal to a, what curve does the ellipse approach as a limit? What is the limit of e ? of the directrices ? 89. Every equation of the form Ax^ + By^ + 2 €fx + 2 Fy + C = 0, in which A. and S have the same sign, represents an ellipse whose axes are parallel to the coordinate axes. Equa- tions [35] and [3C], obtained for the ellipse, are all, when expanded, of the form AxP+By'' + 2Gx + 2Fij+O = 0, (1) where A and B have the same sign, and neither of them is zero. Conversely, au equation of this form represents an ellipse the axes of which are parallel to the coordinate axes. A numer- ical example will suggest the general proof. Example. Given the equation ia^ + 9y^ — 16 x -f IS?/ — 11 = 0, to show that it represents an ellipse, and to find its elements. On completing the square for the terms in x, and also for those in y, and trans- posing, this equation becomes 152 THE CONIC SECTIONS [Ch. VIII 4ie»-16a! + 16 + 92/» + 182/+ 9=11 + 16 + 9, that is 4(a; - 2f + 9{y + 1)^ = 36 ; hence ^:^ + (^L+lH' = l. 32 T- 2= This equation is of the form [36], and, therefore, its locus has the geometric property given in Art. 88, and is an ellipse. Its center is the point (2, —1); its major axis is parallel to the x-axis, of length 6; its minor axis is of length 4 ; the foci are the points i?"=(2-V5,-l), 2f=(2 + V5, -1); and the equations of the directrices are, respectively, V5 x=2+-^. V6 EXERCISES Determine for each of the following ellipses the center, semi-axes, foci, vertices, and latus rectum; then sketch each curve : 1. 3a:' + 9jr*-12a!-182^-6 = 0. 3. 16ar' + y= + 60a; + 42/ + 15 = 0. 4. A3? + Bif' + 2Gx+2Fy+G=Q. 89-91] THE HYPERBOLA 153 5. By completing the squares of the x-terms and of the y-terms, and a suitable transformation of coordinates, reduce the equations of exercises 1, 2, and 3 to the standard form [35]. III. THE HYPERBOLA Special Equation of the Second Degree Ax^ — By^ + 2 Gx + 2Fy+C = 90. The hyperbola defined. An hyperbola is the locus of a point which moves so. that the ratio of its distance from a fixed point, to its distance from a fixed line, is constant and greater than unity. This curve is the conic section with eccentricity e > 1. 91. The first standard form of the equation of the hyperbola. Let F be the focus, D'D the directrix, the line of symmetry OFX as the a>-axis, and CY perpendicular to OX halfway between the two vertices A and A' [see Art. 78] the y-axis. Let 2 a be the distance A' A be- tween the vertices. Since the hyperbola differs from the ellipse only in the relative values of e and 1, i.e., in the sign of 1 — e', which is + in the ellipse and — in the hyperbola, the standard equation of the hyperbola can bo derived by the method of Art. 86, and it will be found that with choice of axes and notation as there given, the results given in equations (1), (2), and (3) of that article apply equally to the hyperbola. If now, since 1 — 6* is negative, the sub- stitution b^ = a''{^ — l) is made, equation (3), Art. 86, will FiQ. 69. 154 THE CONIC SECTIONS [Ch. VIU 2 2 become fi-^ = l' [37] the standard equation of the hyperbola. Every equation representing an hyperbola can be reduced to this form, as is shown later. Here, as in Art. 86, CF=ae, therefore the focus F is the point (ae, 0). (4) Similarly for the directrix ; C0 = -, e hence the directrix is the line x = 0. (5) e As above defined, b is real, and its value is known when a and e are known. In Fig. 69, CB=b, CB' = -b, and 6 = aVe^-l. (6) 92. To trace the hyperbola —2-T^=^' Equation [37] shows that : (1) The hyperbola is symmetrical with regard to the aj-axis, i.e., with respect to the line through the focus and per- pendicular to the directrix, called the principal axis of the hyperbola. (2) The hyperbola is symmetrical with regard to the y-axis also; i.e., with regard to the line parallel to the directrix and halfway between the vertices. (3) For every value of x from — a to a, y is imaginary ; while for every other value of x, y is real and has two values, equal numerically but opposite in sign. But for every value of y, x has two real values, equal numerically and opposite in sign. When X increases numerically from a to oo , then y also in- creases numerically from to oo . ni-93] THE HYPERBOLA 155 These facts show that no part of the hyperbola lies between the two lines ijerpendicular to its principal axis and di'awn through the vertices of the curve ; but that it has two open infinite branches, lying outside of these two lines. The form of the hyperbola is as represented in Fig. 70. The segment A'A of the principal axis, intercepted by the curve, is called its transverse axis. The segment B'B of the second line of symmetry (the y-axis), where B'O = OB = 6, is called the con- jugate axis ; and although not cut by the hyperbola, it bears important relations to the curve. From the symmetry of the hyperbola, with respect to these axes, it follows that it is also symmetrical with respect to their intersection 0, the center of the curve. From this symmetry about the y-axis it follows also that there is a second focus, the point i^ = (~ae, 0), and a second directrix on the negative side of the conjugate axis, whose equation is a; + - = 0. [Cf. Art. 87]. The latus rectum of the hyperbola is readily found to be — . [Cf. Arts. 82, 87]. a 93. Intrinsic property of the hyperbola. Second standard equa- tion. Equation [37] states a geometric property which belongs to every point of an hyperbola, whatever the coordinate axes chosen, and to no other point ; and which therefore completely defines the hyperbola. With the figure and notation of Art. 91 equation [37] states that (see Fig. 70), Fia. 70. 156 THE CONIC SECTIONS [Ch. VIII OM MP OA' OB' = 1, a property entirely analogous to that of Art. 88 for the ellipse. It enables one to write at once the equation of an hyperbola with given center and semi-axes, — the axes of the curve being parallel to the coordinate axes. \ Y \ « / if. ^x / Xs ^ X A C 1£, B B' J ^F ^ \ X \ \ Fio. 71. Fig. 'a. For example, if the transverse axis is parallel to the avaxis, as in Fig. 71, and the center at the point C = (A, k), and if P = (x, y) is any point on the curve, then CW MP CA^ CB' = 1, I.e., (a>-K)'^ (?/-fc)^ _l^ [38 a] which is the equation of the hyperbola, with a and 6 as semi- axes. Again, if the transverse axis is parallel to the y-axis, as in Fig. 72, with the center at the point (/i, k), the equation of the hyperbola will be found to be 93] THE HYPERBOLA 157 (OS - fe)' {y - K)^ , ^ _ *•«■' 6' ~ a' -- ^- [38 6] Note 1. That tbe expressions obtained on p. 184 for the distances from the center to the focus and the directrix, of hyperbola [37], are equally true for hyperbolas [38 a] and [38 6] follows from the fact that those expressions involve only a, 6, and e ; moreover, equation (6) of Art. 91 determines e in terms of a and 6 ; hence, for all these hyperbolas e" = 2-i — , the distances from the center to the foci are given by ae = ± VcF+V', and those to the directrices by e ± Vo2 + 62 Note 2. It should be noticed that in equations [37], [38 a], [38 6], the negative term involves that one of the coordinates which Is parallel to the conjugate axis. EXERCISES 1. Find the equation of the hyperbola having its focus at the point (+1, +1), for its directrix the line ^ = 4, and eccen- tricity 2. Plot the curve. [Cf. Art. 78, and Art. 84, Ex.]. Find the equation of the hyperbola having its center at the origin and 2. Semi-axes equal, respectively, to 3 and 5 ; 3. Conjugate axis 16, — the point (20, 5) being on the cuive ; 4. The distance between the foci 10, and eccentricity Vf; 5. The distance between the foci equal to twice the trans- verse axis. [Two solutions for Ex. 2, S, 4, and 5.] 158 THE CONIC SECTIONS [Ch. VIII Find the equation of an hyperbola: 6. With center at the point (4, 5), semi-axes 5 and 6, and the transverse axis parallel to the a^axis ; plot the curve ; 7. With center at the point ("4, ~o), semi-axes 3 and 2, and the transverse axis parallel to the y-axis. Plot the curve. 8. Find the foci and latus rectum for the hyperbolas of exercises 6 and 7. 9. By a suitable transformation of coordinates, reduce the equation of exercise 6 to the standard form — — -^ = 1. 10. Find the foci of the hyperbolas, («) f-f = 1. m ^-f = 1. (r) -1-^ = 1. Plot the curves (fi) and (y). 94. Every equation of the form Ax'^ -f- By'^ + 2 Chc + 2Fy -I- C = O, in wbich A and S have unlike signs, represents an hyper- bola whose axes are parallel to the coordinate axes. When cleared of fractions and expanded, the three equations found for the hyperbola are of the form Ax' + By'+2Gh;+2Fy+C=0, (1) where A and B have opposite signs, and neither of them is zero. Ccnversely, it may be shown that every equation of this form represents an hyperbola, whose axes are parallel to the co- ordinate axes. A numerical example will suggest the general proof. Example : The equation 9 ar" - 42/2 - 18 a; -I- 24 y- 63 = may be written 9 (x — 1)^ — 4(7/ — 3)" = 36 ; ie (x-1)^ (y-3f _^. 03-95] THE HYPERBOLA 159 hence [Cf. 38 a] its locus is an hyperbola, with center at (1, 3), transverse axis of length 4 and parallel to the a^axis, conjugate axis of length 6 ; also, e = ^^-^i its foci the points (1— a/IS, 3) and (1+Vl3, 3), its directrices the lines x = X ^^ and « = 1 H — - — Vl3 V13 EXERCISES Determine for each of the following hyperbolas the center, semi-axes, foci, vertices, and latus rectum : 1. 16/-8a:=-(-64y-36a; + 10 = 0. 2. ar'-5y2-f-153/-10a;H-l = 0. 3. 2y + 6a; + 3ar'=2^-f-7. 4. ^x»-JSj/«-f-2G'a;-|-2i?V+C=0. 5. Reduce the equations of exercises 1, 2, 3, to the standard form — — ^ = 1. Sketch each curve. a' b- 95. Summary. In the preceding articles it has been shown that the special equation of the second degree, A3i?+B}/+2Gx-ir2Fy+C=(i, (1) always represents a conic section, whose axes are parallel to the coordinate axes. There are three cases, corresponding to the three species of conic. (1) Tlie parabola ; either .4 or 5 is zero. In exceptional cases this curve degenerates into a pair of real or imaginary parallel straight lines, and these may coincide. [Arts. 84, 53]. (2) Tlie ellipse: neither A nor B is zero, and they have like signs. In exceptional eases this curve degenerates into a circle, a point, or an imaginary locus. [Arts. 89, 86, 53]. (3) The hyperbola : neither A nor B is zero, and they have 160 THE CONIC SECTIONS [Ch. VIU unlike signs. In exceptional cases this curve degenerates into a pair of real intersecting lines. [Arts. 94, 53]. The ellipse and hyperbola have centers, and therefore are called central conies, while the parabola is said to be non- central.* The equation for each conic has two standard forms, which state a characteristic geometric property of the curve, and to which all other equations representing that species can be reduced. These standard forms are the simplest for study- ing the curves ; but the student must discriminate carefully between general results and those which hold only when the equation is in the standard form. 96. Polar equation of the conic. Based upon the " focus and directrix " definition already given [Art. 78], the polar equation of a conic section is easily derived. Let D'D (Fig. 73) be the given line (the directrix) and the given point (the focus) ; draw ZOR through O and perpendic- ular to D'D, and let O be chosen as the pole and OR as the initial line. Also let P= (j>, 6) be any point on the locus, and let e be the eccen- p^^___ tricity. Draw MP and OK parallel, and ' 'k,^^7\ LP and HK perpendicular to D'D, and "Am\ let OK=l\ then loy '. I B_ ' { -^ OP = e- LP, [definition of the curve] ^v,^^ =e{ZO-\- OM) ; Fig. 73. _.. ^ = g Z'- -|- p COS bX This equation, when solved for p, may be written in the form • It is at times more convenient to consider that the latter curve lias a center at infinity, on the principal axis. 95-97] THE GENERAL EQUATION ICl 1 — e cos 8 [39] which is the polar equation of a conic section referred to its focus and principal axis; e being the eccentricity and { the senii-latus-rectum. If e = 1, equation [39] represents a pa- rabola ; if e < 1, an ellipse ; and if e > 1, an hyperbola. Equation [39] shows that if e < 1, i.e., if the equation repre- sents an ellipse, there is no value of 6 for which p becomes in- finite. Therefore there is no direction in wliich a line may be drawn from the focus to meet an ellipse at an infinite distance. If e = 1, i.e., if the equation represents a parabola, there is one value of 0, viz., 6 = 0, for which p becomes infinite. Therefore there is one direction in which a line may be drawn to meet a parabola at infinity. If e > 1, i.e., if the equation represents an hyperbola, there are two values of 0, viz., $ = ± cos"' (1 : e), for which p becomes infinite. Therefore there are two direc- tions in which a line may be drawn to meet an hyperbola at infinity. rV. THE GENERAL EQUATION dx* + 2 Hxy + J5J/2 + 2Gx + 2Fy+Cs:0 Reduction to the Standard Form 97. Condition that the general quadratic expression may be factored : the discriminant. The most general equation of the second degree in two variables may be written in the form Ax' + 2Hxy+By''+2Gx-{-2Fi/+C = 0. (1) If factorable, this equation represents two straight lines. [Arts. 37 and 53]. It is required to find the relation that must exist among the coefficients of this equation in order that its first member may be separated into two rational factors, each 162 THE CONIC SECTIONS [Ch. Vlii of the first degree, i.e., it is required to find the condition that the equation may be written thus : (a,a; + 6i!/ + Ci)(a2a; + 622/ + C2)=:0. (2) Evidently if equation (1) can be written in the form of equation (2), then the values of x obtained from equation (1) are either -Ci- 6,V — Cj- M or X =x y «! ttj and are therefore rational. Solving equation (1) for x in terms of y, by completing the square of the a;-terms ; ^V + 2 A(Hy + G)x + {Hy + Of = -ABf~2AFy-AC+iHy + G)^ (3) i.e. Ax + Hy+6 = V(JP-AB)y' + 2{HG-AF)y+Cf-AC; (4) and finally, a; = - ^y--,±-, ■V{IP-AB)y'+2{HG-AF)y+G'-AC. (r,) A A A But since x is, by hypothesis, expressible rationally in terms of y, therefore the expression under the radical sign is a perfect square ; and therefore (HG-AFy-{H'-AB)(G'-AC)r=0, (6) i.e., ABC + 2FGH- AF^ - BG^ - CH^ = O. [40] If this condition among the coefficients is fulfilled, then equation (1) has for its locus two straight lines. The expression ABC + 2 FGH- AF- - BG'- - CH' is called the discriminant of the quadratic, and is usually represented by the symbol A. 97] THE GENERAL EQUATION 163 Note. The analytic work just given fails if ^ = 0. In that case equation (1) may be solved for y instead of solving it for x, and the same condition, viz., A = 0, results. If, however, both A and B are zero, then tlie above method fails altogether.* In all cases, however, A = is the necessary and sufiBcient condition that the given equation can be factored. To illustrate the use of equation [40] examine the equation of Art. 54 : f 2a? — ay — 3j/' + 9a5 + 4y + 7 = 0. Here A=2, B=-3, 0=7, H=-i, G=^, and F=2; hence A = — 42 -9 — 8 + ^f^— J = 0; therefore the first member can be factored. EXERCISES Prove that the following equations represent pairs of straight lines : 1. 6y^-in/ — aP + 30i/ + 36 = 0. 2. !c'-4:Xi/ + 3y'' + 6y-9 = 0. 3. a? — 2xifseca + y' = 0. 4. For what value of ft will the equation a?—3xy + y- + 10x — 10i/ + 7c = represent two straight lines ? Suggestion. Place the discriminant (A) equal to zero, and thus find k = 20. Find the values of ft for which the following equations represent pairs of straight lines. Find also the equation of •For consideration of these special cases see Tanner and Allen's "Elementary Course in Analytic Geometry," pp. 112-113. t Ordinarily, however, it is better to use the method of Art. 97 rather *an the resulting equation [40] in solving similar uroblems. 164 THE CONIC SECTIONS [Ch. VIII each, line, the point of intersection of each pair of lines, and the angle between them. 5. 6x' + 2kxy + 12y'' + 22x + 31y + 20 = 0. 6. 123^ + kxy + 2y' + llx — 5y + 2 = 0. 7. y' — kxy — 5x-{-5y = 0. 8. Find the conditions that the straight lines represented by the equation Aay' + 2 Bxy + Cy^ = may be real ; imagi- nary ; coincident ; perpendicular to each other. 9. Show that 1 a? — i&xy — 7 y'^ = Q is the equation of the bisectors of the angles made by the lines 12 air -\-7 xy — 12 y^ = 0. Does the first set of lines fulfill the test of exercise 8 for perpendicularity ? 98. The general equation of the second degree with A^O. It remains to show that the equation Aa? + 2Hxy + Bf + 2Gx + 2Fy+C=(i, (1) with A :^ 0, represents a conic section. To prove this it is only necessary to show that, by a suitable change of the coordinate axes, equation (1) may be reduced to the form of equation (1), Art. 95 ; i. e., that the xy-texTo. may be removed. Choose new axes OX' and OY', making the angle 6 with the corresponding given axes ; then [Art. 74], x = x' cos 6 — y' sin 6, and y = ar' sin tf + y' cos 6. (2) Substituting these values (2) in equation (1), it becomes J(!b' cos e — y' sin ef + 2 Hix' cos d-y' sin «)(»' sin O-^-y' cos 6) + B{x' sin e-ify' cos 6f+ 2 G(x' cos O — y' sin d) + 2i?'(x'sin(9 + ycose)+O=0, (3) which, being expanded and rearranged, becomes : 97, 98] THE GENERAL EQUATION 165 x'^(A cos=' ^ 4- 2 Zf sin e cos fl + 5 sin' ff) + a; V'(- 2 -4 sill 6 cos e — 2 fl" sin' e + 2H cos'' tf + 2 B sin ^ cos «) + t/'^(^ sin= e - 2 if sin e cos (9 + B cos' 6) + x'{2 G cos e + 2 F sin 6) + y'{-2GsmO + 2Fcose)+C=0. (4) This transformed equation (4) will be free from the term containing the product x'y' if 6 be so chosen that — 2 ^ sin e cos e - 2 IT siu^i d + 2 if cos^ « + 2 £ sin 6 cos 5 = 0, i.e., if 2 if(cos' — sin" d) = (^ - B)2 sin fl cos fl, i.e., if 2B'.cos2fl=(^-JB)sin2e, or finally, if tan 2 « = ^^ . (5) ^ — S Moreover, it is always possible to choose a positive acute angle d so as to satisfy this last equation whatever may be the numbers represented by A, B, and H. Having chosen so as to satisfy equation (5), and having substituted the resulting values of sin 6 aud cos 6 in equation (4), that equation reduces to A'x'" + B'y" + 2 G'x' + 2 F'y' + C= 0, (6) (wherein A', B',--- represent the new coefBcients) and therefore represents a conic section with its axes parallel to the new coordinate axes. But equation (6) represents the same locus as equation (1) ; hence in rectangular coordinates, if A :^ 0, evei-y equation of the form Ax^ + 2 Hxy + By^ + 2Ga5+2J'y+C = represents a conic section ivhose axes are inclined at an angle to the given coordinate axes, where is determined by the equation tan2e=-p^. [41] -.1 — IS 166 THE CONIC SECTIONS [Cii. VIII It is to be noted that the constant term C has remained unchanged by the transformation given above. Note. In the proof just given it is assumed that the given axes are at right angles. This restriction may, however, be removed ; for if tliey are not at right angles, a transformation may be made to rectanf^lar axes having the same origin [cf. Art. 75, and Ex. 26, p. 133], and the equation will have its form and degree left unchanged ; after which the proof already given applies. Example. Given the equation -a!^ + 4xy-3r'-4V2a; + 2 V2.y-ll = 0, (1) to determine the nature and position of its locus. Turn the axes through an angle 6, i.e., with tan2e = - = oo, so that 2^ = 90°, and 6 = 45°. Now, substituting x = — ^^ and y = — ^~-, (2) equation (1) becomes x'''-3y"-2x' + 6y'-ll=0, (3) which represents the same locus as is represented by equation (1). Equation (2) may be written in the form (xf-iy-3(y'-iy=9, which represents an hyperbola. [Cf. Art. 93.] Its center is at the point (1, 1) ; the transverse axis is parallel to the x'-axis; with a = 3, b = VS, and e = |\/3; Also, F=(l + 2 VS, 1) i^=(l-2V3,l) 98] THE GENERAL EQUATION 167 and the directrices have the equations a;' = H-fV3 and = l-tV3, Fio. 74. respectively ; and the latus rectnm is 2. These results refer to the new axes, of course, and the locus is that represented in Fig. 74. Example 2. Given the equation 4ar' + 4xv + y2_l8a; + 26y + 64 = 0, (5) to determine the nature and position of its locus. Here tan 2 fl =|, from which it follows, Art. 12, that sin 2 ^ = I and cos 2 e = 4 ; thence, because cos 2e=(ios^e- sin^ e = 2 cos= 6 — 1 = 1-2 sin" 9, 1 2 it is easily deduced that sin = — and cos 6 = V5 VB Then 2x' —v' , x' + 2y' ■■ =7^, and y = — ' — ^• V5 V5 168 THE CONIC SECTIONS [Ch. Vlll Substituting these values in equation (5), it becomes 5 x"' - 2\/5 x' + 14V5 ?/' + 64 = 0, this is the equation of a parabola whose vertex is the V5' "-^=^' and whose focus is the point J" =(-^, V The axis is paral- lel to the negative end of the y'-axis, and the latus rectum is These results refer to new axes ; the VB locus of the above equation referred to the old axes is indicated in Art. 79. EXERCISES 1. For the hyperbola in Fig. 74 find the coordinates of the center and of the foci, and also the equations of its axes and directrices, all referred to the axes OX and T. HiKT : Make use of the equations of transformation (2) of Art. 98. By first removing the xy-terva, determine the nature and position of the loci represented by the following equations. Also plot the curves. 2. ar'-2v'3a;y + 3j^-24x-8V3j/ + 16V3 = 0. 3. x'-4:V5m/ + 2y''+V5x + 10y = 0. 4. 2ar' + 8xy + 8 2/' + a;+2/ + 3 = 0. 5. x' + 2xy-y'+Sx + iy-8 = 0. Note. The next five Articles, 99-103, may be omitted for a briefer course. 98-100] THE GENERAL EQUATION 169 99.* Center of a conic section. As already defined [Arts. 87, 92, 95], the center of a curve is a point such that all chords of the curve passing through it are bisected by it. It has also been shown that such a point exists for the ellipse and hyper- bola, i.e., that these are central conies. If the equation of the conic is given in the form Ax? + 2Hxy + Bf + 2Gx + 2Fy + C = 0, (1) the necessary and sufficient condition that the origin is at the center, is G = and jP = 0. For if the origin is at the center, and (x^, y^) is any given point on the locus of equation (1), then (~Xi, 'y^) must also be on this locus (because these two points are on a straight line through the origin and equidistant from it) ; hence Axi' + 2 Hx,y, + By,' + 2 Gx, + 2Fij, + O=0, (2) and Ax,' + 2 Ha^y, + By,- - 2 Gx, -2Fy^+ C= 0. (3) By subtraction iGx, + 'LFyi=:0', i.e., Gx, + Fyi=0. (4) But equation (4) is to be satisfied by the coordinates x, and y, of every point on the locus of equation (1), and the neces- sary and sufficient conditions for this are ©=0 and F=0. 100.* Transformation of the equation of a central conic to paral- lel axes through its center. Let the equation of the given conic be Ax' + 2Hxy + Bf+2Gx + 2Fy + C=Q, (1) and let the coordinates of its center be a and /S. Transform equation (1) to parallel axes througli the point («, /?) : then x = x'+a and y = y'+0. This substitution gives • See note, p. 168. 170 THE CONIC SECTIONS [Ch. VIII A(x' + ay+2 H{x' + «) («/' + /8) + B{y' + pf + 2G(x' + a)+2F{y'+P)+C=0; i.e., Ax" + 2 Hx'y' + By'^ + 2 x\Aa + Hfi+G) + 2y<{Ha + Bp + F) + Aa'' + 2HaP+B^ + 2Ga + 2Ffi+C=0. (2) Now, by Art. 99, Aa + Hp+G = Oa.ndHa + Bfi + F=0; (3) solving these equations gives B^-AB H^—AB which are the coordinates of the center of the locus of equation (1). It is to be noted here that the new absolute term, i.e., the term free from x' and y' in equation (2), may be obtained by substituting a and /3 for x and y in the first member of equar tion (1) ; also that the terms of second degree are unchanged by the transformation of this article. 101.* Test for the species of a conic. In Art. 100 the center (a, /3) is a finite point unless H' — AB = ; therefore, if H' — AB = 0, the conic is a parabola. That is, for a parabola the terms of second degree, viz., Aa? + 2 Hxy + By^, form a perfect square. In the transformation of Art. 98 the new coefiicients found were A' =A cos^d + 2 jKTsin OcosO + B sin« fl; (1) B' =Asm^e-2Ha\necose + Bcos^e; (2) 2H' = 2H 0, A' and B' have opposite signs, and the cwve is an hyperbola; If H^— AB< Q, A' and B' have like signs, and the curve is an ellipse. 102.* Reduction of the given equation to tbe standard equation of the conic. By means of the invariants of Art. 101, shorter methods than those of Art. 98 may be used for reducing equa- tions of the second degree to their standard forms : Case I. Central Conies. J^^O,a.nA H^—AB=^0, (1) Find « and p, by equation [42]. (2) Find O by the relation (y=Ga + Fp+C;^ i.e., by Eq. 42, C= -^ * See note, p. 168. t For C'= An? + 2 JEToiS + B/S"^ + 2 ©a + 2 J?^ + C = o(^a + If^e + (?) + /3(flii + B/J + J?) + ©a + ii'/S + C = Go + Jf'iS + C by equations (3), Art. 100. 172 THE CONIC SECTIONS [Ch. VIII (3) Calculate A and £' from the two invariants, A -\-B'=.A-\-B, -A'B' = H'-A£, taking A' — B' with the same sign as H* (4) The transformed equation then is A'x' + By+C = 0. (5) To plot, construct 2 6 from equation [41]. Case II. Non-central conic. A^tO, and H^—AB = 0. (1) Factor A3^ + 2 Hxy + By^ = {ax + byf. (2) With tan 6= , rotate the axes through the angle 0, [A.rt. 98] : either A' or B' will vanish with H'. (3) Change the resulting equation to form of [33, p. 140] or [34, p. 141]. 103.t The equation of a conic through given points. The general equation of a conic may be written A!c' + 2Hxy + By^+2Gx + 2Fy+C=0, (1) and contains five parameters, the five ratios between the coeffi- cients A, H, B, O, F, C. Since five equations, or conditions, will determine those parameters, in general five points will determine a conic. That is, in general, o conic may be made to pass through five, and only five, given points. If, however, the conic is to be a parabola, one equation is given ; viz., H^ — AB = 0, hence only four additional conditions • From Art. 101, equation (4), since fl^'=0, ^'-B'= V(yl -B)2+41?^ ; from equation [41], sin 2 9= '^ ^ = ^^ ■ V(^-B)2 + 4ifi A'-B' But B is cliosen in the first quadrant, hence sin 2 is 4- ; then S and A' — B' have the same sign, t See note, p. 168. 102, 103] THE GENERAL EQUATION 173 are needed or are possible. In general, a parabola may he made to pass through four points, only. A circle has two conditions given, viz., A = B, H=0; there- fore, in general, a circle may be made to pass through three points, only. A pair of straight lines has one condition given, A=0, therefore, in general, a pair of straight lines may be made to pass through four points, only. The method to be followed in obtaining the equa- tion of the required conic has been used in Art. 57, and may be indicated for finding the equation of the parabola through four given points, -Pi = i^h, yO, Pi = {^2, Vi), Pi = (a's, Vs), and P^ = (x^, t/^). The equation must be of the form (1), therefore, Ax,' + 2Hxa, + By,' + 2Gx, + 2Fy,+C=::0, Ax^ + 2 Hx^.^ -f By^ + 2 Gx.,-\-2Fyi+ C=Q, Axi + 2 Hx^, + By,' + 2Gx^ + 2 Fy,+ C=0, Ax^' + 2 Hxtyi + By,' + 2 Gx^ + 2 Fyt+ C=0; also, H'-AB = 0. The required ratios between the coefficients of equation (1) may be found from these equations. EXAMPLES ON CONICS Find the center (or the vertex), the axes, and the species, of the following conies: 1. ar + 5xy + f + 8x — 20y + 15 = 0; 2. {x-yy + 2x-y=l; 3. ^x' + 2y--2x + y-l = 0; 4. Sxy-3y' + ny + 33?-x-10 = 0; 17-1 THE CONIC SECTIONS [Cu. VIU 5. 9x'-6sn/ + 9x + y'-3y + 2=0; 6. 3a-' + xy + 3y' = 0; 7. 3ar' + 32/' + lla;-5y + 7 = 0; 8. x' + 2xy-f + 8x + iy-8=:0; 9. f + xy — 23^ + 6 x + 3y = 0; 10. 9a^ + 16y= + 24a;3/ — 60a; — 803/4-100 = 0. Trace the following conies : 11. 33^ + 2xy + 3f—16y + 23 = 0; 12. i!e' + 9y'-12x-24:y-ll = 0; 13. 33e'-3y^ + 8xy-10y + 6x + 5 = 0i 14. (x + y)(x + y + i)+A = 0. 15. What conic is determined by the points (2, 3), (0, ~3), (2, 0), (5, 6), and ("5, -5) ? 16. Find the equation of the parabola through the points (4, 3), (0, -4), (6,1), (-6,2). 17. Find the equation of the conic through the points (5, 3), (4, 4), (2, 6), (7, -1), (0, 0). V. SECANTS, TANGENTS, NORMALS, DIAMETERS 104. Tangent to a conic in terms of its point of contact. The equation of a tangent to a conic may be found by the method of Art. 61. That Article should now be reread with care. 105. Tangent to a curve at a given point : second method. A second form of the method of Art. 61 for finding a tangent to ;i curve is given by an example. Given two points Pj = (a^, yj) and P^ = (xj + ft, yi + k) on the conic Az'+2Hxy + By'+C=0. (1) 103-106] SECANTS, TANGENTS, NORMALS, DIAMETERS 175 Y Fio. 75. k Now, the slope of the secant is - ; and as the secant moves so that P2 approaches Pi along the curve, this slope approaches as a limit the slope of the tangent P^T; while h and A; ap- proach as a limit ; ~k' m, = limit when A = 0, and fc = 0. Now, from Eq. (1), Ax,' + 2 //X12/1 + By,' +C = 0. (2) A{x, + hy + 2H (x, + h) (2/1 + fc) + -8(2/1 + fc)» + C= 0; (3) i.e., Ax^ + 2 Hx^y, + By^ + 2 liAx, + 2 kBy, + 2 fcZ/aii + 2 Zt/Zy, -l-yl/i2 + 21fAA;-f£fc2^C,^0. (4) Subtracting (2) from (4), and factoring, 7t(2 Ax,+ 2 ////!+ .lA +/rfc) + A-(2//x,-|-2B?/,-|-7/A 4--B)fc) = 0. (5) Hence * = _ "^ ^ =^^ + '^ Hy, + Ah + Hk h 2Mx, + 'JByi + Hh + Bk (6) 176 THE CONIC SECTIONS [Ch. VIII Then . . "'"i.* . „ ft = 0, fc =0 2Axi + 2Hy,_ Ax, + Hiii 2HXi + 2 By I Hx^ + By^ Now the equation of the tangent becomes Multiplying and rearranging, equation (8) becomes (^a^ + Hy^) X + {Hx, + By^) y - {Ax^^ + 2 Hx^y^ + By^^ = ; (9) which, by (2), may be written Ax^x + H{xyi + x^y) +Byiy + C= 0, (10) the required equation. EXERCISES Derive in full, by the secant method, the following equations of tangents, the point of contact being 7*, = (a-,, y^ : \- For the parabola f = 'i:px; viy = ^pix-\-x{). [45] 2. For the ellipse ^2 + f-I = l; ^ + ^=1' C*®] 3. For the hyperbola ^ - 1' = 1 ; ^ " ^ = 1- C*^] 4. For the hyperbola 2 a5y = c; xyi+a!|j/ = c. [48] 5. For the Qomc Aa? + 2 Uxy + Bf + 2 Gx + 2 Fy + C = 0; Ax^x + II{y^x + Xiy) + Byiy + G(x+ x^) + F{y + y;)+ C=(i. Notice that the equation of the tangent for each conic through a given point of contact (a^ y-^ may be written as follows : in the Vtl I 'V ft equation of the conic substitute XjX for a^, y,y for i/", .> — — for x + x^ , , V + Vx , xy, — ^- for X, and ■ for y. 105,106] SECANTS, TANGENTS, NORMALS, DIAMETERS 177 106. Nonnal to a conic. The equation of a normal to a conic can he obtained at once from that of the tangent, by the method oi Art. 62. Example. To find the equation of the normal to the ellipse ^ + |- = 1 ut the point (3, 2). 18 H The equation of the tangent at the point (3, 2) is || + ^=l; i.e., 2x + 3y = 12. The perpendicular linetlirough (3, 2) is 3 x—2y = S, which is, therefore, the required normal. EXERCISES 1. Is the line 4:x + 9y = 3o tangent to the ellipse 2a,-2 + 3!/^ = 35? 2. Find the equation of a tangent to the conic a^ + 5y^ — Sx+lOy— 4=0, parallel to the line y=4a!+10. [Art. 69]. Write the equations of the tangent and normal to each of the following conies, through a point (x^, y{) on the curve: 5. a^ = 4:p(y + 5); sketch the figure. 6. 4 ar^ — 5 2/^ — 12 a; = ; sketch the figure. 7. x' + 3 f — 3 X + 12 y + 9 =^ ; sketch the figure. 8. Derive, by the secant method, the tangent to the ellipse ia^ + y- + Sx — 2y + l = 0, the point of contact being (x^, y^. Write the equations of the tangents and normals to each of the following conies, at the given points ; also sketch each figure: 9. 9 ar' + 5 2/= 4- 36 a; + 20 2/ + 11 = 0, at the point ("2, 1) ; 178 THE CONIC SECTIONS [Cii. VIII 10. 9 a-^ + 4 y^ + 6 a; + 4 2/ = 0, at the positive end of the latus rectum ; 11. y- — Gy — Sx = 31, at the negative eud of the latus rectum ; 12. ^ + ^ = 1, at the point ( VT, 1) ; 5 5 13. 4 ar' + 3 y'' = 16, at the point (1, -2). 14. Find the equation of the tangent and normal to the parabola y^ — 6y — 8x — 31=0 passing through the point (~4, ~1) not on the curve. 107. The equation of the tangent to a conic in terms of its slope. The equation of a line having the given slope m is y = mx+k; (1) it is desired to find that value of A; for which this line will be- come tangent to a given conic, e.g., to the parabola whose equation is f = ipx. (2) Considering equations (1) and (2) as simultaneous, and eliminating y, gives the resulting equation, (mx + ky = ipx, (3) which has for its roots the abscissas of the two points in which the loci of equations (1) and (2) intersect. These roots will be equal, and therefore the points of intersection will be coinci- dent, if (mlc-2py-mV = 0, i.e., if k^P. m (4) Therefore, in [49] is, for all values of m, the equation of the required tangent. Similarly, the tangent with slope m 106-108] SECANTS, TANGENTS, NORMALS, DIAMETERS 179 a^ , 2/' for the ellipse, _ + ^ = l is y = mx± Va-in^ + 6* ; [50] ?l-t^^ i. for the hyperbola, — — f- = l is y = mx ± Va^rn^ - b'^. [51] a" 108. Diameter of a conic section. The locus of the middle points of any system of parallel chords of a given conic is called a diameter of that conic, and the chords which that diameter bisects are called the chords of that diameter. This definition leads at once to the equation of a diameter, as will be shown for the ellipse. Let m be the slope of the given system of parallel chords of the ellipse whose equation is and let y = mx + c (1) (2) be the equation of one of these chords, which meets the curve in the two points Pj = (xj, i/j) and P., = (x^, y.?). Let P = (x', y') be the middle point of this chord, so that 180 THE CONIC SECTIONS [Cb. VIII 0^=^^, y'=yi±yi. (3) The coordinates of Pi and P2 are found by solving (1) and (2) as simultaneous equations, therefore the abscissas a^ and ajj are the roots of the equation (a'm' + b'^x' + 2 a'cmx + a'c' - a'b" = ; (4) hence, by Art. 8, x'=— /'/''" , (5) Then, by Eq. (2) y' = ^^.- (6) Now, by varying the value of c, equations (5) and (6) give the coordinates of the middle point for each of the chords of the given set. It is required to find the locus of P' for all values of c, i.e., to find an equation satisfied by x' and y' and not dependent upon the value of c. If x' is divided by y', then c is eliminated from the equations (5) and (6), giving Therefore the coordinates of the middle point of every chord of slope m satisfy the equation X a^ y b' i. e., V=- -^ as ; [52] which is therefore the equation of the required diameter. Likewise it may be shown that the equation of the diameter of the hyperbola is 1/ = —^ — «> [53] = 22. tn using the first standard equation of each conio. of the parabola is 1/ = —*^; [54] 108, 109] SECANT8, TANGENTS, NORMALS, DIAMETERS 181 109. Equation of a conic that passes through the intersections of two given conies. Let the given conies loe S, = A,x' + Bj,^+2O^x + 2F^+C, = 0, (1) and S2 = A^ + B.,i^ + 2G:^ + 2Fa+C2=0; (2) then, if k be any constant whatever, Si + kS., = (3) represents a conic whose axes are parallel to the coordinate axes [Art. 95], and which passes through the points in which the conies Si = and iSj = intersect each other [Art. 38] ; i.e., Si + kSi = represents a family of conies, each member of which passes through the intersections of Si = and S^ = 0. The parameter k may be so chosen that the conic (3) shall, in addition to passing through the four points in which Si = and iS2 = intersect, satisfy one other condition; e.g^., that it shall pass through a given fifth point. Moreover, if ^i = and Si = are both circles, then Si+kS2=0 is also a circle. [Cf. Arts. 65 and 66]. EXAMPLES ON CHAPTER VIII 1. Find the equations of those tangents to the conic 4^^ — 5 3/^=20, which pass through the point (0, ~2). [Use equation y + 2 = mx]. 2. Find a conic through the intersections of the ellipse 4 ar" + 3/^ = 16 and the parabola y' = 4:X + 4, and also passing through the point 0, 0. What species of conic is it ? 3. Show that the curves 4a;2 + 9 2/^= 36 and 2a^-3 2/== 6 have the same foci, and that they cut each other at right angles. 4. Find the vertices of an equilateral triangle circum- scribed about the ellipse Qar' + IB^/^ =144, one side being parallel to the major axis of the curve. 182 THE CONIC SECTIONS [Ch. VIII 5. Find the normal to the conic 3af' + 2/^ —2x — y=^, making the angle tan~^ (— i) with the avaxis. 6. Show that the equation of the secant line that cuts the ellipse 4 a;^ + 9 3/" — 36 = in the points Pj = («„ y^ and Pj= (x^, 3/2) may be written 4 {x—Xi){x — x^ + 9 (3/ — 2/1) (2/ — ^2)= 4 or' + 92/^—36. Determine the species of the following conies; and also their foci, directrices, centers, semi-axes, and latera recta : 7. f={x-Z){x + A.); 8. Zo^-2y''-&x-^y = l; 9. a;'-|-a' = 4ax-|-4py -l-4/)6; 10. 2x» + 3/+16a; + 12y-f-38=0; 11. 9y'-4ar'=36y; 12. a? + 2f—2 = a(2x — i.y + 9a). 13. Show that the polar equation of the parabola, with its vertex at the pole, is p = . „ „ • 14. Show that if the left-hand focus be taken as pole, the polar equation of the ellipse is p = ^ ^ ~ — '— ■ l — ecosO 15. Derive the polar equation of an hyperbola, with its pole at the focus, eccentricity 3, and the distance of the focus from the directrix equal to 9. 16. From equation [39], trace the parabola. 17. From equation [39], trace the ellipse. 18. By means of equation [39], prove that the length of a chord through the focus of a parabola, and making an angle of 30° with the axis of the curve, is four times the length of the latus rectum. CHAPTER IX SPECIAL PROPERTIES OF THE CONICS 110. In the present chapter, some of the intrinsic proper- ties of the conic are to be studied, i.e., properties which belong to the curves and are entirely independent of the position of the coordinate axes. For this purpose, it will, in general, be easier to use the simplest form of the equation of the curve in each case. THE PARABOLA y^ = 4^px 111. Recapitulation. The following data may be assumed for the further study of the parabola : The eccentricity is e = 1 ; the standard equation y' = ipx; the focus is the point F= (jp, 0); the directrix is tlie line a;= — jj; the axis of the curve is 2/ = ; the equation of the tangent at (a;,, y^), y,.?/ = 2p(x + a^) ; or, in terms of its slope, y = mx + i-; m the equation of the diameter y = — ^, m where m is the slope of the chords bisected. 112. Construction of the parabola. The two conceptions of a locus given in Art. 28 lead to two methods for constructing 183 184 SPECIAL PROPERTIES OF THE COMICS [Ch. IX Fig. 77. a curve : (1) by plotting points to be connected by a smooth curve, and (2) by the motion of a point constrained by some mechanical device to satisfy p^^^ the law which defines tlie curve. These two methods may be used in construct- ing a parabola. (1) By separate points. Given the focus F and the vertex O, draw the axis OFX, the directrix D'D cutting this axis in Z, and also a series of lines per- pendicular to the axis at Ml, Mi, Ms, etc., respectively. With F as center and ZMi as radius, describe arcs cutting the line at M^ in two points Pj and Qi ; similarly, with F as center and ZM^ as radius, cut the line at M2 in P^ and Qj; and so on. The points thus found evidently satisfy the defini- tion of the parabola [Art. 79]. If these are then connected by a smooth curve, it will be approximately the required parabola. (2) By a continuously mov- ing point. Let D'D be the directrix and F the focus. Place a right triangle with its longer side KII in coincidence with the axis of the curve, and its shorter side KJ in coinci- dence with the directrix. Let one end of a string of length KH be fastened at H, and the other end at F. If now a pencil Fig. 78. 112, 113] THE PARABOLA 185 point be pressed against the string, keeping it taut while the triangle is moved along the directrix, as indicated in the figure, then, in every position of P, FP=KP, therefore the pencil will trace an arc of a parabola. 113. Subtangent and subnormal. Construction of tangent and normaL Let ^1 = (»i. .Vi) be any given point on the parabola f:=i:px. (1) Draw the ordinate MPi, the tangent TP,, and the normal P^N. Then [Art. 63], the subtangent is TM, the subnormal is NM, the tangent length TPj, and the normal length NP^. The tangent at Pi has for its equation M = 2p(a! + a^), (2) hence its avintercept is — x-^, \_=AT'\. But AM = r=ZNPiQ, since PtN±P^T. EXERCISES 1. The focal distance of any point of the parabola t/* = 4:px iap + x. 2. Show that (-2., —£.] is the point of contact of the \m- m J tangent y = mx + ^ with the parabola, m 3. The equation of the normal to the parabola y' = ipx, with slope TO, is y = mx — 2 pm — pm\ 190 SPECIAL PROPERTIES Of THE CONICS [Cii. IX 4. Through any point in the plane two tangents may be drawn to the parabola. 5. Through any point in the plane three normals may be drawn to the parabola. 6. The line joining any point in the directrix to the focus of a parabola is a perpendicular to the chord of contact corre- sponding to that point, [Fig. 80]. 7. A perpendicular let fall from the focus upon a tangent meets that tangent upon the tangent at the vertex, [Fig. 80]. 8. Write the equations of the normals drawn through the point (0, ~3) to the parabola y^ = —10x. 9. The circle on a focal chord as diameter is tangent to the directrix, [Fig. 80]. 10. The angle between two tangents to a parabola is one half the angle between the focal radii of the points of tan- gency. 11. The product of the segments of any focal chord of the parabola i/' = ipx equals p times the length of the chord. 12. Two tangents are drawn from an external point Pj = (xy, yi) to a parabola, and a third is drawn parallel to their chord of contact. The intersection of the third with each of the other two is halfway between P, and the corresponding point of contact. 13. The area of a triangle formed by three tangents to a parabola is one half the area of the triangle formed by the three points of tangency. 14. The tangent at any point of the parabola will meet the directrix and latus rectum produced, in two points equidistant from the focus. 114, 115] THE PAKABOLA 191 15. The normal at one extremity of the latus rectum of a parabola is parallel to the tangent at the other extremity. 16. The tangents at the ends of the latus rectum are twice as far from the focus as they are from the vertex. 17. The circle on any focal radius as diameter touches the tangent drawn at the vertex of the parabola. 115. Some properties of the parabola involving diameters. The equation of the diameter of the parabola [Art. 108], y = ^, (1) m shows at once that every diameter of the parabola is parallel to the axis of the cwve. Conversely, since any value whatever may be assigned to TO, each value determining a system of parallel chords, equa- tion (1) may represent any line parallel to the ovaxis, and therefore evei-y line parallel to the axis of a parabola bisects some set of parallel chords, and is a diameter of the curve. Again, each of the chords cuts the parabola in general in two distinct points, and the nearer these chords are to the extremity of the diameter the nearer are these two points to each other and to their midpoint. In the limiting position, when the chord passes through the extremity of the diameter, the two intersection points and their midpoint become coinci- dent, and the chord is a tangent. Therefore the tangent at the end of a diameter is parallel to the bisected chords. It follows from the preceding properties, or directly from equation (1), that the axis of the parabola is the only diameter perpendicular to the tangent at its extremity. EXERCISES 1. Find the diameter of y' = 16 a; which bisects the chords parallel to the line 3x — 4y4-10 = 0. 192 SPECIAL PROPEKTIES OF THE CONICS [Ch. IX 2. A diameter of the parabola y'-' = — Sx passes through the point (2, ~3) ; what is the equation of its corresponding chords ? 3. Find the equation of the diameter of the parabola y'= — Ax + 8 which bisects the chords 2y — 3x = k. 4. Find the equation of the tangent to the parabola (y + 6)^ = — 4:(x + 2) which is perpendicular to the diameter y + 4 = 0. 5. The tangents at the extremities of a chord of a parabola intersect upon the corresponding diameter. 6. Show how the properties of Art. 115 give a method for constructing a diameter to a set of chords, and in particular for constructing the axis of a given parabola. 7. Show how the problem to construct a tangent and normal to a given parabola at a given point can be solved when the axis is not at first given. [Of. Art. 113]. REVIEW EXERCISES Find the equation of a parabola with axis parallel to the avaxis : J.. Passing through the points (0, 0), (4, 2), (4, ~2); 2. Passing through the points (^, 0), (4, - 1), (i^, — 2) ; 3. Through the point (0, ~6), with its vertex at the point (-2, -1). 4. A parabola whose axis is parallel to the y-axis passes through the points (-^1, "l), ("2, "10), and (3, "5); find its equation. 5. Find the vertex and axis of the parabola of Ex. 4. Find the equation of a parabola : 6. If the axis and directrix are taken as coordinate axes. 115] THE I'AUAliOLA 103 7. With the focus at the origin, and the ^-axis parallel to the directrix. 8. Tangent to the line 3 y = — 4 a; — 12, the equation being in the simplest standard form. 9. If the axis of the parabola coincides with the a>-axis, and a focal radius of length 5 coincides with the line 8y — 6x + 3 = 0. 10. Two equal parabolas have the same vertex, and their axes are perpendicular ; find their common chord and common tangent. [Cf. Ex. 10, p. 187]. 11. At what angle do the parabolas of Ex. 10 intersect ? 12. Two tangents to a parabola are perpendicular to each other ; find the product of the corresponding subtangents. Find the locus of the middle point : 13. Of all the ordinates of a parabola. 14. Of all chords that meet at the foot of the axis, z [Fig. 80]. 15. From any point on the latus rectum of a parabola, per- pendiculars are drawn to the tangents at its extremities ; show that the line joining the feet of these perpendiculars is a tangent to the parabola. 16. A double ordinate of the parabola y^ = ipx is 8p ; prove that the lines from the vertex to its two ends are perpendicular to each other. Two tangents of slope m and m', respectively, are drawn to a parabola; find the locus of their intersection: 17. If mm' = k ; 18. Ii- + — = k; m m 19. Ifl_l. = fc. 194 SPECIAL PROPERTIES OF THE CONICS [Ch. IX 20. Find the locus of the center of a circle which passes through a given point and touches a given line. 21. The latus rectum of the parabola is a third proportional to any abscissa and the corresponding ordinate. 22. Find the locus of the point of intersection of tangents drawn at points whose ordinates are in a constant ratio. 23. What is the equation of the chord of the parabola y'=—ix whose middle point is at ( 3, 1) ? 24. Find the locus of the center of a circle which is tangent to a given circle and also to a given straight line. 25. Find the equation to all parabolas which are touched by the straight lines y = ± - • 26. At the point P a normal is drawn ; find the coordinates of the point where it meets the curve again, and the length of the intercepted chord. THE ELLIPSE ^ + 2^=1. 116. Recapitulation. In further study of the ellipse the fol- lowing data may be assumed : The standard equation, — + 2- = i . or b^ 2 ' the eccentricity, e* = its foci, -Fi s (— ae, 0), F^ = (ae, 0) ; its directrices, x= — -,x=-; e e its tangent at P„ S? + M = 1 j y = mx± ■Vd'm' + V ; am 115-117] THE ELLIPSE 195 its tangent with slope m, its diameter, for chords whose slope is m. Clearly, if the axes are equal, so that b = a, the curve takes the special form of the circle, with eccentricity e = 0, the two foci coincident at the center, and the directrices infinitely distant. 117. The sum of the focal distances of any point on an ellipse is constant ; it is equal to the major axis. The ellipse — + 2. = i has its foci at the points or b^ Fi = (— ae, 0) and Fj = (ae, 0) ; with 6* = o' - aV. [Cf. Art. 86]. Let Pi = (a;,, y,) be any point on the curve, D Y B Fia. 81. then FiPi = e(ZO + OM) = e(-+x\ = a 4- esB,. 196 SPECIAL PROPERTIES OF THE CONICS [Ch. IX Also F^Pt, = e{OZ'- OM) = a — ej\. Hence, by addition, F^P^ + F^Pi = 2 a ; i.e., the sum of the focal distances of any point on an ellipse is constant ; it is equal to the major axis. This property gives an easy method of finding the foci of an ellipse when the axes A'A and B'B are given. For FiB + F2B = 2a; but FiO= OF2, FiB = FiB = a. Hence, to find the foci, describe arcs with B as center and a = OA as radius, cutting A'A in tlie points Fi and F2 ; these points are the required foci. 118. Construction of the ellipse. The property of Art. 117 is sometimes given as the definition of the ellipse; viz., the ellipse is the locus of a point the sum oftchose distances from two fixed points is constant. This definition leads at once to the equation of the curve [ef. Ex. 32, p. 56], and also gives a ready method for its construction. / F, Fio. 82. / (a) Construction by separate points. Let A'A be the given sum of the focal distances, i.e., the major axis of the ellipse ; and Fj and F^ be the given fixed points, the foci. With either 117, H8] THE ELLIPSE 197 focus as center, and with any radius A'R ^ A' A describe an arc; then with the other focus as center, and radius BA, describe an arc cutting the first arc in two points. These are points of the ellipse. In the same way as many points as desired may be constructed; a smooth curve connecting these points is approximately an ellipse. (/8) Construction by a continuously moving point. Fix two upright pins at the foci, and over them place a loop of string, equal in length to the major axis plus the distance between the foci. Press a pencil point against the cord so as to keep it taut. As the pencil moves around the foci, it will trace an ellipse. EXERCISES 1. Construct an ellipse with semi-axes 4 cm. and 2 cm. 2. Construct an ellipse with semi-axes 5 cm. and 10 cm. 3. Construct an ellipse with the distance between the foci 20, and the major axis of length 24. 4. Find the equation of a tangent, and also of a normal, to the ellipse x' + 4 y'' = 16, each perpendicular to the line 4:X + 3y=5. 5. By employing equation y = mx ± -Vahn' + b'-, find the tangent to the ellipse 16 ar* -|- 25 y^ = 400, passing through the point (3, 4). 6. By the method of Ex. 16, p. 188, show that an ellipse is concave toward its center. 7. Through what point of the ellipse —-\-^ = l must a w b tangent and normal be drawn, to form with the avaxis an isosceles triangle ? 198 SPECIAL PROPERTIES OF THE CONICS [Cii. IX 8. Write the equations of the tangent and normal at the positive end of the latus rectum of the ellipse 9 a:^ + 16 y* = 144. Where do these lines cut the aj-axis ? 9. Tangents to the ellipse Aa^+3y' = 5 are inclined at 45° to the X-axis ; find the points of contact. 10. Find the equation of an ellipse (center at the origin) of eccentricity |, such that the subtangent for the point (3, ^) is (- ¥)• 11. Find the equations of the tangents from the point (2, 3) to the ellipse 9 x* + 16 y^ = 144. Find also the equation of the normal from (2, 3). 12. Find the tangents to the ellipse 7 a^ + 8 y' = 66 which make the angle tan"' 3 with the line x — y+1 = 0. 13. Find the product of the two segments into which a focal chord is divided by the focus of an ellipse, — using Art. 96. 14. Find the points on the ellipse Va? ■\- o?y' = dV, such that the tangent makes equal (numerical) angles with the axes ; such that the subtangent equals the subnormal. 119. Auxiliary circles. Eccentric angle. The circumscribed and inscribed circles for the ellipse (Fig. 83) are called auxiliary circles, and bear an important part in the theory of the ellipse. Let the equation of the ellipse be 2^+^=1. (1) The circle described on its major axis as diameter is called the major auxiliary circle ; its equation is a?Jrf = d?; (2) and the circle on the minor axis as diameter is the minor auxiliary circle ; its equation is r' + ?/'=6» (3) 118, 119] THE ELLIPSE 199 If Z AOQ is any angle at the center of the ellipse, with the initial side on the major axis, and the terminal side cut- ting the auxiliary circles in B and Q, respectively ; and if P Fio. 83. is the intersection of the abscissa LR with the ordinate MQ, then P is a point on the ellipse. For the coordinates of P are I.e., 0M= OQ cos <^ and MP = M'R = OR sin <^, a; = acos A). 4. Find the locus of the foot of the perpendicular drawn from the center of the ellipse v? -\- 4: y' = 4: to a variable tan- gent y = mx -I- V4 m^ -f 1. 5. Prove that tlie line joining the point (3, 2) to the origin bisects the segment of the line 3 a; -f 8 7/ = 4 intercepted by the ellipse 3? + 4.i/=ii, 208 SPECIAL PROPERTIES OF THE CONICS [Ch. IX 6. Find the locus of the center of a circle which passes through the point (3, 0) and touches internally the circle x' + y^ = 100. 7. Find the length of the minor axis of an ellipse whose major axis is 50 and whose area is equal to that of a circle whose radius is 10. 8. One focal radius is 6, the other is 3, for a point whose abscissa is 3; find the major axis of the ellipse, distance between foci, and area. 9. A line of fixed length moves so that its ends remain in the coordinate axes ; find the locus generated by any point of the line. 10. Find the locus of the middle points of chords of an ellipse drawn through the positive end of the minor axis. 11. With a given focus and directrix a series of ellipses are drawn ; show that the locus of the extremities of their minor axes is a parabola. 12. Show that the line xcos a + ysma=p touches the ellipse ^ + & = ^ 'f jj* = a^ cos^ a + 6' sin' a. 13. Find the locus of the foot of the perpendicular drawn from the center of the ellipse — + ^ = 1 to a variable tangent. 14. Prove, analytically, that if the normals to an ellipse pass through its center, the ellipse is a circle. 15. Find the locus of the vertex of a triangle of base 2 a, and such that the product of the tangents of the angles at its base is — • r 124, 125] THE HYPERBOLA 209 16. The ratio of the subnormals for corresponding points on the ellipse and major auxiliary circle is — • 17. From an external point T whose coordinates are (A, A) a line is drawn to the center C meeting the ellipse in R; find the ratio of Crto CR. 18. Normals at corresponding points on the ellipse, and on the major auxiliary circle, meet on the circle a?-\-y'= {a+Vf. The tangents drawn from point P to an ellipse make angles $1 and Oi with the major axis ; find the locus of P : 19. When 6x + 6i-=2a, a constant. 20. When tan 0-^ + tan 0.2 = c, a constant. THE HYPERBOLA, ^-^=1 125. Recapitulation. From the discussion already given of the hyperbola, and its close analogy to the ellipse, the follow- ing data may be assumed here : The standard equation, ^-^=1; the eccentricity, e'-^'+/;^nde>l, a' the foci, F, = (-ae,0), F,=(ae,0); the directrices. x = > x = -; e e the tangent at P^, a' b' the tangent with slope m. y = mx± s/m'a' — 6'; the diameter. b^ bisecting chords whose slope is m. 210 SPECIAL PROPERTIES OF THE CONICS [Ch. IX In spite of the close similarity between these lesults for the hyperbola and the ellipse, the nature of the hyperbola appar- ently differs widely from that of the ellipse, consisting, as it does, of two open infinite branches instead of one closed oval. It is desired now to show some of the most important proper- ties of the hyperbola which correspond to similar properties in the ellipse ; and also to prove some special properties which are peculiar to the hyperbola. 126. The difference between the focal distances of any point on an hyperbola is constant ; it is equal to the transverse axis. Let Pi = (xi, y{) be any given point on the curve ; it may be shown as in Art. 117 that the focal distances are FiPi = eXi + a, (1) FsPi = exi — a, (2) where e > 1. Subtracting equation (2) from equation (1) gives i?'iPi-FjPi = 2o; hence, the difference between the focal distances of any point on an hyperbola is constant ; it is equal to the transverse axis. If the foci are not given, they may be constructed as follows, provided the semi-axes of the curve are known : plot the points A = (a, 0) and B=(0, b); then with the center of the hyper- bola as center, and the distance AB as radius, describe a circle; it will cut the transverse axis in the required foci i?*, and F^, for ^B = VaM^ = Vfl??=±ae. 127. Construction of the hyperbola. The property of the pre- ceding article might be taken as a new definition of the hyper- bola, viz., the hyperbola is the locus of a point the difference of whose distances from two fixed points is constant. This defini- tion leads at once to the equation of the curve [cf. Ex. 33, p. 57] and also to a method for its construction. 125-128] THE HYPERBOLA 211 (a) Construction by separate points. Let A' A be the given difference of the focal distances, — i.e., the transverse axis of the hyperbola, — and Fi and F.^ the given fixed points, the foci. With either , , „ focus, say Fj, as a "1 '' ''"X" center, and a radius A_ A R A'R % A! A, describe " ^ "*" *"k an arc ; then with --V--' ■■J—- the other focus as a ... ' , center, and a radius AR describe an arc cutting the first arcs in the two points P,. These are points of the hyperbola. Similarly, as many points as desired may be obtained and then connected by a smooth curve, — approximately an hyperbola. (/3) Construction by a continuoushj moving point; the foci being given. Pivot a straightedge LM at one focus Pj, so tliat FiM is greater than the trans- ijl/ verse axis 2a; a.t M and the other itp^ focus F.2 fasten the ends of a string of length I, such that FiM=l+2a; aJ I then a pencil P held against the Fig. 87. ^ string and straightedge (see Fig. 87), so as to keep the string al- ways taut, will, while the straightedge revolves about F^, trace one branch of the hyperbola. By fastening the string at the first focus and the straightedge at the second, the other branch of the curve can be traced. 128. The tangent and normal bisect internally and externally the angles between the focal radii of the point of contact. Let F^ and F^ be the foci of the hyperbola -j-'^= 1, PiT the tangent, and P^N tlie normal at the point Pi = (xj, y-^. 212 SPECIAL PROPERTIES OF THE CONICS [Cb. IX Then the equation of P^T is ^ -^ = 1, and the length of the intercept OT of the tangent is Fia. 88. Now, in the triangle FiPiFi, and FiT=FiO+OT=ae + \ TF, = OF, - 0T= ae - ! »i ah = -(exi-a); but iJ-iP, = ex, + o, [Art. 126] and P^i = eoc\ — a. Hence F-^ T:TFt = F^P^ : P^F^, and, by elementary geometry, the tangent bisects internally the angle between the focal radii. Then, since the normal is perpendicular to the tangent, the normal Pi JT bisects the ex- ternal angle F^Pi W. 128, 1291 THE HYPERBOLA 213 These facts suggest a method, analogous to that of Art. 121, for constructing the tangent and normal to an hyperbola at a given point. 129. Conjugate hyperbolas. A curve bearing very close re- lations to the hyperbola is that represented by the equation i.e., by a? 7.2 ^> (1) (2) Fio. 89. in which a and h have the same values as iu equation (1). This curve is evidently an hyperbola, and has for its trans- verse and conjugate axes, respectively, the conjugate and transverse axes of the original, or primary, hyperbola. Two 214 SPECIAL PROPERTIES OF THE CONICS [Ci;. IX sucli hyperbolas are called conjugate hyperbolas; they are sometimes spoken of as the x- and y-hyperbolas, respectively. It follows at once that the hyperbola (2), conjugate to the hyperbola (1), has for its eccentricity for foci the points (0, ± he'), and for directrices the lines ^ e' Two conjugate hyperbolas have a common center, and their foci are all at the common distance Va^ + 6^ from this center; i.e., the foci all lie on a circle about the center, having for radius the semi-diagonal OS of the rectangle upon their com- mon axes, the sides of which are tangent to the curves at their vertices. Moreover, when the curves are constructed it will be found that they do not intersect, but are separated by the diagonals OS and OK, extended, of this circumscribed rec- tangle, which they approach from opposite sides. These diag- onals are examples of a class of lines of great interest in analytic theory ; they are called asymptotes. EXERCISES 1. Construct an hyperbola, with vertex at the point (3, 0) and focus at the point (5, 0). 2. Construct an hyperbola, given the distance from di- rectrix to focus as 4 cm. How many such hyperbolas are possible ? 3. Write the equation of an hyperbola conjugate to the hyperbola 9 y* — 16 x* = 144, and find its axes, foci, and latus rectum. Sketch the figure. 129] THE HYPERBOLA 215 4. Write the equations of the tangent and normal to the hyperbola 3 ar* — 4 j^^ = 11 at the point (—3, —2), and find the subtangent and subnormal. 5. Yor what points of an hyperbola are the subtangent and subnormal equal ? 6. Given the hyperbola 9 y* — 4 af = 36, find the focal radii of the ends of the latera recta. 7. A tangent which is parallel to the line 5aj + 42/ + 10 = 0, is drawn to the hyperbola x' — if = 9; what is the subnor- mal for the point of contact? 8. What tangent to the hyperbola tt; — IQ = 1 ^^^ its avintercept 4 ? 9. Find the two tangents to the hyperbola 4 a^ — 2 ^' = 6 which are drawn through the point (0, 5). 10. Prove that if the crack of a rifle and the thud of the ball on the target are heard at the same instant, the locus of the hearer is an hyperbola. 11. Find the equation of an hyperbola whose vertex bisects the distance from the focus to the center. 12. If e and e' are the eccentricities of an hyperbola and its conj ugate, then e' + e'' = e^e"*. 13. If e and e' are the eccentricities of two conjugate hyper- bolas, then ae = be'. 14. Find the eccentricity and latus rectum of the hyperbola se'^iiif + a^. 15. Find the tangents to the hyperbola 4 ar' — 15 j/' = 60, which, with the tangent at the vertex, form a circumscribed equilateral triangle. Find the area of the triangle. 16. Find the lengths of the tangent, normal, subtangent, and subnormal for the point (2, 3) of the hyperbola 3 a^— ^=3. 216 SPECIAL PROPERTIES OF THE CONICS [Ch. IX 130. Asymptotes. If a tangent to an infinite branch of a curve approaches more and more closely to a fixed straight line as a limiting position, when the point of contact moves farther and farther away on the curve, and becomes infinitely distant, then the fixed line is called an asymptote of the curve.* More briefly, though less accurately, this definition may be stated as follows: an asymptote to a curve is a tan- gent whose point of contact is at infinity, but which is not it- self entirely at infinity. It is evident that to have an asymp- tote a curve must have an infi- nite branch ; and this branch may be considered as having Fra. 90. two coincident, and infinitely distant, points of intersection with its asymptote. This property will aid in obtaining the equation of the asymptote. The hyperbola V i = l. (1) is cut by the line y = mx -f- c, (2) in points whose abscissas are given by the equation (o%' - W)3? + 2 d?onx + aV + a V = 0. (3) If line (2) is an asymptote, the roots of equation (3) must both be infinite ; therefore, by Art. 7, (4), aW-6^ = and 2a^cm=0, (4) h hence m = ± - and c = 0. • This definition implies that the distance between a curve and its asymptote becomes infinitely small. McMahon & Snyder, " Differential Calculus," Chap. XIV. 130, 131] THE HYPERBOLA 217 Substituting these values in equation (2) gives y = -x, and y= x; (5) and these equations represent the asymptotes of the hyper- bola: they are the Hues OS and OK in Fig. 90. Therefore, the hyperbola has two asymptotes; they pass throngh its center, and are the diagonals of the rectangle de- scribed upon its axes. Since the equation of the hyperbola conjugate to (1) is and thus differs from equation (1) only in the sign of the second member, •vrhich affects only the constant term in equation (3), therefore the equations (4) determine the value of m and c for the asymptotes of the conjugate hyperbola also. It follows that conjugate hyperbolas have the same asymptotes. The equations of the asymptotes may be combined, by Art. 37, into the one equation which represents both lines, viz., $-$ = 0. [58] 131. Relation between conjugate h3rperbolas and their as3mip- totes. It has been seen that the standard forms for the equa- tions of the primary hyperbola, its asymptotes, and its conjugate hyperbola are, respectively, (1) (2) (3) 3? f ■■% a"' =0, a? a'' y' = — 1, 218 SPECIAL PROPERTIES OF THE CONICS fCa. IX It will be noticed at once that these three equations differ only in their constant terms; and that the equation of the primary hyperbola (1) differs from that of the asymptotes (2) by the negative of the constant by which the equation of the conjugate hyperbola (3) differs from equation (2). Moreover, this relation between the equations of the three loci must hold when not in their standard forms, i.e., what- ever the coordinate axes. For, any transformation of coor- dinates will affect only the first member of equations (1), (2), and (3), and will affect these in precisely the same way. After the transformation, therefore, the equations of the loci will differ only by a constant (not usually by 1, however, when cleared of fractions) ; and the value of the constant in the equation of the asymptotes will be midway between the values of the constants in the equations of the two hyperbolas. Example 1. An hyperbola having the lines (1) a! + 2y-^3 = and (2) Zx + iy + Z = for asymptotes, will have an equation of the form (a?-|-2y+3)(3a;+4y-|.5) + fc=0, (3) while the equation of its conjugate hyperbola will be (a!-f-2y + 3)(3a!-l-4y + 5)-fc = 0. (4) If a second condition is imposed upon the hyper- bola, e.gf., that it shall pass through the point (1, "1), then the value of h may be easily found thus: since the curve passes through the point (1, ~1), therefore by equation (3), (l_2+3)(3-4-f5)-f A; = 0; .-. fc=-8, and the equation of the hyperbola is (a!4-2j, + 3)(3a!-f-4y + 5).-8 = 0, that is, 3a^-|-10a;j/ + 8/-hl4a;-f-22y + 7 = 0; (5) 131, 132] THE HYPERBOLA 219 and the equation of the conjugate hyperbola is 3a;' + 10a!y-|-8/ + 14a; + 22y + 23 = 0. Example 2. If the asymptotes of an hyperbola are chosen as the coordinate axes, their equations will be a! = and ^ = 0, respectively; or, combined in one equation, a;y=0. (1) Therefore the equation of the hyperbola, — which differs from that of its asymptotes by a constant, — is xy = k, (2) wherein the value of the constant h is to be deter- mined by an additional assigned condition concerning the curve ; e.g., that it shall pass through a point, such as the vertex, \ 2 ' 2 J' The equation becomes a^=^— » (3) which is the equation of the hyperbola referred to its asymptotes as axes, ordinarily oblique. 132. Equilateral or rectangular hyperbola. If the axes of an hyperbola are equal, so that a = b, its equation has the form a?—y' = a*, (1) and its eccentricity is c= V2. Its conjugate hyperbola has the equation (i^ — y*= —a*; (2) with the same eccentricity and the same shape; "while its asymptotes have the equations « = ± ». (3) 220 SPECIAL rUOPERTIES OF THE CONICS [Ch. IX and are therefore the bisectors of the angles formed by the axes of the curves ; hence the asymptotes of these hyperbolas are perpendicular to each other. The hyperbola whose axes are equal is therefore called an equilateral, or a rectangular hyper- bola, according as it is thought of as having equal axes or asymptotes at right angles. EXERCISES 1. Find the asymptotes of the hyperbola 9 a? — 16^=144, and the angle between them. 2. If the vertex lies one third of the distance from the center to the focus, find the equations of the hyperbola, and of its asymptotes. 3. If a line ff = mx + e meets the hyperbola — — ^ = 1 in a' b one finite and one infinitely distant point, the line is parallel to an asymptote. 4. Show that, in an equilateral hyperbola, the distance of a point from the center is a mean proportional between its focal distances. 5. Find the equation of the hyperbola passing through the point (1, 0), and having for asymptotes the lines a'+y + 3 = and x — y + 7=0 6. Write the equation of the hyperbola conjugate to that of Ex. 5. 7. Find the equations of the asymptotes of the hyperbola a?-iy^-2x-16y-19 = 0; also find the equation of the conjugate hyperbola. 8. Find the equation of the asymptotes of the hyperbola 132, 133] THK HYPERBOLA 221 9. Find the equation of the hyperbola conjugate to 10. Prove that a perpendicular from the focus to an asymp- tote of an hyperbola is equal to the semi-conjugate axis. 11. The asymptotes meet the directrices of the avhyperbola on the a!-auxiliary circle, and of the conjugate hyperbola on the ^-auxiliary circle. 12. The circle described about a focus, with a radius equal to half the conjugate axis, will pass through the intersections of the asymptotes and a directrix. 13. The line from the center to the focus F of an hyper- bola is the diameter of a circle that cuts an asymptote at P; show that the chords CP and FP are equal, respectively, to the semi-transverse and semi-conjugate axes. 133. Diameters. The equation of the diameter to the hyper- bola is y = -?-«. [53] This equation shows that every diameter of the hyperbola passes through the center. Conversely, it is true, as in the case of the ellipse, that every chord of the hyperbola through the center is a diameter. That chord of the original set which passes through the center is the diameter conjugate to [53] ; and its equation is y = tnx. [59] Let m' be the slope of a diameter, and m that of its conju- 222 Sl'ECIAL PROPERTIES OP THE CONICS [Ch. IX gate ; the essential condition that two diameters should be conjugate to each other is that [of. Art. 123] mm'=^ [60] 134. Properties of conjugate diameters of the hyperbola. («) It is clear that the condition [60] holds also for the hyperbola which is conjugate to the given hyperbola; for, replacing a' by — a^ and — h^ by 6^ leaves equation [fiO] unchanged. Hence, diameters which are conjugate to each other for a given hyperbola are conjugates also for the conjugate of that hyperbola. (j8) The axes of the hyperbola are clearly diameters of the curve. They are perpendicular to each other, and therefore satisfy the relation mm' = — 1. Compare this condition with that of equation [60] ; it follows that the transverse and conjugate axes of the hyperbola are the only pair of perpendicular conjugate diameters [cf . (/S), Art. 124]. If a = 6, the condition [60] reduces to mm' = 1 ; therefore [Art. 12], in the rectangular hyperbola the sum of the angles which a pair of conjugate diameters make with the transverse axis is 90°. (y) Since in equation [60] the product m,m' is positive, it follows that the angles which conjugate diameters make with the transverse axis are both acute, or both obtuse. Moreover, if m < ± -, then m' > ± - : a a 133, 134] THE HYPERBOLA 223 and the diameters lie on opposite sides of an asymptote ; i.e., two conjxigate diameters lie in the same quadrant formed by the axes of the hyperbola and on opposite sides of the asymptote. [Cf. Art. 124, (a)]. (8) An asymptote passes through the center of an hyperbola, hence may be regarded as a diameter. Its slope is m=±-, .'. m'=±— ; a a hence, an asymptote regarded as a diameter is its ovm conjugate; it may be called a self-conjugate diameter. This is a limiting case of (y) above. («) It follows from this last fact that if a diameter intersects a given hyperbola, then the conjugate diameter does not inter- sect it, but cuts the conjugate hyperbola. It is customary and useful to define as the extremities of the conjugate diameter its points of intersection with the conjugate hyperbola. With this limitation, it follows from (a) of this article that, as in the ellipse, each of two conjugate diameters bisects the chords par- allel to the other. (Q As a limiting case of this last proposition, also, it is evi- dent that the tangent at the end of a diameter is paraMel to the conjugate diameter. By reasoning entirely analogous to that given in Art. 124, for the ellipse, properties similar to those there given may be derived for the hyperbola. They are included in the following exercises,, to be worked out by the student. EXERCISES 1. Fini the equation of the diameter of the hyperbola 4: x'— 5rf= 20 which bisects the chords y= — 4 a; + &. Find also tlie conjugate diameter. 224 SPECIAL PROPERTIES OF THE CONICS [Ch. IX 2. Find, for the hyperbola of Ex. 1, a diameter through the point (~1, "1) ; also find its conjugate diameter. 3. Find the diameter of the hyperbola 9 a^ — 16 j/* = 25 which is conjugate to the diameter a; = 5 y. 4. Find the equation of a chord of the hyperbola 4 a^ — 3 1/^ = 36, which is bisected at the point (~8, ~4). 5. Show that, in an equilateral hyperbola, conjugate diam- eters make equal angles with the asymptotes. 6. The difference of the squares of two conjugate semi- diameters is constant; it is equal to the difference of the squares of the semi-axes. 7. The angle between two conjugate diameters is sin~*— — -• ab 8. Tangents at the ends of a pair of conjugate diameters in- tersect on an asymptote. REVIEW EXAMPLES 1. Write the equation of an hyperbola whose transverse axis is 2V3 and the eccentricity equal to 2. 2. Find the equation of the diameter of the hyperbola 16 a;^ — 9 y* = 144, the coordinates of the extremity of a diam- eter being (4, ■*^). Find the equation of the conjugate diameter. 3. Assume the equation of the hyperbola, and show that the difference of the focal distances is constant. 4. Find the locus of the vertex of a triangle of given base 2 c, if the difference of the two other sides is a constant, and equal to 2 a. 5. Find the locus of the vertex of a triangle of given base, if the difference of the tangents of the base angles is constant. l:J4] THE HYPERBOLA 225 6. rind the asymptotes of the hyperbola xy — ix — 5y =0. What is the equation of the conjugate hyperbola ? 7. Show that two concentric rectangular hyperbolas, whose axes meet at an angle of 45°, cut each other orthogonally. 8. The portions of any chord of an hyperbola intercepted between the curve and its conjugate are equal. Suggestion. Draw a tangent parallel to the line in question. 9. For the hyperbola Aa? — By' +0=0 show that e = A/l +"^- Is there any restriction upon the values of A andB? [Cf. Ex. 3, p. 207J. 10. Prove that the asymptotes of the hyperbola xi/ + 4y + 4 a; = are a; + 4 = and y + 4 = 0. 11. Find the coordinates of the points of contact of the comjnon tangents to the hyperbolas, 4 a^ — 4 3/* = 3 a^, and 2xy = a\ 12. If a right-angled triangle be inscribed in a rectangular hyperbola, prove that the tangent at the right angle is perpen- dicular to the hypotenuse. 13. Show that the line ^ = 9ftx-f-2i;V— m always touches the hyperbola xy = 1^; and that its point of contact is (vfe'*^" 14. If an ellipse and hyperbola have the same foci, they in- tersect at right angles. 15. Find tangents to the hyperbola a;' — 4 ^"^=1 which are perpendicular to its asymptotes. 16. Find normals to the hyperbola fa-3)'' _ Cv-2) ^ ^ which are parallel to its asymptotes. 226 SPECIAL PROPERTIES OF THE CONICS [Ch. IX 17. Show that iu an equilateral hyperbola conjugate diam- eters are equally inclined to the asymptotes. 18. Show that- two conjugate diameters of a rectangular hyperbola are equal. 19. Find the equation of the tangent to xi/ = 4 at the point (•''■ij ,Vi) by the secant method. 20. Show that, in an equilateral hyperbola, two diameters at right angles to each other are equal. 21. A variable circle is always tangent to each of two fixed circles; prove that the locus of its center is either an hyperbola or an ellipse. 22. Find the common tangents to the hyperbola -5 — 'i = 1 and its mid-circle «* + ^ = a6. 23. In the hyperbola 2oa? — 16^ = i00, find the conjugate diameters that cut each other at an angle of 45°. 24. The latus rectum of an hyperbola is a third proportional to the two axes. 25. Find the equation of a line through the focus of an hy- perbola and the focus of its conjugate. 26. Show that the a^axis is an asymptote of the hyperbola 2xy + 3y' + iy = 9. What is the equation of the other asymptote ? Of the con- jugate hyperbola? 27. If two tangents are drawn from an external point to an hyperbola, they will touch the same or opposite branches of the curve according as the given point lies between or outside of the asymptotes. 28. If P be the middle point of a line AB which is so drawn as to cut off a constant area from the corner of a square, the locus of P is an equilateral hyperbola. PART II SOLID ANALYTIC GEOMETRY CHAPTER X COORDINATE SYSTEMS. THE POINT 135. Solid Analytic Geometry treats by analytic methods problems which concern figures in space, and therefore involves three dimensions. It is evident that new systems of coordinates must be chosen, involving three variables ; and that the analytic work will therefore be somewhat longer than in the plane geometry. On the other hand, since a plane may be considered as a special case of a solid where one dimension has the particu- lar value zero, it is to be expected that the analytic work with three coordinate variables should be entirely consistent with that for two variables, — merely a simple extension of the latter. The student should not fail to notice this close analogy in all cases. 136. Rectangular coordinates. Let three planes be given fixed in space and perpendicular to each other, — the coordinate planes XO Y, YOZ, and ZOX. They will intersect by pairs in three lines, X'X, Y'Y, and Z'Z, also perpendicular to each other, called the coordinate axes. These three lines will meet in a common point O, called the origin. Any three other 227 228 COORDINATE SYSTEMS. THE POINT [Ch. X yY' ^N /M -^X Fig. 91. planes, LP, MP, and NP, parallel respectively to these coordi- nate planes, will intersect in three lines, N'P, L'P, MP, which _ will be parallel respectively to the axes ; and these three lines will meet in, and com- pletely fix the position of, a point P in space. The di- rected distances N'P, L'P, and M'P thus determined, i.e., the perpendicular dis- tances of the point P from the coordinate planes, are the rectangular coordinates of the point P. They are represented respectively by x, y, and z. It is clear that x = N'P = LL' =NM'=OM; y = L'P = MM' = LN' = ON; z = M'P= NN ' = ML' = OL. It is generally convenient, however, to consider x=OM, y = MM', and z = M'P. The point may be denoted by the symbol P= (x, y, z). The axes may be directed at pleasure; it is usual to take the positive directions as shown in the figure. Then the eight portions, or octants, into which space is divided by the coordinate planes, will be distinguished completely by the signs of the coordinates of points within them. If the chosen coordinate planes were oblique to each other, a set of oblique coordinates for any point in space might be found in an entirely analogous way. Unless otherwise stated, rectangular coordinates will be used in the subsequent work. 136, 137] COORDINATE SYSTEMS. TUE POINT 229 Fig. 92. 137. Direction angles: direction cosines. In another useful method of fixing a point in space the axes of reference are chosen as in rectangular coordinates, and any point P of space is fixed by- its distance from the origin, called the radius vector, and the angles a, p, y, which this radius vector makes with the coordinate axes, respectively. These angles are ^ called the direction angles of the line OP, and their cosines, its direction cosines. The point may be concisely denoted as the point P={p,a,p, y). Simple equations connect these coordinates with those of the rectangular system ; for, projecting OP upon the axes OX, OT, and OZ, respectively (cf. Figs. 91, 92), a;=pco8o, 2/ = pcosP, « = pcos7. [61] 0P'=0M'^ + M^=0M^ + MM''' + 3rF^, p2=a!« + I/2 + »2. [62] Moreover, the direction cosines are not independent, but are connected by an equation ; for, by combining the above equations, Again, I.e. I.e., p' = p2 cos" a + p^ cos" P + p^ cos" y, cos* a + COS^ P + COS* 7=1. [63] Such a relation was to have been expected, since only three magnitudes are necessary to determine the position of a point, and therefore the four numbers p, a, p, y could not be independent. Any three numbers, a, b, c, are proportional to the direction cosines of some line : because if these numbers are considered 230 COORDINATE SYSTEMS. THE POINT [Ch. X as the coordinates of a point, then by equation [61] the direc- tion cosines of the radius vector of that point are b cos o=- Va2+62+c«' , cos p= Va«+62+c« , C0S7 = ^a^+b^+c^ [64] These direction cosines are proportional to a, b, c; and are found by dividing a, b, and c, respectively, by the constant y/a? + b' + c'. Direction cosines are useful in giving the direction of any line in space. The direction of any line is the same as that of a parallel line through the origin, therefore the direction of a line may be given by the direction angles of some point whose radius vector is parallel to the line. 138. Distance and direction from one point to another. Let OX, OY, OZ be a set of rec- tangular axes, and P^ = (xj, j/i, z^ and Pj = (x^ !/2, Zj) be two given points. Then the planes through Pi and Pj parallel, respectively, to the coordinate planes, form a rectangular parallelepiped, of which the required distance PiPj is a diagonal. Prom the figure, since Z P1QP2 = 90" and Z M^'RM^ = 90°, therefore T^ = P^ + QP^ = M^'^ + QP^ = i?', - x,y + (yi- y,y + (z, - z,y. FiQ. 93. 137-139] COORDINATE SYSTEMS. THE POINT That is, if d be the required distance, 231 d =y/{x.i - a^i)* + (j/2- 2/1)* +(82- «i)2- [65] Moreover, since the direction of the line PjP^ is given by the angles a, /8, y, which it makes, respectively, with the lines PiX', P\Y', and P^Z', drawn through Pj parallel to the axes, therefore the projection of d (= P^P^) upon these lines in turn gives P^Pi cos a = P^X', PiPn cos /3 = PiT*, P^P^ cos y = PiZ', I.e., d cos a = ai — ah, d 00s ji = y^ — yi, dcosy = «2 — Zi; and, finally, COSa^^i^S, c«8p = l^i9i^, C«8,=?^9^. [66] dad If d is taken positive, these equations determine the direc- tion from Pj to Pj. 139. Angle between two radii vectores. Angle between two lines. Let Pj = (pi, a,, /3i, yi) and P„ = (p^, ctj, /S^, y2) be two given points, and the angle included by the radii vectores pi and />2. Then, projecting upon OPi [Art. 13], proj. OP2 = proj. OMnMi'Pi, i.e., pj cos = OMj cos aj + M^Mi cos j8i + M^P^ cos yj. But OJW2 = P2 cos ttj, J!f2J!f2' = Pi cos /Sj, and M2P2 = p2 cos y, ; hence, ^2 cos 6 = Pi cos oj cos «! + /jj cos jSj cos /Si + p^ cos ya cos yi ; I'.e., cos e = cos aj cos ag + COS Pi COS Pa + COS 71 cos 72, [67] and this relation determines the required angle 0. Fig. 94. 232 COORDINATE SYSTEMS. THE POINT [Ch. X It follows, since any two straight lines in space have their directions given by the direction angles of radii vectores which are parallel to them, respectively, that formula [67] applies as well to the angle 9 between any two straight lines in space, whose direction angles are given. Two special cases arise, that of parallel and that of perpen- dicular lines. If the two given lines are parallel, evidently Oi = og, Pi = P2, 71 = 74? [68] and formula [67] reduces to equation [63]. If the lines are perpendicular, cos 6 = 0, and equation [67] reduces to cos oj COS oj + COS Pi COS Pa + cos yi cos Yg = 0. [69] EXAMPLES ON CHAPTER X 1. Prove that the triangle formed by joining the points (a, 6, c,), (6, c, a), and (c, a, 6), in pairs, is equilateral. 2. The direction cosines of a straight line are proportional to -3, -2, +1 ; find their values. [Cf. Art. 137.] 3. Find the angle between two straight lines whose direc- tion cosines are proportional to 3, 3, and 3, and 7, ~4, and 6, respectively. 4. Prove that if Pj bisects the line PjPj then a;j= — ^ — , ^8=- — ^ — ' *8 = — g ^^^^ 5. Find the direction angles of a straight line which makes equal angles with the three coordinate axes. 6. A straight line makes the angle 30° with the avaxis, and 60° with the 2-axis. What angle does it make with the 2/-axis ? 7. Prove analytically that the straight lines joining the mid- points of the opposite edges of a tetrahedron pass through a common point, and are bisected by it. 139] COORDINATE SYSTEMS. THE POINT 233 8. What is the length of a line whose projections on the coordinate axes are 5, 6, 7, respectively ? 9. Find the radius vector, and its direction cosines, for each of the points (~1, "2, ~3), (0, "1, -1), and (a, b, c). 10. Prove that if P, divides the line PjPj in the ratio ^, so mj that PjPg : PjPj = m, : wij, [cf. Art. 24], nix + "*2 ' m-i + wij »Mi + wig J? jt/r .96 ^' then aBj = - yt=- «3 = [71] 11. Find the coSrdinates of the points dividing the line from (0, 0, 0) to (3, 4, 5) externally and internally in the ratio 1 : 5. 12. Prove analytically that the straight lines joining the midpoints of the opposite sides of any quadrilateral bisect each other. 13. Show that the equations of transformation from one set of rectangular axes to a parallel set are x = x'-\-Ti, y = y' + 7c, z = z'+J. 14. Show that the equations of transformation from one set of rectangular axes to another through the same origin, making the direction angles «i, ^i, yi, a^, p^, yj, and a^ /Sj, yj respec- tively with the old axes, are a; = a;' cos «[ + y' cos kj + z' cos Oj, y = x' cos /3i + y' cos ^82 + 2' cos ya, Z = x' cos y, + y' cos ya + z' cos yg. CHAPTER XI THE LOCUS OF AN EQUATION. SURFACES 140. Attention has been called to the close analogy between the corresponding analytical results for the geometry of the plane and of space. In geometry of one dimension, restricted to a line, the point is the elementary conception. Position is given by one variable, referring to a fixed point in that line; and any algebraic equation in that variable represents one or more points. In geometry of two dimensions, how- ever, it has been shown that the line may be taken as the fundamental element. Position is given by two variables, referring to two fixed lines in the plane; and any algebraic equation in the two variables represents a curve, i.e., a line whose generating point moves so as to satisfy some condition or law. Correspondingly, in geometry of three dimensions the surface is the elementary conception. Position is given by three variables, referring to three fixed surfaces, since any point is the intersection of three surfaces ; and it can be shown that any algebraic equation in three variables represents some surface. The study of the special equations of first and second degree in three variables will be taken up in the two succeeding chapters. Here it is desired to consider briefly three simple classes of surfaces: (1) planes parallel to the coordinate planes; (2) cylinders, i.e., surfaces which are generated by 140, 141] THE LOCUS OF AN EQUATION. SURFACES 235 a straight line moving parallel to a fixed straight line, and always intersecting a fixed curve ; and (3) surfaces of revolution, i.e., surfaces generated by revolving some plane curve about a fixed straight line lying in its plane. 141. Equations in one variable. Planes parallel to coordinate planes. From the definition • of rectangular coordinates it follows that the equations a; = 0, y = 0, z = 0, represent the coordinate planes, respectively, and that any algebraic equation in one variable and of the first degree, as y = — 3, represents a plane parallel to one of them. Similarly an equation in one variable, and of degree n, will represent n such parallel planes, either real or imaginary. For, the first member of any such equation, as j9„a;» + pix-i + p^-' +■■■+ p„_ia; + p„ = 0, (1) can be factored into n linear factors, real or imaginary, Po(x — Xi)(x — Xi)(—)(x — x„) = 0; (2) and by the reasoning of Art. 37, equation (2) will represent the loci of the n equations a; — a;i = 0, a;— X2 = 0, ••-, x~x„ = 0, each of which is a plane, parallel to the z/z-plane, and real if the corresponding root is real. In the same way, an equation in y or a only will represent planes parallel to the zx- or xjr-plane. A7>y algebraic equaiion in one variable represents one or more planes parallel to a coordinate plane. It follows at once, by Art. 36, that two simultaneous equa- tions of the first degree, each in one variable, represent the intersection of two planes parallel to coordinate planes ; there- fore, represent a straight line parallel to tlie coordinate axis of 236 THE LOCUS OF AN EQUATION. SURFACES [Ch. XI the third variable ; e.g., y = b, z = c, considered as simulta- neous equations, represent a straight line parallel to the ovaxis. 142. Equations in two variables. Cylinders perpendicular to coordinate planes. Consider the equation 2x + 3y=6, (1) in two variables only. In the ar!/-plane it represents a straight line AB. If, now, from ^ any point P of AB a straight line be drawn parallel to the «-axis, the X and y coordinates of every point Q on this line will be the same as for P, and therefore satisfy equation (1). Moreover, if the line PQ be moved along AB, and always parallel to the sf-axis, still the coordinates of evej:y point in it satisfy equation (1). As the line PQ is thus moved, it traces a plane surface perpendicular to the aiy-plane; and, as the coordinates of a point not on this surface evidently do not satisfy equation (1), this plane is the locus of equation (1). Again : the equation y' + 2^ = r* (2) Fio. 96. r / pf\ \ / Q / \ u Fia. 97. 141-143] THE LOCUS OF AN EQUATION. SURFACES 237 represents in the z/a-plane a circle. It is therefore satisfied by the coordinates of any point Q, in a line parallel to the a)-axis, through any point P of this circle ; and also by the coordinates of Q as this line PQ is moved along the circle and parallel to the ovaxis. The circular cylinder thus traced by the line PQ, perpendicular to the yz-plane, is the locus of the given equation. Similarly, it may be shown that the locus of the equation T.-h = ^' (3) is a cylindrical surface traced by a straight line parallel to the y-axis, and moving along the hyperbola whose equation in the a»-plane is equation (3). And in general, it is clear by analogy that any algebraic equation in two variables represents a surface whose elements are parallel to the axis of the third variable, and which has its form and position determined by the plane curve rep- resented by the same equation. Such a surface is cylindrical. If, now, a cylinder has its axis parallel to a coordinate axis, a section made by a plane perpendicular to that axis is a curve parallel to, and equal to, the directing curve on the coordinate plane, and is represented in the cutting plane by the same equation. Thus, the section of the elliptical cylinder whose equation is 3 a^ + y* = 5, cut by the plane z = 7, is an ellipse equal and parallel to the ellipse whose equar tion is 3 x? + 3/- = 5. 143. Equations in three variables. Surfaces. A point is de- termined by three coordinates, which will vary according to some definite law if the point moves so as to trace some defi- nite surface. Algebraically, the coordinates must then satisfy an equation having three variables: that is, the locus of any algebraic equation in three variables is a surface. 238 THE LOCUS OF AN EQUATION. SUKFAOES [Ch. XI 144. Curves. Projections of curves. Traces of surfaces. Two surfaces intersect in a curve in space ; and since every algebraic equation in solid analytic geometry represents a surface, a curve may be represented analytically by the two equations, regarded as simultaneous, of surfaces which pass through it. Thus it has been seen that the equations y = h, z = c separately rep- resent planes, but considered as simultaneous they represent the straight line which is the intersection of those planes. But by the reasoning of Art. 38, the given equations of a curve may be replaced by simpler ones which represent other surfaces passing through the same curve. In dealing with curves it is often useful to obtain, from the equations given, equations of cylinders through the same curve ; i.e., it is generally useful to represent a curve by two equations, each in two variables only. Since the cylinder is taken perpendicular to a coordinate plane, its equation when considered as of a plane curve, repre- sents the projection of the space curve upon the coordinate plane. Example. The curve of intersection of the two surfaces, (1) a^+2/2 + «2-25 = and (2) ar' + /-16 = 0, is also the intersection of the surface a? + ^ + z'-25 - (9^ + 2/2-16) = 0, i.e., «= ± 3, (3) with the surface (2). The curve is therefore com- posed of two circles of radius 4, parallel to the xy- plane at distances -|- 3 and — 3 from it. Here equation (2) is itself the equation of the pro- jection of the space curve upon the a;y-plane; while z= ±3 gives the projection upon each of the other coordinate planes. 144, 145] THE LOCUS OF AN EQUATION. SURFACES 239 Similarly, the curves of intersection of a surface with the coordinate planes may be used to help determine the nature of a surface. These curves are called the traces of the surface. The intercepts upon the axes are of course also useful. Thus, the trace of the surface x' + j/' -\- z' = 25 on the y«-plane, is if + z' = 25; on the zx-plane, is ar* + 2^ = 25 ; on the a!y-plane, is a^ + ^ = 26. Each of these traces is a circle of radius 6, about the origin as center ; and the surface is a sphere of radius 5 with center at the origin. Since three surfaces in general have only one or more separate points in common, the locus of three equations, con- sidered as simultaneous, is one or more distinct points. 145. Surfaces of revolution. Analogous to the cylinders are the surfaces traced by revolving any plane curve about a straight line in its plane as axis. From the method of formation, it follows that each plane section perpendicular to the axis is a circle, — the path traced by a point of the gen- erating curve as it revolves ; and the radius of the circle is the distance of the point from the axis before revolution be- gins. These facts lead readily to the equation cf any surface of revolution, as a few examples will show. (a) The cone formed by revolving about the z-axis the line 2x + 3z = 15. (1) Any point P of the line (1) traces during the revolution a circle of radius LP, parallel to the a;y-plane. The equation of that path is x- + i/ = LP\ 240 THE LOCUS OF AN EQUATION. SURFACES [Ch. XI Z Fig. 98. But in the a%-plane, before revolution is begun, LP is the abscissa of P and may be found in terms of the third coordi- nate, z, which does not change as P moves : by equation (1), 2 ' so that the equation of the path of P is ■3z)' ^^f = ^ (2) But P is any point of line (1) ; hence equation (2) is sat- isfied by every point of the line during its motion, i.e., of the surface generated by the line; which is the required conical surface. (6) Tlie sphere formed by revolving about the z-axis the circle x-2-|-z2 = 25. (3) In this case, any point P of the curve traces during the revolution a circle of radius NP, parallel to the «y-plane. The equation of this path is therefore Fio. 99. 14u] THE LOCUS OF AN KQUATIOX. SURFACES 2-41 lUit ill the ass-plane, by equation (3), NP=x = ^25-z^. Hence, substituting above, a,-^-|-.V-+2==25; (4) which is the equation of the required spherical surface. Of the various surfaces of revolution those of particular interest are generated by revolving about their axes the various conic sections, giving the cones, spheres, paraboloids, ellip- soids, and hyperboloids of revolution. The student may verify the equations of the following surfaces: * The sphere : with center at the origin, and radius r, x' + f+^^.r'; (5) with center at (a, h, c), by Art. 138, equation (5) becomes (x-ay + (y-by+{z-cy = r'. (6) The cone : the surface generated by the right line z = mx + c, rotated about the «-axis, o^ + f = ^^^- (7) mr The oblate spheroid: the surface generated by the ellipse — I — = 1, rotated about the minor axis, a' b^ -. + ^ + S = l- (8) a' or V The prolate spheroid: the surface generated by the ellipse — J — = 1 rotated about the major axis, a^ ¥ ' a" h^ h^ *See Chapter XITI, where diagrams are given for the corresponding cases of the general quadric, with elliptical instead of circular sections. 242 THE LOCUS OF AN EQUATION. SURFACES [Ch. XI The hyperboloid of one nappe : the surface generated by the hyperbola -^ ~ r^ = !> rotated about the conjugate axis, or a" 0* The hyperboloid of two nappes : the surface generated by the hyperbola -5 — t^ = !> rotated about the transverse axis, ^-l-l^l. (11) a" 1^ b" ^ ^ The paraboloid of revolution : the surface generated by the pa- rabola a? = ipz, rotated about its axis, aP + f=4:pz. (12) EXAMPLES ON CHAPTER XI What is the locus of each of the following equations ? 1. 25 a?* -10 35+1 = 0. 4. a3?=bxy — cy\ 2. ar' + 4as = 0. 5. a:» + ar'-4ar — 4 = 0. 3. a?—2xy-\-f+Sx—Sy=0. 6. 9r' + 9y = 9. What are the curves of intersection of the surfaces represented by the equations : 7. a;-l = 0, a;^ + t^ + 2' = 10? 8. 4ar'-9jr' = 0, « = 4? 9. ar' + y' + 2= = 9, 4a!=' + 3^ = 4? 10. 9(a!» + y»)-«» = 25-10», « = ±5? " ' 9 16 Determine the traces upon the coordinate planes of the follow- ing surfaces: 12. ar' + 2/' = 4sr'; 13. a? - 3y^ + 2z^=6. 145] THE LOCUS OF AN EQUATION. SURFACES 243 Find the equation of : 14. The paraboloid of revolution one of whose traces is 15. The cone of revolution one of whose traces is z=y — 10 and whose axis is the axis of y. Find its vertex. z' Q? 16. The oblate spheroid one of whose traces is s ''" "q ~ ■'■• 17. The prolate spheroid one of whose traces is 7 9 18. The surface of revolution whose axis is the avaxis and one of whose traces is y^x = — 4. 19. The hyperboloid of two nappes one of whose traces is 16/-9a:'=144. 20. The sphere having the minor axis of the ellipse i.3?-\- 93^ — 6y — 4a; = as diameter CHAPTER XII EQUATIONS OF THE FIRST DEGREE Ax + By+Cz + D = PLANES AND STRAIGHT LINES I. The Plane 146. Every equation of the first degree represents a plane. A plane is a surface such that it contains every point on a straight line joining any two of its points. Let P, =(xi, yi, Zi) and P2 = («2» Vn, ^s) be any two points of the surface whose equation is Ax + By+Cz + D = 0, [73] 80 that Axi + Byi + Czi + D = 0, (1) and Ax, + By^ + Czj + Z) = 0. (2) Now, if P3 = (xj, y^, «,) be any point on the straight line from P, to Pj at a distance d, from Pi and dj from P^ then, by equation [71], djKj + diXj _ di,Va + diiyi ^^^djV+djZj^ -g> rfi + f^ ' di + d2 di + d. But this point lies on the surface represented by equation [72] ; for, on substituting its coordinates in equation [72], the latter becomes -4--(.4x2 + By, + Cz, + D)+ -A^ {Ax, + By^ + Cz^+ D) =0, U6-148] THE PLANE 2-45 which is a true equation, since each number in parenthesis vanishes separately by equations (1) and (2). Hence every point of the line PiP^ is on the locus of equation [72], and that locus is therefore a plane. Every algebraic equation of the first degree in three variables represents a jAane. 147. The intercept equation of a plane. A plane will in gen- eral cut each coordinate axis at some definite distance from the origin, and this distance is called the intercept of the plane on the axis. If a, b, c be the intercepts on the x-, y-, and «-axes, respectively, made by the plane whose equation is Ax+By+Cz+D = 0, (1) then (a, 0, 0,), (0, 6, 0), (0, 0, c) are points of the plane, and therefore Aa + D = 0, Bb + D = 0, Cc + D=0, ^ = --a' ^ = -f' ^=-?- (2) Hence equation (1) may be written a b c i.e., - + ^ + - = 1; [73] a b c and this is the equation of the plane in terms of its intercepts. From equations (2) it is clear also that the intercepts of plane (1) are a = --, b = --, c=--. (3) 148. The normal equation of a plane. A plane is wholly determined in position if the length and direction of a perpen- dicular upon it from the origin be known ; and this method of fixing a plane leads to one of the most useful forms of its 246 EQUATIONS OF THE FIRST DEGREE [Ch. XII equation. Let OQ be the perpendicular from the origin to the plane ABC, let p be its length, and let a, p, y be its direction angles. Let P = (x, y, z) be any point of the plane (other than Q), and draw its coordinates OM, MM', and M'P. Then, projecting upon OQ, proj. OJf+ proj. MM' + -pTo]. Jlf'P=proj. OP, that is, a!C08a + j/co8P + «co8-y =i». [74] This is called the normal equation of the plane. By reasoning analogous to that of Art. 46 it can be shown that the equation Ax + By + Cz + D = 0, when written in the normal form, becomes A ... B VA^+W+C' ^/A' + B^-\-C^ and therefore that A :* + ■ ^/A^ + B'+C^ -D y 008 'Y: >/^2 4 JS« + C« c C08P= ' B P = [76] Hence, to reduce equation (1) to the normal form, it is only necessary to transpose the constant term to the second member 148, 149J THE I'LANK 247 of the equation, and then divide both members by the square root of the sum of the squares of the coefficients of the variable terms. The sign of the radical may be determined so that any chosen direction angle, as «, shall be less than 90° : then, the sign of the radical is the swine as the sign of the x-tertn, i.e., in general, positive. 149. The angle between two planes. Parallel and perpendicu- lar planes. The angles formed by two intersecting planes are the same as the angles formed by two straight lines perpen- dicular to them respectively ; i.e., are the same as the angles between the respective normals from the origin to the planes. If Aix + Ba+Ciz + Di = 0, (1) and A^ + B^+C^ + Dt = 0, (2) are two planes, then by equations [67] and [75], if $ be the angle between the two planes, and hence between the two normals, pi>iie= A1J2 + B1B2+C1C2 [763 V^l2 + Bi^ + Ci« V^2« + -B28 -I- Ci^ There are two cases of special interest. (a) Parallel planes. If the planes (1) and (2) are parallel, their normals from the origin will have the same direction cosines, and differ only in length ; therefore, by equations [75], the equations of the planes must be such that the coefficients of the variable terras are the same in the two equations, or can be made the same by multiplying one equation by a con- stant. In other words, if the planes (1) and (2) are parallel, then — = ^ = — ; [77] A^ Bg Ci i.e., the plane Ax + By+ Cz+ K=:0 (3) 248 eql'atioxs of tue eiiist degree [Ch. xn is parallel to the plane Ax + By+Cz + D = 0, (4) for all values of the parameter K. (b) Perpendicular planes. If the planes (1) and (2) are per- pendicular to each other, then cos 6 = 0, and AxAi + BiB.i+CiCi = 0; [78] and conversely. II. The Straight Line 150. Two equations of the first degree represent a straight line. Every equation of first degree represents a plane, and two planes intersect in a straight line; hence the locus of the two simultaneous equations of first degree, Aa; + -B^+Ci2 + A = 0, A^ + B^+C^JrDi = Q, (1) is a straight line. As suggested in Art. 144, it is generally more simple to represent the straight line by equations in two variables only, standard forms, to which equations (1) can always be reduced. 151. Standard forms for the equations of a straight line. (a) The straight line through a given point in a given direction. Let Pi = (a^, 3/i, «,) be a given point, and a, yS, y the direction angles of a straight line through it. Let P=.(x, y, z) be any other point on the line, at a distance d from Pi. Then by equation [66], d cos a = a; — a^ d cos P = y — yi, d cos y = z — z^ (1) hence a: - aor ^y - Vy ^z-z^ _ ^^^^ cos a COS P COS 1 which are the equations of a straight line in the first standard forms, called the symmetrical equations. 549-152] THE STRAIGHT LINE 249 (6) ITie straight line through two given points. Let Pj = -(*i> yi> %) ^■"d P2 = (x2, j/j, Za) be the given points. Any straight line passing through Pj has [79] for its equations. If the line passes also through Pjj, then ■n_ y2— yi _ g2-gi . (2) cos « cos fi cos y ' and hence, dividing equations [79] by equations (2), to elimi- nate the unknown direction cosines, ac2-xi Vi-Vi «2-«i These are the second standard forms for the equations of a straight line. (c) The straight line with given projections on the coordinate planes. If the equation of the projections of a given line upon the aa> and ^z-planes are, respectively, f = ns +d,i y then, considered as simultaneous, these also axe the equations of the given line in space. 152. Redaction of the general equations of a straight line to a standard form. Determination of the direction angles. (a) Third standard form : projections. This reduction may be illustrated by a numerical example. Given the equations 3x + 2y + z — 5 = 0, x + 2y-2z = 3, (1) representing a straight line. Eliminating z, y, and x, succes- sively, the equations 7a; + 62/-13 = 0, 2x + 3z-2=^0, 4y-7z-A = (2) are obtained, each representing a plane through the given line 250 EQUATIONS OF THE FIRST DEGREE [Ch. XII and perpendicular to a coordinate plane. Therefore these equations are also the equations of the projections of the line, upon the xy-, zx-, and yz-planes, respectively. Any two of them will represent the line (6) First standard form: direction angles. The method of reducing the general equations of a straight line to the first standard form, and finding its direction angles, can also be illustrated by a numerical case. Considering still the line whose equations are (1) above and taking the equations of any two of its projections, e.g., 2x+3z-2 = 0, 4:y-7z-i = 0; (3) these have one variable, z, in common. Equating the values of this common variable from the two equations, gives _ — 2a; + 2 _ 4y — 4 '~ 3 ~^T~' which, may be written, to correspond with equations [79], ^_1 y_l ^_0 Now, although the denominators — f , ^,1 of equations (4) are not direction cosines of any line, yet, by equations [64], they differ from such direction cosines only by the divisor Vf + lf+1, i.e., by iVlOT. or, rewriting equations (4) in the form x-l _ y-l z-0 ^lR--—7— = —A—> W Vioi Vioi Viol they correspond entirely to equations [79]. Therefore the line passes through the point (1, 1, 0), and its direction angles are given by the relations 152] THE STRAIGHT LINE 251 6 „ 7 4 cos «= 1^=, COS a = , , cosy = • VlOl ^ VlOl ^ VlOl The method given above is evidently entirely general. EXAMPLES ON CHAPTER XII 1. Find the equations of a line through the points (-1, -2, -3) and (3, 0, 1). 2. Find the equation of a plane through the points (0, 0, 0), (0, 1, -1), and (5, 4, 3). 3. Write the equations of the straight line through the point (ce, &, c), and having its direction cosines proportional to I, m, n. 4. What are the projections of the line of Ex. 1 upon the coordinate planes ? Where does the line pierce those planes ? 9. Find the equations of a straight line through the point ("1, 2, ~3) and perpendicular to the plane a; + 3y4-3« = 0. Eeduce to the intercept and normal forms, and determine which octant each plane cuts : 6. 2x + Sy-\-10z = 7; 7. ay + bz—l(i = cx. 8. Reduce the equations of the line 2x — 5y — z=7, 5y + 2z — l = x to the symmetrical form, and determine its direction cosines. 9. What does equation [74] become if the plane is perpen. dicular to the «y-plane ? What if parallel to the ay-plane ? 10. Derive the formula for the distance of the point Pi = (sfi) Vi} 2i) from the plane Ax + By + Cz + D = 0. ^A^ + B2 + C* Hint : Pass a parallel plane through Pi, and find the distance p for each plane. [Cf. Art. 51.] 252 EQUATIONS OF THE FIRST DEGREE [Ch. XII 11. Wi'ite the equation of a plane parallel to the plane 3a!-42/ + 10«-10=:0, and passing through the origin ; through the point (~4, 6, 6). 12. Write the equation of a plane perpendicular to the plane 3x + 5y — z + 6=0, and passing through the two points (3, 1, 2) and (0, ~2, ~4). 13. Find the distances of the points ("7, +2, "3) and (~3, "3, 1) from the plane 2a;+5y — z — 9 = 0. Are they on the same side of the plane ? [See Ex. 10]. 14. At what angle does the plane ax + by + cz=0 cut each coordinate plane ? Each coordinate axis ? 15. Find the equation of a plane through the point (1, 1, 1) and perpendicular to each of the planes 2x — 3y + 7z = l, x — y — 2z = 2. 16. Write the equation of a plane at the distance d from the origin and inclined equally to the three coordinate planes. 17. Write the equations of a straight line through the point ("4, ~5, ~6) and parallel to the line 2x—3z + y = 0, x + y + z = 0. 18. Find the projections on the coordinate planes of the line 3x — 3z + y — 2 = 0, x + y + z + l = 0. 19. Prove that the planes x-4y = 0, y — 5 = 0, (B — 20=30 intersect in a Una Find its equations. CHAPTER XIII BQTJATIONS OF THE SECOND DEGREE QUADRIC SURFACES 153. The locus of an equation of second degree. The most general algebraic equatioa of second degree in three variables may be written Ax^ + By^+ Cz' + ST^z + ZOacz + iMxy + iLas + SMy + 2J!fz + JB:= \o c) a If now the equations " — = Jc and (1) (2) (3) be formed, wherein k is any constant, each will represent a plane; and the two planes will intersect in a straight line, which may readily be shown to lie on the given surface. By varying k a family of such straight lines will be obtained. The unparted hyperboloid is an example of a ruled surface, i.e., of a surface such that through every point of it a straight line can be drawn which will be wholly on the surface.* The hyperbolic paraboloid and the cone are further examples of ruled sur- faces, as may be shown by treating their equations like the above. 161. The hyperboloid and its asymptotic cone. The hyperboloid a' b^ d" and the cone are closely related. It is clear that, since the equations differ only in the constant terms, the surfaces can have no finite points in common ; while as the values of y and z are increased indefinitely, the corresponding values for x from the two equations become Fig. 108 * See Salmon's "Geometry of Three Dimensions," Cliap. VI. 264 EQUATIONS OF THE SECOND DEGREE [Cii. XIII relatively nearer. In fact, the hyperboloid may be said to be tangent to tlie cone at infinity, and bears to the cone a relation entirely analogous to that between the hyperbola and its asymptotes. In the same way, the cone ^ — 'j^— j = is asymptotic to the hyperboloid — — 7^ = 1. a c EXAMPLES ON CHAPTER XIII 1. Derive Eq. [85] directly from the definiticn of Art. 156. 2. Derive Eq. [86] directly from the definition of Art. 157 3. Derive Eq. [89] directly from the definition of Art. 159, 4. Show analytically that the intersection of two spheres is a circle. 5. Find the equation of the tangent plane to the sphere {x — 0)'+ (j) — &)'+(« — c)' = r', at any point of the sphere. 6. Show that the equation Ax^x + By^y + Cz^z + K= rep- resents a plane tangent to the quadric, -4x^-1- By^+ C«'+ K=0, at the point (a^, y„ «i) on the quadric. 7. Find the equation of the cone with origin as vertex and a? z' the ellipse 5^ + =1 ^^ t^^ plane y = 2, as directrix. 8. Find the equation of a sphere having the line from Pi = (xi, y„ z,) to Pg = (x^ yj, Zj) as a diameter. 9. Show that a sphere is determined by four points in space. 10. Write the equation of the quadric whose directing curves have the equations : u^i-"- "' M-^ CHAPTER XIV CURVE TRACING. HIGHER PLANE CURVES* 162. Definitions. In Cartesian coordinates a curve whose equation is reducible to a finite number of terms, each involv- ing only positive integral powers of the coordinates, is called an algebraic curve ; all other curves are called transcendental. Algebraic curves the degree of whose equations exceeds two, and all transcendental curves that lie wholly in a plane, are called higher plane curves.* In plotting plane curves, as suggested in Art. 30, the "point-wise" method is often puzzling, as well as long and tedious. It is usually helpful to find the intercepts and a few of the most easily obtained points ; but then application should be made of the symmetry and other peculiarities of the equation. In this way tangents, diameters, and asymptotes often can be found, which may aid very effectively in determining the gen- eral shape of the locus. Among the following curves are given many that are usually studied in more detail by calculus methods, including some of special historic interest. * The material in the following pages is intended merely to give brief suggestions for further drill in curve tracing. For a fuller treatment of Higher Plane Curves, with historical notes, see Tanner and Allen's " Elementary Course in Analytic Geometry." 266 266 CURVE TRACING. HIGHER PLANE CURVES [Ch. XIV 163. Illustrative examples. Diameters and tangents. If the given equation is of the second degree in one of its variables, a diameter and tangents of the locus may sometimes be obtained, and the curve traced approximately, by the method illustrated below. (a) Theconic 2x' + 6xy + iy--8x-13y + 12 = 0. (1) Y r ~ - N s ^h >. s P. p "N s \ s. s, H \ f} B •s <] P \ P. sA ti inffe Qt U ) A \ \ ^i , O [^ "r X ■^■f-^ N P ta mt tei n (. ») ^, 3 F >A ^-. ", ' \ rt> p.i •V .,. 3 F 1.-^ > s V . \ S ' V hi- ", R '\ f'> _ - ^ 1 k Fig. 109. Solve for x : x=-^y + 2±iV(y+4){y-2). Plot the straight line : x=-iy + 2: (2) (3) this line is a diameter of the locus. For, given any ordinate, say y = 4, then for the point D, on line (3) the abscissa is x^= — 4, while for the the corresponding points Pg and Pg' on the locus (2), Xs=-i + 2=-2, and a;j'=-4-2^-6; 163] CURVE TRACING. HIGHER PLANE CURVES 267 SO that the chord P^ P^ is bisected by the point A- Similarly, for any ordinate y^, the point D^, of the line (3) bisects the corresponding chord P/ Pi of the locus (2) ; hence, by Art. 108, line (3) is a diameter. iv i/io „ The length of the chord is in each case'-eqad) toithe radical, V(2^i+4)(2/,-2); e.g., P,'Pj = 4, and Pe'A = 2ViO. Again, plot the lines : y-2 = 0, (4) and 2/4-4=0, (5) suggested by the factors in the radical. Now if y^ = 2, one point, only, of the curve is found, viz., A, which lies also in line (4) ; hence, line (4) is a tangent to the locus. Similarly, line (5) is a tangent. If yx be taken between + 2 and — 4 the radical becomes imaginary. Hence, no points of the locus lie between these tangents. Numerically, x increases indefinitely as y increases beyond — 4 in the negative direction, or beyond + 2 in the pos- itive direction. "Finally, particular points can be found by calculating R = ^ V(y + 4)(y — 2) and plotting on either side of the cor- responding point D of the diameter, as in the figure. y B P 3 \y/l =±1.3 Pi 4 J Vie = ± 2.0 Pa 6 ■x/lO = ± 3.2 P. -5 ±1.3 Pi -6 ±2.0 P. -8 ±3.2 P. (6) The curve f = 2a — X 268 CURVE TRACING. HIGHER PLANE CURVES [Cii. XIV From the equation it follows that : (1) The origin is a point of the curve (because there is no constant term). (2) The curve is symmetrical with respect to the a!-axis, only (because the y-term is of even degree). (3) It lies wholly in the first and fourth quadrants (because the negative values of x give imaginary values for y). (4) It lies wholly to the left of the line a; — 2 a = 0, but ap- proaches this line more and more closely as y increases (because if x>2a, then y is imaginary, and as a; = 2 a, ^ becomes infinite). (5) The points (a, ± a), ±0.29 a\ fe±2.6a\ Fio. 110. etc., are on the curve. This curve is represented in figure 110. It is the Cissoid of Diodes, and was invented for the purpose of solving the famous problem of the " duplication of the cube." 8 a" (c) TTie curve a» + 4a» (1) The curve lies wholly in the first and second quadrants. (2) It is symmetrical with respect to the y-axis, only. (3) If a; = 0, y = 2 a, which is the maximum value for y. (4) As X increases indefinitely, the curve approaches the a;-axis. (5) The points (± a, 1.6 a), ( ± 2 a, a), (3 a, .6 o), etc., are on the curve. The curve is shown in figure 111, tangent to the line y — 2a = 0. It is called the Witch of Agnesi. 163] CURVE TRACING. HIGHER PLANE CURVES 269 Y Fia. 111. EXERCISES Trace approximately the loci of the following equations 1. 2a?-\-y^-4.xy + 12x-&y-irZ = Q. 2. a^ + 2y^~2xy-x-2y + 2=0. 3 ie' + y' + 2xy+x-y+l = 0. 6 ' xi^vi = a^. 4. y = x-3 S. «' + y^ X — o Show in detail the nature of the loci of the following equations : 7. a^y = a?; the cubical parabola. Fig. 112. 270 CURVE TRACING. HIGHER PLANE CURVES [Cu. XIV Y 8. ay'^a?: the semicubical parabola. FiQ. 113. 9. oV=aV — a^. FiQ. 114. 164. Asymptotes. As has been seen with the hyperbola, asymptotes, if any exist, are of decided help in tracing curves. Their equations may by found by a method similar to that of Art. 130, making use of the following algebraic theorem. If in an equation of the form ax" + 6x"-'+ cx"-=+ ••. +p = the coeflBcients of the two highest powers of x approach zero as a limit, then two of the roots become infinite.* * For, let y = - so that y is a root of the equation X a + by + cy^ + dy' + \-py° = 0. 163, 164] CURVE TRACING. HIGHER PLANE CURVES 271 The method may be illustrated by an example: Given the hyperbola !icy — 3y — x — 3 = 0i let y = mx + c be the equation of an asymptote, if one exists ; then these two curves have two points of intersection at infinity. Eliminating 2^, map + (c — 3 m — I)* — 3o— 3 = 0. Two roots of this equation are infinite ; hence m = and c — 3m — 1 = 0: i.e., c =» L Therefore, y = • a? + 1, i.e., y = 1 is the equation of an asymptote. There is one further possibility, viz., that there is an asymptote whose equation is not included in the form y = mx + c, because the line is perpendicular to the a>-axis. [Art 45]. In this case, x = 'k, where k is constant, and the corre- sponding ordinate y is infinite. Now if x=k, then y= "*" ; A* ^ o and y approaches infinity as k approaches 3. Hence x = 3 is the equation of an asymptote. The hyperbola can now be traced, as in Art. 163, Ex. 6. Kow if a and b approach zero, this equation approaches the form y\c + dy+ — + p3/»-») = 0, which has two roots equal to zero. But as y approaches zero, the cor- responding value of X becomes infinite. That is, as a and b approach zero, two of the ar-roots become infinite. 272 CURVE TRACING. HIGHER PLANE CURVES [Ch. XIV EXERCISES 1. Show that a; + y + a = 0, is an asymptote to the curve r' 4- y» _ 3 axy = 0, and trace the curve. This curve is the Folium of Descaites. Fia. iia. Fio. U6. 2. Trace the curve a^ = (y + 1)*(9 — J^. This curve is an example of the Conchoid of mcomedes. The conchoid may be used to solve another famous problem of the ancients, the "trisection of an angle," as well as to accom- plish the " duplication of the cube." 3. Trace the curve 4. Trace the curve a;* + y* = o* ; the hypocycloid. a^ + yJ = o*. 104, 165j CURVE TRACING. HIGHER PLANE CURVES 273 5. Trace the curve FiQ. 117. a) [b] 165. Exponential and logarithmic curves. An exponential curve has an equation of the form x=a'. Since this equation may be written y = log. X, the locus may also be called a logarithmic curve. Trace y = logjo x. Y ao,i) Fio. 118. 274 CURVE TRACING. HIGHER PLANE CURVES [Cb. XIV (1) The curve lies wholly in the first and fourth quadrants (since there are no logarithms of negative numbers). (2) The line a; = is an asymptote. (3) As X increases, y increases, indefinitely. (4) The points (1, 0), (10, 1), (100, 2), (0.1, -1) are on the curve. EXERCISES 1. Trace y = log, x. Use the fact that ^°^''' = i^^°^"" = oiDl^°^«'* = 3.32 logio X. 2. Trace y = log, X (where e = 2.71828). log.a! = logioe 3. Trace logio X = 2.302 logio X. y = e. 166. Trigonometric curves. Trigonometric curves may be plotted by points, using trigonometric tables ; or they may be found by using line functions, from a circle with radius 1. Fia. lao. 165-167] CURVE TRACING. HIGHER PLANE CURVES 275 E.g., trace y = sin x. Draw a circle tangent to the ^-axis on the negative side, with its center Q on the avaxis, and radius 1. Lay off from the origin on the avaxis the distances 2ir = 6.28, ir, -, etc., and at the respective points of division erect perpendiculars. In the circle lay o£E central angles of - radians, - radians, Z 4 ^ radians, etc., thus getting points D, B, A, etc., from which o let fall perpendiculars upon OX, viz., AM^, BM^, CM,, etc. These lines represent the sines of the corresponding angles. Therefore, for a; = ^, 3/ = M^A ; for x = '^, y=MiB , etc. : and 8 4 similarly, for x=^, y=-MiA; for x = —,y = -MiB, 8 4 and so on. The curve therefore consists of an infinite series of waves, as in the figure. The largest value of y is called the Ampli- tude of the wave. EXERCISES Trace the loci : 1. y = cos X. 2. y = tan x. 3. y = sec x. 4. y = cot X. 5. X = sin y. 6. y = tan~'x. 167. Loci of combined equations. If two curves are plotted on the same axes, and to the same scale, a new curve can be obtained by adding the ordinates of those points of the curves which have a given abscissa, to get the ordinate of the corre- sponding point of the new curve, with the same abscissa. For example : plotting (1) y = e'and (2) y = e-' 276 CURVE TRACING. HIGHER PLANE CURVES [Ch. XIV then adding corresponding ordinates, the curve y = C + e-' is obtained. This curve is called a catenary, and is the curve formed approximately by a cord, or chain, suspended freely between two points. Plot the curves : 1. y = sin X + cos x. 2. y = 2x+ sin x. EXERCISES 3. y = e' — sin 2 x. 4. y = a? + 2x — Z. 5. Show by the method of Art. 167 how to solve graphically the equation a^+(2a--3) = 0. Solve graphically the equations ; 6. 3ar'-6a; + 2 = 0. 7. Sin 2 05 = 2 cos x. 167, 168] CURVE TRACING. HIGHER PLANE CURVES 277 168. Polar coordinates. A plane curve in polar coordinates is traced by methods similar to those used with Cartesian coordinates. A few examples will illustrate the method. Fig. 12'2. (a) Trace r = a6. The angle 6 mnst, of course, be measured in radians, so that r and $ may be plotted in units of length. (1) r increases indefinitely as 6 increases indefinitely. (2) The points (0, 0), f 1.57 a, |\ (3.14 a, w), U.71a, ^\ (6.28 a, 2 it), etc., are on the locus [approximately]. (3) If (r„ ^i), and (r^, fli + 2ir), be points on the locus, then r2=a0i+2air=ri+2ira., i.e., any radius vector is divided by the curve into equal segments, each of length 2 na. (4) For negative values of 6 the locus is symmetrical with the part shown by the full line in the figure, with respect to the line e = |- This curve is the Spiral of Archimedes. (6) Trace r=a sin 2 6. 278 CURVE TRACING. HIGHER PLANE CURVES [Cii. XIV Fia. 123. (1) The pole is on the curve. (2) The maximum numerical value of r is obtained if sin 2 e = 1, i.e., if ^ = 7, 4' 4 etc. (3) r increases from to a as ^ increases from to ^: then r decreases to 0, as increases to ^- A loop is formed, sym- metrical with respect to the line =~ (4) The curve consists of four such loops ; and is symmetri- cal with respect to the four lines 6 = 0, tf = — , d = J, and 4 2 The locus is the Four-leaved Rose. EXERCISES Trace in detail the following loci : 1. r-=a sin B. 2. r(l — cos 6) = 2 a : a parabola. 3. r = 3cosfl + 2. 168] CURVE TRACING. HIGHER PLANE CURVES 279 / i ! \ i / '"'^1^^ I ■\/ / * 1 ,A -...um. -' \ \>^ J) R _^j>< \ Fig. 124. Fig. 125. 4. r = - : the reciprocal spiral. 5. t^=^aB : the parabolic spiraL V Fig. 126. 6. r2 = -: thelituus. e 7. r cos 2B = a. 8. r^ cos 2 d =a'' : an equilateral hyperbola with pole at the center. 9. r sin B tan S = 4 a : a parabola. 10. r = 6 — a cos B : the lima90ii of Pascal. Show the three forms, for 6 < a, b= a, and b > a, respectively. •J80 CURVE TRACING. HIGHER PLANE CURVES TCh. XIV 11. 12. 13. 14. Fig. 127. T = a? : the logarithmic spiral. r=-a sin d-\-h cos B : a circle. r = a sec' - • 2 2 a sec ^ r = 1 H- tan > 15. r=o(l— cos d) : the cardioid. Fig. 128. 16. r^ = a^sin2tf: the lemniscate. Fig. 129. JOS, lOy] CUKVE TRACING. HIGHER PLANE CURVES 281 17. r = asin'- FlG. 130. 169. Parametric equations. Some curves are conveniently- represented in Cartesian coordinates, but with a third number — the parameter — as the independent variable. Thus, the parametric equations of the ellipse are x=a cos <^, y = b sin , where is the eccentric angle [Art. 119]. Among the important parametric equations are those of the cycloid : 2ir X Fig. 131. x = a( — sin ^), i/ = a(l — cos ^). 282 CURVE TRACING. HIGHER PLANE CURVES [Ch. XIV The equations for the simple hypocycloid — the astroid — are x=a cos' , y = a sin' <^; and the equations for the epicycloid, of two arches, X = -— cos <^ — - cos 3 <^ ; y = -^ Sin <^ - - sin 3 0. Trace the curves : 1. x^=a sin <^, 7/ = a cos ^. 2. a; = 5 + 2 cos ^, 2/ = 2 sin ^. 3. a; = 6 cos ^, y = a sin ^. 4. a; = 5 4- 2 cos ^, y = 4 + 3 sill <^. Fia. 132. EXERCISES 5. a! = 2<^ — 2sin^, y = 2 — 2 cos <^.- 6. a; = 3 sin <^ — sin 3 ^, y = 3 cos ^ — cos 3 ^. 7. a; = 6 cos — 4 cos' 0, y = 4 sin' 0. 8. a; = < — a, 9. a; = 2* sin <^, y = 2*cos0. ANSWERS Pages 6-7. Art. 6. 1. (2), (3), (4), and (6) are identities. 3. Roots imaginary. 4. 1= - 3 + m 4(2Tm) ''' 4(2 + m) ^~ 31 ">= - 82 m - 39 ; roots are equal 6. (1) imaginary ; (2) imaginary ; (3) real and nnequal. 6. (1) m = 2or-V; (2) m= -^±^VZri5; (3) m = 3or-6; (4) m= ±a y/Tf. »• (1) <'=|; (2) c=^; (3) c=:±2ViO. 10. Ex. 2. x= -l±iVT3; Ex. 3. x=^±^V^^; Ex. 5. \x= ±V^^, 11. a; = ± 4 or ± 3 ; a; = — 2 or — 5. 1. Xl+Xi= - 1 + m (1 + m) I a; + Pages 9-10. Art. 9. ; xia^! = - 4 ; 3 { x + V26 + 32m + 16m'' 1 . 2(l + 7n) 2(1 +m) i' 3 , \/2S + 32m + 16TOM 2(1 +m) 2(1 + to) i U ANSWERS 2. (1) (x-4)(i-l); (2) (x + 4)(*_2); (5) (3w* + 2)(w*-32); (6) (11 +6y)(l -Sj/). a ., J. » _ "t + 3 . , ^ _ 4OT+3 . „ „ if ^ -41 , 2 , '■ '='+^ = -2i;rM''=''^ 2^rH'^^='^'^"'=-3r=^3T^^; one root becomes inBnite ilm= — } ; '^•»+'^^+4-^-j(;;rri)^-«^'»*-«''"-3«}' {'= + 4STl)-'i0^^-2^'»"-«^'»-3»}- 4. The roots are equal If m = — s — ~ ; the roots are real for all Teal values of m ; one root becomes infinitely great if tn = 2 ; one root becomes zero if m = ; the fa<;tors are (»-2){log.-^-?H±^-^-^V20™«-20,„ + 9}; 6. Real and equal ; imaginary. 7. (1) z=VT,y = H; (2) !/ = -!. * = ¥; (3) j._ 3± V9 + 12C I c y_ 3±V9H-12e . (4) a = - 3 ± Vl4, y = V- 6(3 ± V14) ; 8, 9. (1) 6 = ±a-\/§7; (2) h=-; (3) 6 = ±2V29. m Pages 14-15. Alt. 12. 1. 15"= i'*"' =0.2618*'* approximately ; 60°= -"* =1.0472'* approxi 12 o xnately ; etc. 2. (I)"' = 45= ; (^)''*=08"; (1)'" = 14° + 19' + 26.2"; etc. ANSWERS iii 8. (1) Bin « = ± -|r , cos 9 = ± -i= , cot » = 1, sec » = ± ^/Tn Vw VIo CSC 9 = ± ^ ; (2) sin a; = ± -L> «<»*=-—. tan z = T 1, cot a; = T 1, CSC 05 = ± v2 ; etc. 4. sin 30° = I, COS 30° = i V3, etc. : sin 45° = cos 45° = -;=. etc ■ sm 60° = cos 30° = J V3, cos 60° = }, etc. ; sin 90° = 1, cos 90°=0, etc. ; sin 135° = — , cos 136° = - -L, etc. ; 8in(- 45°) = L cosf - 46°) 1 V2 -v^ V2 = — , etc. V2 6. tan 3500° = tan 80° = cot 10° ; - cso 290° = esc 70° = sec 20°; Bin(- 369°) = — sin 9° ; — cos iii= - cos - ; cot (-1215°) = cot 45°. 6 5 6. (1) sin e = — cos 210° = cos 30° = sin 60°, hence one value ^V2, aTeais32J. 8. G,-?); (8, 21); (-7, -24). 4. (7,9). 6. |V62l8: 8. Slopes are : 1, '-^2, LiL?!. 1 — o 1 + a 11. V(A + 1)2 + (k- 1)2 = V(ft- 1)2+ (*-2)2, t-.>38S. IS. Distinct points if & < 1. Coincident points if 6 = I Imaginary points if 6 > 1. 14. c = ±6V^. 16. a? + y2-26 + A;(y2-4x) = 0. 16. (3i-2y + 12)(y»-4x)(fl« + jf»-a2)=0. 24. (3, 0) and (- S, 0). 25. (0, 1). 26. 4 1 + y = 11, x - y = 9, and 9x + y = 21. 27. y — x = l,x + y = 3,x-y = S,x + y + l=0. 28. Center is at (1, 0), radius is 2, eq. of circle is x^ — 2 x + y^ =3. 29. («) x + y = 0; (/3) 3a! + !/ = 17 ; (y) x-y = 0. 80. 2x-3y=12. 31. x-=4y. 82. 9xH 25y2— 54x-150y+81=0. 83. 163fi-t/'-90x + 6y + ni= ■^, tan-i -f^, tan-J f 17. 90°, 45°, 45°. 18. Parallel if 6 = ; perpendicular if &» = a« + 1, 19. Distance between lines is -=^. 20. — 2 — . 2I. Vl7 2V34 V'629 22. They make numerically equal hut opposite angles with the x-axis. 23. tan-i(-H); 7y = 9z-l, 7z + 9y = 73. 24. 45°; (8, -6); y=3z + 9, 2y = z+8 ANSWERS ix Pages 83-84. Art. 51. JL , _n_ -3a6 ^ -6 „ 24 VIT 2vl3 Va^ + ft" v^ V5 7. --^. 8 -^- ^ g J^ 10 yi - toj:, - & V58 ■ V29' ^' + 52 • V58' vT+^T-' 11. AltUudes are : -5L, JL, JL; the area is 36. VlO V34 Vl3 IS ' 'i - 2 yi - 11 _ , 3 zi - 4 yi - 5 V5 5 14. y — x = 12, and 7 2 + 7 y = 36. Page 85. Art. 52. 1. »-y + 6 _, 3z-10y + 10 V2 Vi09 2. Other bisector is 7 a; — 7 y + 24 = 0. \b + 3a)' ^ \a + 2b)' „ 38±15v^ . 16±25-k/2 a = — — , = — = • 142 142 Page 87. Art. 53. 1. 3 y + I + 6 =0, 2 !/ - 2 + 6 = ; (f , - V) ; 45°. 2. 2-3y+l=0, a + y + l=0;(-l, 0); tan-i 2. 5. X — y(seo a + tan a) = 0, a; — yfsec a— tan a) = ; (0, 0) ; a. 4. x + Sy + 6 = 0, x + Sy — l=0; lines are parallel. 6. 6x-5y + li = 0,6x + 5y = &6; G, 7). Pages 88-91. Examples on Chapter IV. 1. The third vertex is (2 ± 4 V3, ± 4^/3) ; equations of the two sides are : (1 ± V3)y + (1 =F V3)x - 2 =F 6^3 = 0, and (1 ± V3)x + (1 T y/S)y - 1 T bVE = 0. 2. The area is 72, and the fourth vertex is at any one of the follovring points: (6, - 10), (14, 2), (- 2, 14). 8. (6V^±2)y-(4V3 T3)z + 12V3T9 = 0. 6. a; = 1. 6. 6z-2y = 50. 7- y = (2 ± V3)x T V3. 8. y = 0ora: = 6. X ANSWERS 11. If the base be chosen as x-axis, and its middle point as origin, then the equation of the locus of the vertex is ox — 2 c' = 0. 12. 2l-5s( + 7 = 0,z-j/ = l; 1:1:1. 13. (7VT0-3V53)y+ (VSS - 2-\/i0)x = ll-\/53 + VlO ±4^530. 14. 4V63(3y-a: + ll)=3\/10(2a;-7y + l). 15. y=y/ax + 3-V3. 16. x + y = 0. 17. 3x + 2y=13. 18. Center is (7, 2), radius is 5. 19. If the base (= 2 a) coincides with the x-axis, ite middle point at the origin, and if & = the area of the triangle, then the locus of the vertex 18 ay — k = 0. 21. 4 X = 5(axes chosen as in Ex. 19). 22. (a) X = 0, y = 0, X - 2/ = ; (j3) 7 x + 2/ = 0, 2 x - y = ; (7) X + a = 0, y + 6 = 0. 23. m = 3. 26. 9y-4x = 24, 4x-y =8. 27. 20y-9x = 96. 28. x = 6. 29. x2 - y2 = 0. 30. x = 4, X = 6, y = 6, y = 8. 32. tan-'fi^^^ZE2\. P+Q Page 93. Art. 55. 1. a^ + y' - 12 X + 8 y + 27 = 0. 2. 4 x" + 4 yS 4- 24 y + 27 = 0. 5. x» + y* - 8 X + 8 y + 16 = 0. 4. x" + y^ = 26. 6. x« + y» + 8y+7 = 0. 7. / 3=" + J/' ± 2 Ax = 0; ( x» + y» ± 1 x^ + y" + 2A(x-y)+A2 = o. Pages 95-96. Art. 57. 1. r = fv^; a, 2). 2. r = iv'62; (|, J). 3. r = fv'5; (J, 0). 4. r = };(0,}). 6. r = ±y/WT^:(-±,-^]. 2 a \ 2 a 2 a/ 6. r = 2a ; (0, 0). 7. A point circle ; imaginary circle. 8. x2 + y2 - 22 X - 4 y + 25 = 0. 9. (a + 6)Cx» + y")-(a2+ ft") (a; + j^)=0. 10. x' + y2-Ux-4y — 5 = 0. 11. x2 + y2+7x + 6 = 0. 12. x= + y2-6x-y + 3 = 0. 13. x2 + y2-40(x + y) +400 = 0, x^ + y^- 8(x + y) + 16 = 0. 14. 9(x* + y2) -42 y + 47 = 0, center is [0, -V r=— ; or 8(x'' + y'') +8x-48y + 73 = 0, center is l-\,z\,r = —- ANSWERS XI Page 99. Art. 59. 1. x + 2y±2Vl = 0. 2. 3y = 3a;+2±2V.:. 5. y = 2x + 13±6y/l. 4. x±y=±ry/2. S. (--^, -^)- 6. (1,0) and (0, 1). 7. 12±20v'6. 8. jVio. 9. (0, o). 10. J =—5, 3/=v^a; + 10, y = — VS x + 10, area, 75 v'3, f or one of the fouT triangles. Pages 100-101. Art. 60. 1. y = ix±SVn. 2. 3y + x±2VW=zO. 3. 3y z=y/3x±6V3; y+y/Sx = ±6. 5. j/ = 5x±6>/26. 6. 2/ = x±V2. 7. y + J'=Tn(z+ G)±^G^+F-^-c ■ y/l + m^. Pages 104-105. Art. 62. 2. (a) 4 a; + 6 2/ = 41; (/S) 5x- 12 y = 152 ; (7) 3z + y + 19 = ; (S) y + 2x=0. 3. (a) 5x = 4y; (/3) 12a; + 5y + 7=0; (7)x-3j/=7; («) x-2y=0. 5. a: - 4 = 0. 6. x + 3 y = 20 ; 31 x - 27 y = 260. 7. 13x»+13y2-130x-78j,- 519 = 0. 8. -l+l=i. 10. (XI, yi)=(2, 6) or =(f|, - f4). Page 107. Art. 64. 1. 28f ; 26| ; 11}^ ; - 4fj. 2. Sf ; - 2i ;J ; + 4. 8. 6|; -5i; 5; +3. 4. 2x!' + 2y2 = 51; (-3, ±y/^^). Page 109. Art. 67. 1. x-4i/-3=0. 2. (- 1, -1), (7, + 1); V68 ; ^ V65 . 146. 3. x-y = 0; J ^2(0 + 6)2-16 0. 4. (-2,-1). 7. 90°. 8. tan-i Vl9. xii ANSWERS Pages 110-114. Ezamples on Chapter V * iJ + y2_14x — 4y-5 = 0; (7, 2); VBI. 2. (a, b); this family consists of all circles of radius Va' + 6', and having their centers on the circle whose equation is x'' + y^ = a^ + b\ 3. Gi = Gi, and Fi = F2; Gi' + Fi^ - Ci = Gj^ + p^i _ d. i. m = E^ = iF. 6. 2a2 + 2j/2+ 13a; = 0. 6. (2 ± ZyTH,, - S) . 7. The circles are : x^ + y'^—\ix — iy = - ^f^, x^ + y^ + 2x + 6y = •VWt'. *" + !/' - 6 a; - 6 2/ = B!l- The radical axes aie: &x- 2y = 35, 4x + 3y = 9,8 x+ Uy = 1; rad- ical center {i^', — Y)- 8. x^ + y^-2x + y = 15, x^ + y^ - 6x + 2y = 15, x' + y^ -lOx — 5y+21 =0; ix-y = 0; ix + 1y = i2, i'^ + 3y = 21; Qh ^). ». yi(x' + y^) = y{xi' + yi^)- lo. z + y + 2 = o. U. 27x2 + 27y2-384j;-134y + 264=0. 12. 5z» + 5y« = 9. 15. az — 6y = ft'', by — ax = o', az + 6y = 0. 14. (z-16)2+(y-l)2 = 74, (z -6)' +(y - 15)> = 74. 16. X* + y« + 4 az - 4 ay + 4 a= = 0. 16. z" + y^ = ZiZ + yiy. 20. (i, f). 21. Gi : J!*! : : Gj : ^2. 22. 2x2 + 2y2-|-40z- 85y - 3 = 0. 28. y = 2, z = 1, 4 1 - 3 y = 10, 3 X + 4 y = 5. 24. The radical axis is 2 x 4- y = 2, the ratio is 3 : 7. 25. (';^i 2), ^ ; this point is the radical center. 26. (.^ _ i)(z2 + y2) _ 2(G + ar^)x -2(_F+ bi^)y -C+ r^(jifi + 6«) = 0, where r is the given ratio and (a, ft) the fixed point. 27. V4 f^ _ 2(o - ft)" ; r = 5^. V2 28. |-— , — ); I — r. — r); 3) 0; and for three other points in each case. ^"^ ■*^' ^V2 ^2/ 29. Equations of sides of inscribed triangle are : 2z + 6r=0, z± V3 y = 5 ; equations of sides of circumscribed triangle are : z = 5, z = ± VS y — 10 ; and the length of the sides are, respectively, SVS and 10\/3. SO. x« + y2 + 2z + 4y = 13,orx2 + y'-2x-4y + 5 = 0. 81. z» + y« = 2(c»-6o2). 32. x«+y» = aa;. "■ ('-|)'-(»-?)'-f- 84. 5z» + 6y«-50x-46y + 22 = 0. S6. Yes. 86. x« + y'+ 10z + lOy =35, orz!! + y2-20z-8y + 31=0. ANSWERS xiii 37. 8,_3=^?^^(a:-2). 88. Sx + 2y = 13, 2x-3y = ; x= ±Viiy+6,y = 0. 89. 46°. 40. Equation of diameter is 5 1 + 4 y = 1, equation of chords is 5 y = i X + k. 41. a;2 + y2 = n^ + rjS. Page 117. Alt. 69. 3. (6,0°), (6, 60°). 4. (-2, ^), (-2, ^^"^ , (2, :=1^]; (3,265°), (3, -105°), (-3, -285°); (5, 360°), (5, -360°), (5, -0°); (0, 240°), (0, - 120°), (0, - 300°). 5. On circumference of a circle of radius 7 and center at the pole ; same circle ; on the line through the pole making an angle of 26° with the initial line ; on the initial line ; on the initial line. 6. p = 7;p=-7;9 = 25°;e = O''' ; 9 = - 180°. 7. Parallel to the initial line. Pages 119-120. Art. 70. 1. -JlS-iacos^, VlO, Js-icoB^^- 4. ^■ \ 12 ' ' V 12 4 6. 14- yt\/3; V'l9+ V89+ V73-24v'3. 6.(0,0). 7. (± a, 0). 8. p = tane. 9. (6, cos-'|). 10. p and S of the points of in- tersection satisfy the equations : />= ^2 = 9(1 + Vp^ — I ) and = sin"' ( - ] ■ 14. p-2cose + k(_pcoa0-\) = O. Pages 122-123. Examples on Chapter VI. 2. psin 9 = i; 3; /)C0s9 = i:3. 8. p cos 9 = 5. 4. psin(0-9)=pisin(0-ex). 6. (pi, fl,) = ^10, ^V 9. p2— 8pcos(9--W9; (2±vl3, 0). 10. p2 — 20 p sin 9 + 96 = ; sin* 9 = H. 11. p._16psin9 + 48 = 0;8. la. („) (_JL_, ^); (p) g v^, |). Page 126. Art. 73. 1. 2x~3y = 0. 2. x'' + y'^ = 30. 3. y^-4x^ = i. 4. y = mx. xiv ANSVVKKS Page 127. Art. 74. 4. 9x2 + 26 3/2 = 225. 5. x^-iy^ = a^. Pages 129-130. Art. 76. 1. 2 p = a. 2. p sin 9 — cot tf + 4 a = 0. S. 25 cos 2 tf = 1. 4. «itan9 = n. 6. p(cos « + -s/3sin 9)= 2. 6. p2coa2e = 49. 7. ^2 ^ y2 = qS. 8. z2 _ y2 = 4. 9. IJ/ = 10. 10. (z2 + y2)2 = 2 a^iy. U. a;2 + y2 = i0x. 12. l]y = 2a. 13. y2 + 4^=4*2. Pages 130-133. Examples on Chapter VII. 1. xy = 4. 2. VS r + 6 !/ = 0. 8. x^-xy = 0. 4. v^(y2_j.y_2) + x + 3y = 0. 6.(-3,-l). 7. x + y = 0, 3 a:— 2y = 0. 8. y = 0. 9. tan-i(i)- "• tan e =- - ; tan 9 = ^. 13. lane = 5. 14. Any point on the line .4a: + .By + C=0. 15.(2,1). 16. The new origin may be any point on the locus of the given equation. 17. 2 x2 + 3 j,2 = 6. 18. x'^+6xy + 9^ + 40-1/10 y = 0. 19. (a) ca;2 + y2)2 = a2(j^ _ yi) . (^3) a:^ _ y2 = a" ; (y) (a:2 + y2)8 = 4 fc2a:V- 20. p2 = ft2 CCS 2 9. 21. p2cos2ff = 49. 22. p = 4a. 23. (3+ 4 V3) y2 + 24oy — 8 -v/Saiy = 0. 25. p cos (9 — a) =p. Pages 141-142. Art. 83. 1. a;2 + y2 _ 2 zy + 16x + 16 y + 32 = 0. 2. a;2 + 4x!( + 4y2 + 12z— 6y-9 = 0. 3. y^ = 16x. 4. X»=— 4oy. 6. x2 + 4a;+i6y = 76. 6. y2_8y + 12x = 8. 1. x^ = ip(i/ -p). 8. 16 ; 4 a ; 16 ; 12. Page 143. Art. 84. 1. (8,2); (2,2);4;x=4; j, = 2. 2. (-1,4); i-l.m:i: 12!f = 47; x + 1 = 0. 3. ni,^-]; (^^]; i; 48x = 29; 6y = 5. V48 6;' V 10 ' 6J ' 3' ANSWERS XV 4. (- 6, a) ; (- 6 + A. «) ; 1 ; a; = T (6 + I'j) ; y = a- G'-F-'-AC\ 2F A' 5 /_§ G'-AC \ . (_G G^- F-^-AC \ \ A' 2AF }' [ A" 2AF ) ffl + -F' - AC „^_ff 2AF ' A \ 2BG ' -bJ' V 2iSe ' Bj' 2G B' 2BG ' ^ B Fagea 150-151. Ait. 88. 1. 7a;2 — 2x!/ + 7y2-42i-427/ + 135 = 0. 2. 3a;2 + 4y2+ 18a:=81. 8. 16a;2 + 15y2- 16y = 48. 4. ^ + l!! = l; _§-. 5. 3k2 + 5j/2=32. 6. f-+^=\. 3 7 -^"^ 81 (2 7. *! + £ = !. 8. i2t + i£ = x. 9. 16x2 + 25j/>' = 625. 45 81 81 45 10. | + f = l. n. I?-^ + ii^ = l; (3±V7,-2). 18. %|%i^' = l; (0,7±7Vif). foci are: ^ g - fe ± v(. +^6)' - (« + ^^^ =-:^) 15. A circle. Pages 152-153. Art. 89. 1. (2,1); n = 3, h= V'A; (2±V0, 1); (5, 1), (- 1, 1); 2. 2. (1, - 1) ; 6 = 1, a = 2 ; (1, -1 ± V3) ; (1, 1), (1, - 3) ; 1. 14. XVI ANSWERS 3. (-2, -2); 6 = -I=,a = 7; (- 2, - 2 ± ,55^210) ; (-2, -2±7); t|. ^15 V ^ -B J ^A ^B' \ A ^ AB ' B )' (^±J?;_Zy g^^M, where ir^-»g^ + ^^°--^^g. V ^ X^' Bj" B AB 93 14 ^|49 Pages 157-158. Art. 93. 1. a:2-3!/2_2x + 30s/-62 = 0. 2. ^_l?=l. 3. -2^ ^=1. 4. 2a:2-3y2 = 30. 9 25 ^^§^ 64 a'i 3a2 25 36 7. (y±5)f _(2^= ^ j_ g (4± ^gj^ 5)^ ^ . (_4^ _ 5± ^i3j_ ^_ 25 3(1 '94 10. («) (±3,0); (« (iVliJ, 0); (7) (0, ±5). Page 159. Art. 94. 1. ^^, -2); a = fV6, 6 = iVS; (^, -2±|V2); (^, -2±|Vo); |V6. 2_(.5,|);a = ^,6 = ^, (5±AVi70;|); (5±i^/5^,|); iV51. 8. (-1,1); arrVS, 6=3; (-l±2v^, 1); (- 1 ± V3, 1); 6V3. ANSWERS xvii Pages 163-164. Art. 97. 5. i=^orW ; the lines corresponding to A; = ^ ^.re Sx+iy+5=0, 2x + Sy +4 = 0, and the angle between them is tan-' (^), their point of intersection is (1, — 2). 6. A =—10, or —^; for A; = —10, the lines are 3 x — j; + 2 =0, 4a;- 2 y + 1 = 0, the angle between them is tan-i (^), their point of intersec- tion is (- ^, - 1). 7. * = l;s^ + 5 = 0, y-!«; = 0;(-5, -5); 45°. 8. B'-AOO; B^-AC<0; :&=AC; A + O = 0. Page 168. Art. 98. 1. Center = (0, v^) ; foci = (± VC, •\/2± VS) ; axes: y—x= ^2, y + x = -\/2; directrices: x + y = _ ■ V2 2. Parabola ; 6 = 30° ; vertex, referred to old axes, is at point I — , - ); and the latus rectum is 4 a/3. V 2 ' 2/' 3. Hyperbola ; 6 = sin-' (§) ; center, referred to old axes, is at the point (lij^, I) ; and a'' = ffj, 6'' = iff. A 4. Parabola ; 8 = sin-i ; vertex, referred to old axes, is at point 297 579\ VS , — I, and the latus rectum is 50 200y 50 5. Hyperbola; 8 = 22^° ; center, referred to old axes, is at the point (-3, - 1) ; and o" = 6« = 11 VI Pages 173-174. Art. 103. 1. Hyperbola ; 8 = 45° ; coordinates of center (old axes) are : 5. Parabola ; = 4S° ; the new equation Is : ('-iei)'=-x("-^> S. Ellipse; 8 = 0; center (old axes)=(i, —-Y, a'' = Jf , 6* = H i foci on new ^-axis. ^ XVIU ANSWERS 4. Two straight lines : 3x — y + 6 = 0, x + Sy — 2 = 0. 6. Two parallel lines : Sx — y + 1 =0, Sx — y + 2=0. 6. A "point ellipse," or two imaginary straight lines through the origin. 7. Circle ; center (old axes) = (— J^, f) ; r" = {J. 8. Hyperbola ; center (old axes) = (— 3, — 1) ; 8 = 22J° ; a* = b^= 11x^2. 9. Two straight lines : 2x + y = 0, x — y = S. 10. The straight line 3 x + 4 y = 10, taken t\vice. 16. The hyperbola 107 x^ — 164 xy + 63 y^ - 139 x + 139 y — 160 = 0. 16. The parabola 9x^- iSxy + 6iy^+9ix + 12y — 736 = 0. 17. Pair of straight lines x^ +8 xy + T i/^ — 66 y — 8 x = 0. Pages 177-178. Art. 106. 1. Yes. 2. 2y = 8a:+ 13, 2y = 8a;— 41. xix yiy y,x Xiy / a'-6' \ x,x yiy_. yix , xiy f efl-i-b^ \ 6. Xix = 2p(y + yi + 10) ; 2px + Xiy = Xi(2p +'»i). 6. 2(2xi-3)x-5!/iy-6a;i = 0; 6yiz + 2(2a;i - 3)!/=Sffi('3i(;i-2). 7. (2 xi - S)x + 6(yi +2)y = 3(a;i -43/1-6); 6(yi + 2)a!- (2 xi - 3)y = 12 Ki + 4 xij/i + 3yi. 8. 4(ii + l)x + (yi-l)y = yi-4xi-l. 9. y = l;x + 2 = 0. 10. Point isf — 5 + sv^ — - + ^^ ) and the equations are SOy + 18 V53; + 6V5 - 9\/l0 + 15 =0 and 54 y - 18-\/5«-6\/i0-6 V6+27=0. 11. i + y + 4 = 0; y=a;+2. 12. y/ix-i- y = 6; x— s/iy =rO. 13. 21 — 3y = 8;3x + 2y + l=0. 14. y+(2± V2)a; + 9±4-v/2 = 0; x -(2 ± v^)y = -2 ± V^. Pages 181-182. Examples on Chapter viiX. 1. y = ±|Vl0x-2. 2. 4x2 — 3 y2 f 16 x = 0, an hyperbola. 4. (0, VWf), (V3+ vl9, -3) and (_ Vs-'x/W, - 8) ; or (0,- V57) (V3 + VTli. 3) and (- V3 - Vl9, 3). 6. 18y + 6x = 5; 18y + 6x = 17. 7. Hyperbola; (- i ± J V^,0) ; x = -J ± } v^ ; (-J, 0) ; }, J ; 7. ANSWERS XIX 8. This equation may be written : ^^ ~ ' — {y±_2_— i^ which shows I 4 that it is an hyperbola ; that the center is at (1, — 1) ; that the trans- verse axis is iVd and is parallel to the x-axis, etc. 9. This equation may be written : (x — 2 a)^ = ip ly+b + — — ] , — a parabola. ' 10. This equation may be written : i^-^t — L ^ 12-+ — L = i, — an ellipse. 11. This equation may be written : — — + '^ ~ ' = 1, — an hyperbola. 9 4 12. This equation may be written: „)^ ~ °\ , + Y t "\ = li —an „. 2(6 a= + 1) 6 o2 + 1 ellipse. \ ' y 16. p = 1 — 3 cos 9 Pages 186-188. Art. 113. 3. y = i v^a; + V3p, and y = — i VS x - VSp ; 1 : 2. 4. 12y = 9x+16p; (^-f^,^)- 6. »=(i ± J v/5)«+l =Fi VI?. 6. X ± y = 6; the directrix and the x-axis. 7. x-|.2y-|- 10 = 0; 2j;-y = 30. 8. 8a;-4y — 9 = 0. 9. 4z-2y-|-p = 0; y/Sy = x + Sp. 10. x + y+p = 0. 11. y^=p(x-p). 18. 6, 6v^. 14. y2=--jr*- Pages 189-191. Art. 113. 8. The value of y -|- 3 = mx, where m are the roots of the equation 5 m' + 10 TO -t- 6 = 0. Pages 191-192. Ait. 115. 1. 3 y = 32. 2. 3 j; - 4 a: = fc. 3. 3 y -1- 4 = 0. 4. i -H 2 = 0. Pages 192-194. Review Exercises. 1. y2 = i. 2. !/2-(iy =8x-2o. 3. (2/ + 1)* = 8(x -f- 2). 4. x"- 2 a; -I-!/ -(-2 = 0. 5. (1,-1); 1=1. 6. y' = ip{x-p). 7. f = ip{x+p). 8. y'-^^^x. 9. 3/2=2x, y^=- 18(j:-6). 10. y = x; x-f- y -(-p = 0. 11. 90=; tan-' (|). 12. ip'^. XX ANSWERS IS. y'^=px. 14. y^ = 2px. 17. kx=p. 18. y = lcp. 19. y^ = ipx+jl^k^. 20. A parabola whose focus is the given point and whose directrix is the given line. 22. *y2 = p(l + ky X, if k is the given constant ratio. 23. y + 2 K + 5 = 0. 24. A parabola whose focus is the center of the given circle, and whose directrix is a line parallel to the given line and at a distance from it equal to the radius of the given circle. 26. y^ = ip{x-ip). \ xi yi I xi Pages 197-198. Art. 118. 4. 3x-4y=±4Vl3; 3a!;-4y=±-I?-- 6. j/=4 or2 j+3a;=17. V73 72 7. Through the points for which i 8. V7x + 4y = 16; 4a:-v/7j^ = }V7; Vv^; -ft-v^- 9. The points for which z = ±t1i-\/105. 10. — + ^=1. ^* 25 9 11. 3a: + 8y =24; 3a; = 2!/. 12. y=-ia;±3, y=-2x±V89. IS. — yt a*(l — e2 cos* fl) 14. The points for which z = — ^ ° ; the same. Pages 206-207. Art. 124. 1. 27y + 20i=0. 2. by = Sx. 4. (a) a^y + 6»a; = ; (/3) ay + 6a; = 0. 6. a' = 6'=V^. 7. tan-i(-2); 2-v/5j^ = x. 8. (fV2, V2); 3y + 2a; = 0. 9. V26. 10. (± 3, ± -v/S) ; tan-i [ ± -4 ) = 30°, 150°. Pages 207-209. Review Ezereiaes. 1. (±1, 0); a:=±4; }. 2. irm^a^. 4. (a:2 + y2)2 _ 4 j;2 + yn. ANSWERS Xxi e. 864 »;» + 400 y2- 10921 = 8281. 7. 4. 8. 9;f;|7rvf. 9. If the generating point divides the line in the ratio a : b, the equa- tion of the locus is — + ^ = 1. .= ("-IT 10. i5 + l__iL = l, IS. (z'' + !/»)« = a2x» + 6!'y». Or D^ 4 i 15. If the base coincide with the z-azis, its middle point at the origin, the equation is 62a;2 + cV = a-ft" 17 "V^+— ■ 19. Cy* — x2 + a^! - 62) tan 2 a + 2 xy = 0. 20. 2 ay = c(a» - o"). Pages 214-215. Art. 129. 8. ^-^=1;8,6;(±5,0);|. Id y 4. 9z-8j+ll = 0;8a; + 9j/ + 42 = 0;-i^;-J^. 6. (±-^=,±--^=r)- 6. |;Ji^. 7. ±5. 8. None. 9. j/ = ± f Vilx + 5. 11. 32»-y3 = 3a». 14. iVE; a. 16. y=±(-^-lV 2V3(3-v'5)j there are three other solutions. "^^ 16. iy/6; SVSj |; 6. Pages 220-221. Art. 132. 1. 4j> = ±3z; 2tan-i| = tan-i^*. Z. S3fi-y^ = 8a'; y = ±2->/2x. 6. x^-y^+10x + 4y = n. 6. x^- i/' + 10x + iy + 5S = 0. 7. a-23^ = 5, i + 2y + 3=0; a;2_4j,2_2x-16y-ll=0. 8. 7y + x = 0, j/4-7z = 0. 9. 4x2-9y3+8i - 18y + 31 =0. Pages 223-224. Art. 134. 1. 5j/ + i = 0; j/ + 4x = 0. 2. y — x; by = ix. S. 16y = 45x. 4. 3 J/ — 8 X = 52. xxii ANSWERS Pages 224-226. Review Examples. 1. ^_l!? = i. 2. 16y = 16a;. 4. ^ 2i_ = i if the middle point of the base is the origin and the base of the triangle on the x-azis. 5. 2 xy =: /b(c' — x'), if k is the constant and axes chosen as in Ex. 4. 6. (x-5)(y-4) =0; xy-6y-4x + 40 = 0. 11 /'-r 3av^ , +4 y + 9 = 0. Pages 232-233. Examples on Chapter Z. Q 2 1 vi4 vi4 -yn ^ 5. a = ^ = 7 = co8-i-L. 6. 90°. 8. y/TlQ. V3 1 2 o 9. pi= V14, cosai= — ;, cos /Si = ;zr.C08'yi= vH Vii vli Pj=\/1, costt: =0, cos ^2 = -, cos 7a = . \/2 -v/^ p,= Vo" + 6» + ci*, cos as = ■ cosft =- Va* + 62 + c-^ Va" + 6> + c?* c -? ,C08ft=- * cos 7t = - Vo^ + fc^ + c* 11. Internally: (J, |,i); externally: ^^, -1, :^\ Pages 242-243. Examples on Chapter XI. 1. Two coincident planes parallel to the jfz-plane and at the distance + i from it. 2. The yz-p\a.ne, and a plane parallel to it, and at a distance — 4 from it. 3. Two planes parallel to the z-azis, and intersecting the xy-plane in the lines x — y = and x — y + 3 = 0. 4. Two planes intersecting in the z-axis, and intersecting the xy-plane .. 1- b +"v^P— 4ac m the lines y = !^^ — ^-^ x. 2c ANSWERS xxiii 6. Three planes parallel to the j/z-plane, and at distances — 1, +2, — 2 from it. 6. A parabolic cylinder with generators parallel to the z-axis. 7. A circle parallel to the y«-plane, with center on the x-axis and radius 3. 8. A pair of lines respectively parallel to 2 x = ± 3 ^. 9. The projection of this curve upon the xz-plane is the hyperbola 3 a;2 — z2 + 5 = 0, and its projection on the yz-plane is the ellipse Sy^ + 4 2= = 32. 10. Por 3=5, the point (0, 0, 5) ; for a = — 5 it is a circle parallel to the xj-plane, and whose equation is 9 x^ + 9 ^a _ iqO. 11. Solved like No. 9. 12. The point (0, 0) ; x = ± 2 a ; y = ± 2 2. 15. x« - 3 y2 = 6 ; x2 + 2 32 = 6 ; 3 y2 _ 2 22 + 6 = 0. 14. x" + 2,2 = 5 2 + 3. 15. x2 4- 2^ = (j/ - 10)2. 16. ?1 + J^ + 5! = 1. 17. (^-l)%jg+(^-l)'=i. 3 2 7 7 9 18. (ys + 22)x + 4 = 0. 19. 16 j/2 -9x2- 922 = 144. SO. 36x2 + 36i/2 + 3622-36x-24y + 5 = 0. Pages 2S1-2S2. Ezamples on Chapter ZH 1. x-2y = 3, x-2 = 2. 8. 7x— 5y-52 = 0. _ X— a ^ y—b _ s — c I m n 4. x—2y = S (on xy-plane), x — 2 = 2 (on Z2-plane), z — 2y = l (on !/2-plane) ; it pierces the xy-plane at ^2,.^, Oj, the yz-plane a* ^0, — , -2^, and the x2-plane at (3, 0, 1). 13 3 rr A 2x 5y 10 2 7 r + "7?^ + ■v/m Vii3 ■v/m vns 7. ■£_+JL+i. = i; _Jif^_ + _^!L_ + ^ _ 10 10 10 ^02 + ft2 + c2 Vo2 + 62 + c2 Va^ + 62 + c2 c a 6 10 va2 + 62 + c2 xxiv ANSWERS 8. ^L=L§=2=l=i ; -i_, -|_, ^. 9. xcc»tt+ycosp=p; e-p. 13-7 V69 -v/M yM 11. 3x — iy + 10z=0;Sx — iy-i-10z = 28. 12. llx-7y-2z=22. 18. _ J VSO ; - fi VSO ; yes. 14. cos-' ( ° ) for the j/z-plane, cos -' ( — — [ for the xz-plane, and cos"' ( '^ ) for the ay-plane; sin-i { ° — V sin-'( ^ V and sin-'f " ^ re- \Va2-l-62 + c2/ VVa^ + fts + cS/ Wa^ + 62 -t- c^/ spectively for the x-, y-, and z-axes. 16. 13k + 11^ + 2 = 25. 16. x + y + z = dy/Z. ._ g-l-4 ^ y-t-6 _ z + 6 ■ 4 -5 1 ' 18. 2a; — 4a = 3,6iBH-4jf+l = 0, 2y-l-6z+5 = 0. Page 264. Examples on Chapter XIII 5. (xi — a) (x — o) + (^1 — h'){y — b) + (zi — c){z — c)=r'^ is tbo tangent plane at (ii, ji, zj). '• ,^ " ^ + ^ = •>• 25 4 9 S. (._^^)%(,_l^)%(z-^y 4 64^4 64 TANNER'S ALGEBRAS By J. H. TANNER, Ph.D., Professor of Mathematics, Cornell University. High School Algebra, Ji.oo Elementary Algebra . ;Ji.oo THE High School Algebra is adapted to conditions in secondary schools throughout the country. It meets the entrance requirements in elementary algebra of all the colleges, and embodies only those features and methods which agree with the views of the best teachers. It arouses the pupil's interest by showing him early some advantages of algebra over arithmetic, making the transition from one to the other both natural and easy. ^ The topics are presented in the order of their simplicity, and definitions are given only where they are needed, while clearness of comprehension is insured by an abundance of concrete illustrations and inductive questions. The volume contains a large, well-chosen, and carefully graded set of ex- ercises which will develop the pupil's perfect understanding of the principles. ^ In the Elementary Algebra the author has aimed to arouse and sustain the student's interest in the study and to teach him to think clearly and to reason correctly. The exercises are numerous, well-chosen, and carefully graded. Each topic is brought in where it is needed, and is not closed with- out a rigorous demonstration of all the principles involved. Frequent recapitulations and reviews emphasize the main re- sults acquired and retain them in the pupil's memory. Scat- tered throughout the book are many practical questions designed to make the student think and to hghten the duties of the teacher. In addition, attention is directed to the informal and explanatory introductions to topics, and to the general problems and interpretation of their solutions. The work meets the most exacting college entrance requirements. AMERICAN BOOK COMPANY ELEMENTS OF DESCRIPTIVE GEOMETRY By ALBERT E. CHURCH, LL.D., late Professor of Mathematics, United States Military Academy, and GEORGE M. BARTLETT, M.A., Instructor in Descriptive Geometry and Mechanism, University of Michigan. $2.2S Part I. Orthographic Projections. 1 1.75 THIS is a modern treatment of descriptive geometry with applications to spherical projections, shades and shadovirs, perspective, and isometric projections, for the use of technical schools and colleges. Though based upon Professor Church's Descriptive Geometry, and retaining as much as possible the original lucidity and conciseress, this work differs from it quite widely. Among the salient features of the book are the following: The figures and text are included in the same volume, each figure being placed beside the corresponding text; General cases are preferred to special ones; A sufficient number of problems are solved in the third angle to familiarize the stu- dent with its use; A treatment of the profile plane of projec- tion is introduced ; Many exercises for practice have been introduced; Several new problems have been added; The old figures have been redrawn, and many of them have been im- proved ; Several of the more diiBcult elementary problems have been illustrated by pictorial views; In the treatment of curved surfaces, all problems relating to single-curved surfaces are taken up first, then those relating to warped surfaces, and finally those relating to surfaces of revolution. Experience proves this order to be a logical one, as the procedure is from the simple to the more complex. Also the student is more quickly prepared for work on intersections and developments. AMERICAN BOOK COMPANY AN ELEMENTARY TEXT- BOOK OF THEORETICAL MECHANICS By GEORGE A. MERRILL, B.S., Principal of the California School of Mechanical Arts, and Director of the Wilmerding School of Industrial Arts, San Francisco ll.OO MERRILL'S MECHANICS is intended for the upper classes in secondary schools, and for the two lower classes in college. Only a knowledge of elementary algebra, plane geometry, and plane trigonometry is required for a thorough comprehension of the work. ^ By presenting only the most important principles and methods, the book overcomes many of the difficulties now encountered by students in collegiate courses who take up the study of analytic mechanics, without previously having covered it in a more elementary form. It treats the subject without the use of the calculus, and consequently does not bewilder the beginner with much algebraic matter, which obscures the chief principles. fl The book is written from the standpoint of the student in the manner that experience has proved to be the one most easily grasped. Therefore, beyond a constant endeavor to abide by the fundamental precepts of teaching, no one method of presentation has been used to the exclusion of others. The few necessary experiments are suggested and outlined, but a more complete laboratory course can easily be supplied by the instructor. ^ The explanation of each topic is followed by a few well- chosen examples to fix and apply the principles involved. A number of pages are devoted to the static treatment of force, with emphasis on the idea of action and reaction. Four- place tables of the natural trigonometric functions are included. AMERICAN BOOK COMPANY (7S! PLANE SURVEYING ^3.00 By WILLIAM G. RAYMOND. C. E., member Ameri- can Society of Civil Engineers, Professor of Geodesy, Road Engineering, and Topographical Drawing in Rensselaer Polytechnic Institute. IN this manual for the study and practice of surveying the subject is presented in a clear and thorough manner; the general method is given first and afterward the details. Special points of difficulty have been dwelt on wherever necessary. The book can be mastered by any student who has completed trigonometry, two formulas only being given, the derivation of which requires a further knowledge. The use of these is, however, explained with sufficient fullness. ^ In addition to the matter usual to a full treatment of land, topographical, hydrographical, and mine surveying, par- ticular attention is given to system in office-work, labor-saving devices, the planimeter, slide-rule, diagrams, etc., coordinate methods, and the practical difficulties encountered by the young surveyor. An appendix gives a large number of original problems and illustrative examples. ^ The first part describes the principal instruments and deals with the elementary operations of surveying, such as measure- ment of lines, leveling, determination of direction and measure- ment of angles, stadia measurements, methods of computing land surveys, etc. ^ In the second part are treated general surveying methods, including land surveys, methods adapted to farm siRveys, United States public land surveys, and city surveys, curves, topographical surveying, ordinary earthwork computations, hydrographic and mine surveying, etc. ^ Both four-place and five-place tables are provided. They are unusually numerous and practical, and are set in large, clear type. The illustrations are particularly helpfiil. AMERICAN BOOK COMPANY (76 1