■■'VHW\^'->^yj ':4fli w f sili J] .i;t!f![ II ill ill ill ill 11 ;lj:,i;;.; iiliii !l|i!l|ii!"i!l!i! Ill illilili'lil Mi it ill Ull III I I I ! II ! I I ii iiililbiJ! • .iiiiliin PI iii-'i!H|i!ii!l!i!!lii!l i "■I !lil ii ii iJ |l|! 1.1 !l!i|!.:. Mm I !r^ ihiiiiil ' ■■ ii ■;. ' • "\v- Willi-||! ill! iHi i m%gi fmB. lilildn;-!,!!;,!:;; ij|i[li!||iii;'''''i':|!i r^' Gnin? OF the estate of Professor William F. Mever lr^/?lL^H~: Digitized by the Internet Archive in 2007 with funding from Microsoft Corporation http://www.archive.org/details/elementsofplanesOOcandrich THE ELEMENTS OF PLANE AND SOLID ANALYTIC GEOMETRY BY ALBERT L. CANDY, Ph.D. ASSISTANT PROFESSOR OF MATHEMATICS IN THE UNIVBRSITY OF NEBRASKA BOSTON, U.S.A. D. C. HEATH & CO., PUBLISHERS 1904 /:Zj^ -^ C % ^~V IV PREFACE secant method of finding the equation of the tangent, the reasoning is essen- tially the same as in the method here used, but the student seldom or never comprehends its significance. And, furthermore, he never uses the method save in the case of the conic sections, whereas the derivative method is one that he can always use. The subjects discussed in Chapter VI need not be taken at the time or in the order in which they occur in the book. Or, if the teacher prefers to pursue the old established method of teaching each branch of mathematics exclusively, he . may, at his discretion, omit this entire chapter without interfering in any way with the continuity of his course. While this book has been in preparation, my own plan has been (with students who have not previously had the Theory of Equations) to give in substance the theorems contained in §§ 63-71 immedi- ately after the work on curve tracing, or symmetry. The remainder can be given any time after Chapter V has been read. In finding the equations of loci, special emphasis is given to the meaning of the parameters which appear in the final equations, and the significance of a variation in their value, and a full discussion and a thorough geometric inter- pretation of the result are rigidly insisted on from the beginning. The teacher should never lose sight of this vital principle. Polar coordinates and their relations to rectangular coordinates have been introduced at the very beginning. The conic section is first briefly studied geometrically. Its defining property is proved in this way, from which its general equation is shown to be of the second degree. The two central conies are treated simultaneously by using the double sign in the standard equation. In this way much time is saved, and the similarities of the properties of the two conies are presented in a striking manner. As the book is intended for beginners, numerous illustrative examples are given in the first part on Plane Geometry, and also a large number of exercises. The numerical examples have all been prepared especially for this book. An- swers are given to only a few of these, as it is far better to check results in such exercises by constructing an accurate figure. A unique feature in the way of exercises is found in the list of Miscellaneous Problems on Loci that occur in the phenomena of everyday life. These cover a wide range of subjects and should be of interest to students in any department. The study of mathematics should not only develop the power of investigation, but should also cultivate the habit of carefully examining interesting phenomena. I hope these problems will help toward the accomplishment of these ends, and at the same time tend to bridge over the chasm between the theoretical and the practical. They are PREFACE V placed at the end of Part I, so that they may be assigned at any time without seeming to have been passed over. The theory of the second part on Solid Geometry is somewhat fuller, and the examples are considerably more extensive both as to number and character, than is usually the case in elementary books. The chief new feature that has been introduced is the use of the notion of Contour Lines in the tracing of surfaces. This idea, as well as the whole subject of surface tracing, has not hitherto been sufficiently emphasized. Where the proof in Solid Geometry is the same as in the corresponding propo- sition in Plane Geometry the demonstration has not always been repeated. In two instances, viz. § 154 and § 169, an entirely different method of proof has been used. This has not been done simply for the sake of variety, although this would be a sufficient reason, but because the algebraic results obtained in this way admit of a much broader interpretation. The student should be re- quired, as an exercise, to apply these methods of proof to the corresponding propositions in Plane Geometry, and vice versa. As a suggestion to this end, appropriate references are given in all these sections. If this is done, the student will be able to prove for himself the harmonic properties of the conic section. I have put two small sections, I and II, in the Appendix rather than assign them to any particular place in the body of the text. The method of finding the direction of a curve at the origin, given in I, I have found to be helpful as early as in the section on curve plotting in Chapter II. If used at all, it should at least precede the formal study of slope. I wish to thank most heartily all my colleagues in this university who have aided me so kindly in the work, and to acknowledge my special obligation to Professor Ellery W. Davis, who, from the inception of the plan to the completion of the book, has given me much valuable assistance. I am also much indebted to Professor George D. Olds, of Amherst College, and Professor E. V. Hunting- ton, of Harvard University, who have read the entire manuscript with great care and offered many helpful suggestions. A. L. C. Thk University of Nebraska, May 25, 1904. CONTENTS PART I. PLANE GEOMETRY CHAPTER I Coordinates, Distances, and Areas BF.CTIONB PASS — 1-4. Cartesian coordinates. Exercises 1 6. Polar coordinates. Exercises 6 6. Relations between rectangular and polar coordinates. Exercises . 7 -- 7-9. Distance between two points 8 10-13. Areas of polygons 12 Examples on Chapter I 16 CHAPTER II Loci and their Equations 14-20. Illustrative examples and definitions 18 21-22. Plotting loci of equations. Exercises 23 23. Use of graphic methods. Illustrations 28 24-26. Intersection of loci. Exercises 32 27-29. Symmetry of loci. Exercises 34 30-37. Finding the equations of loci. Exercises 38 CHAPTER III The Straight Line 38-49. Standard forms of the equations in rectangular and polar coordi- nates. Exercises ..;...... 46 50-51. Equations of the straight line in oblique coordinates. Exercises . 62 Examples on Chapter III 63 Vlll CONTENTS CHAPTER IV Transformation of Coordinates BE0TI0N8 63. To move the origin to tlie point (A, k) 54. To turn rectangular axes through an angle d Examples on Chapter IV . PAGE 67 CHAPTER V Slope, Tangents, and Normals 55-56. Examples of limiting values of ratios. Exercises 58. Geometric meaning of the derivative of the function / (x) 59. Illustrative examples of derivatives. Exercises . 60-61. General formulae for differentiation .... 62. Tangents and normals Examples on Chapter V 70 72 74 76 78 78 CHAPTER VI Theory of Equations, Quadrature, and Maxima and Minima 63-80. Theory of equations. Exercises 82 81. Quadrature. Exercises 101 82. Maxima and minima. Exercises 105 CHAPTER VII Conic Sections 85. Geometric proof of the defining property of the conic section 86. Classification of the conic sections. Exercises 87. General equation of the conic sections 88-91. Standard equations of the conic sections . 92-93. Tangents . . . . 94-95. Pole and polar. Exercises .... 113 116 117 119 126 128 96-100. CHAPTER VIII The Para-bola Properties of the parabola i/2 = 4 ax. Exercises Examples on Chapter VIII 131 136 CONTENTS IX CHAPTER IX The Circle 8EOTION8 PA6B 101-106. Properties of the circle. Exercises 139 Examples on Chapter IX 146 CHAPTER X The Ellipse and Hyperbola 107-108, Tangent, polar, and normal 160 109. Geometric properties. Exercises 152 110-111. Conjugate hyperbola, and director circle 166 112-114, Auxiliary circle and eccentric angle. Exercises . . . 167 115, Asymptotes — definition and equations 162 116-117, Similar and coaxial conies — diameters 164 118-121. Conjugate diameters. Exercises 168 122-123. Equation referred to asymptotes, and polar equation . . 174 Examples on Chapter X 176 CHAPTER XI General Equation op the Second Degree 124-126. To transform the general equation to the standard forms . . 179 Examples on Chapter XI 187 Examples on loci 188 Miscellaneous problems on loci 189 PART 11. SOLID GEOMETRY CHAPTER XII Systems op Coordinates, the Point, Rectangular Coordinates 127-130. Rectangular coordinates. Exercises 193 131. Polar coordinates and direction cosines . . . . , 197 132. Spherical coordinates 198 133. Cylindrical coordinates. Exercises 199 CONTENTS CHAPTER XIII Loci BHCTION8 PAGK 134-135. The locus (in space) of an equation is a surface .... 200 136-139. To trace the locus of an equation. Contour lines. Exercises . 202 140. To find the equation of a locus 205 141. Surfaces of revolution. Exercises 206 CHAPTER XIV The Plane and the Straight Line 142-143. The plane. Exercises 209 144-149. The straight line. Exercises 211 150-151. Transformation of coordinates 216 Examples on Chapter XIV 218 CHAPTER XV CONICOIDS 152-165. The sphere. Exercises 220 156-157. The cone 224 158. The ellipsoid 227 159. The hyperboloid of one sheet 229 160. The asymptotic cone of the hyperboloid of one sheet . . . 231 161. The hyperboloid of two sheets 232 162. The elliptic paraboloid. . . 234 163. The hyperbolic paraboloid 235 164. The paraboloids the limiting forms of the central conicoids. Exercises 237 165-168. Tangent planes and normals 238 169-170. Poles and polar planes 241 Examples on Chapter XV 243 APPENDIX I. The direction of a curve at the origin 245 TI. Example illustrating § 81 .246 III. Trigonometrical formulse 248 ANALYTIC GEOMETRY CHAPTER I COORDINATES, LENGTHS OF LINES, AND AREAS OF POLYGONS Rectilinear Coordinates 1. Let X'X and F' Y be two fixed, non-parallel straight lines, in- tersecting in the point O. Let P be any point in the plane of these lines. Draw HP and QP parallel to X'X and Y' Y respectively. These distances, RP and QP, determine the place of P within the angle XOY. That is, to every position of P there is one and only one pair of distances, to every pair of distances one and only one position of P. Moreover, the position of P can be found when the lengths of the lines RP and QP are given, and vice versa. Suppose, for example, that we are given EP = a, QP= b, we need only measure OQ = a and OR = h and draw the parallels RP and QP, which will intersect in the required point. 2. The two lines RP and QP, ot OQ and OR, which thus de- termine the position of the point P with reference to the lines 2 COORDINATES [3 X'X and T^Y are called the Rectilinear or Cartesian''^ Coordinates of the point F. QP is called the Ordinate of the point P, and is denoted by the letter y; RP, or its equal OQ, the intercept cut off by the ordinate, is called the Abscissa, and is denoted by the letter x. The fixed lines X'X and F'F are called the Axes of Coordinates, and their point of intersection is called the Origin. When the angle between the axes of coordinates is oblique, the axes, and also the coordinates, are said to be Oblique ; when the angle between the axes is right, the axes and the coordinates are said to be Rectangular. If OQ = a and OR = 6, then at P, x±=a and y = h; at Q, x = a and 2/ = ; at i?, « = and y = h', and at 0, cc = and 2/ = 0. The axis XX is called the Axis of Abscissas, or the x-axis; and F' F is called the Axis of Ordinates or the y-axis. 3. Let OQ and OQ' be equal in magnitude to a, and let OR and OR' be equal in magnitude to h. Through Q, Q', R, and R' draw lines parallel to the axes, and intersecting in Pj, P2, P3, P4. Jy' Now at all of these four points x=a, in magnitude, and y =h, in magnitude. Hence in order that the equations x = a and y=^h * This method of determining the position of a point in a plane is due to the French philosopher and mathematician, Descartes. Hence the name Cartesian. The new method was first published in 1637. " It is frequently stated that Descartes was the first to apply algebra to geometry. This statement is inaccurate, for Vieta and others had done this before him. Even the Arabs sometimes used algebra in connection with geometry. The new step that Descartes did take was the introduction into geometry of an analytical method based 4] COORDINATES 8 shall determine only one point, it is not sufficient to know the lengths of a and 6, we must also know the directions in which they are measured. In order to indicate the directions of lines we adopt the rule that opposite directions shall he indicated by opposite signs. It is agreed, as in Trigonometry, that distances measured in the directions OX (or to the right) and OY (or upwards) shall be considered jxysitive. Hence distances measured in the directions OX' (or to the left) and Y' (or downwards) must be considered negative. Therefore (assum- ing a and b to be positive numbers) at P„ a; = a, 2/ = & ; at Pg, a; = — a, 2/ = 6 ; at P3, ic = — (X, y = — 6 ; 3i>t P^, x = a, y = —b. Thus the four points are easily and clearly distinguished, for no two pairs of values of x and y are the same. If all possible values, positive and negative, be given to x and to y, i.e., if both x and y be made to vary independently from — 00 to 4-00, all points in the plane will be obtained. Moreover, to each pair of values of x and y there corresponds, in all the plane, one and only one point ; to each point, one and only one pair of values. 4. For the sake of brevity, a point is represented by writing its coordinates within a parenthesis, the abscissa being always written Jirst. Thus, in the preceding figure. Pi, Pg, P3, P4, are the points (a, b), (—a, b), (—a, —b), (a, —b), respectively. In general, the point whose coordinates are a; and y is called the point (x, y). When the axes are rectangular it is convenient to distinguish the parts into which the axes divide the plane as first, second, third, and fourth quadrants, as in Trigonometry. Because of simplicity in formulae and equations, it is generally more convenient to use rectangular axes. on the notion of variables and constants, which enabled him to represent curves by algebraic equations. In the Greek geometry, the idea of motion was wanting, but with Descartes it became a very fruitful conception. By him a point on a plane was deter- mined in position by its distances from two fixed right lines or axes. These distances varied with every change of position in the point. This geometric idea of coordinate representation, together with the algebraic idea of two variables in one equation hav- ing an indefinite number of simultaneous values, furnished a method for the study of loci, which is admirable for the generality of its solutions. Thus the entire conic sections of Apollonius is wrapped up and contained in a single equation of the second degree." [A History of Mathematics by Florian Cajori, p. 185.] 4 COORDINATES [4 Accordingly, throughout this book, except when the contrary is expressly stated, the axes may be assumed rectangular. EXAMPLES 1. In what quadrants must a point lie if its coordinates have the same sign ? different signs ? 2. Locate the points (1,- 3), (- 2, 4), (5, 0), (- 1, - 3), (4, 2), (0, 3). 3. Construct the triangle whose vertices are the points (0, 4), (—5, — 1), and (4, - 3). 4. Construct the triangle whose vertices are (4, — 1), (1, 2), (-1, - 3). 6. Construct the quadrilateral whose vertices are the points (3, 4), ( — 1, 4), (— 1, —2), (3, —2). What kind of a quadrilateral is it? Consider both oblique and rectangular axes. 6. Plot the points (8, 0), (5, 4), (0, 4), (- 3, 0), (0, - 4), (5, - 4), and con- nect them by straight lines. What kind of a figure do these six lines enclose ? 7. P is the point (x, y) ; Pi, P2, P3 are its symmetrical points with respect to the X-axis, y-axis, and origin, respectively. What are the coordinates of Pi, P2, Ps ? 8. The side of a square is 2a. What are the coordinates of its vertices when the diagonals are the axes ? 9. The side of an equilateral triangle is 2a. What are the coordinates of its vertices, if one vertex is at the origin and one side coincides with the x-axis ? 10. Where may a point be if its abscissa is 2 ? if its ordinate is — 3 ? 11. Can a point move and yet always satisfy the condition x = 0? y z=0 ? both the conditions x = and y = 0? 12. How must a point move so as to satisfy the condition x=c? y = d? both these conditions, c being a negative and d a positive number ? 13. If a point moves along either of the bisectors of the angles between the axes, what is the relation between its coordinates ? 14. Where may a point be if its coordinates satisfy the condition x^ + y^ =: a^? What is the relation between the coordinates of a point which moves so that its distance from the origin is always 2 ? 16. If a line AB is two units to the left of the y-axis, what are the coordinates of a point whose distance from AB is three units ? 16. If P be any point on the bisector of the angle between the ?/-axis and a line three units above the x-axis, what is the general relation between the coordinates of P ? 5] COORDINATES 5 PoLAK Coordinates 5. Let be a fixed point called the Pole, and OX a fixed line called the Initial Line. Take any other point P in the plane and draw OP. The position of the point P with reference to the line OX is known when the distance OP and the angle XOP are given. The line OP is called the Radius Vector of the point P, and will be denoted by p ; the angle XOP, which the radius vector makes with the initial line, is called the Vectorial Angle of the point P, and will be denoted by ^. .p Then p and 6 are the Polar Coordinates* of P; that is, P is the point (p, 6). As in Trigonometry, it is agreed that the angle shall be positive when measured from OX counter clockwise ; that p shall be positive when measured in the direction of the terminal line of the vectorial angle 0. In determining the position of a point whose polar coordinates are given the following direction will be useful : Suppose I stand at facing in the direction of OX To get to the point (p, 6), I turn through the angle 6 to the left or right according as 6 is positive or negative, then, keeping my new facing, I. go a distance p forward or backiuard according as p \^ positive or negative. '\ ♦ Whenever the position of a point in a plane is determined by any two magnitudes whatever, these two magnitudes are the coordinates of the point. Tims there may be an indefinite number of systems of coordinates. For an explanation of other systems which are in common use see Chap. I of Elements of Analytical Geometry by Briot and Bouquet, translated by J. H. Boyd. t This method of locating points by means of coordinates is not altogether new to the student, neither is it confined to mathematics. For example, when we locate places on the surface of the earth by means of their latitude and longitude, we make use of a system of rectangular coordinates in which the axes are the equator and some chosen meridian. When we say the city B is forty miles north-east of the city A, we locate B with reference to A by means of a system of polar coordinates in which the initial line is tlie meridian through A, and A is the polo. Let the student suggest other familiar examples, if possible. How are places located in cities? in AVashington, D.C. ? 6 COORDINATES [5 EXAMPLES Plot on one diagram the following points : 1. (4,30°), (-3,135°), (3,120°), (-4,-30°). * 2. (5,45°), (-4,120°), (3,-150°), (-6,-240°). 3. (a, iTr), (-a, -J-tt), (a, -fTr), (2a, -fTr), (-!«, -^tt), (a, 0), (2rt, tt). 4. (5, tan-15), (-2, tan-i2), (3, -tan-i3), (- 4, tan-i - 1). 5. ,(«, tan-12), (a, -tan-i3), (-a, tan-if), (-a, - tan-i f), [a, tan-i(-4)]. 6. Plot the points (-6,30°), (2,150°), (2, -90°) and connect them by straight lines. What kind of a figure do these lines enclose ? 7. Plot the points (a, 00°), (&, 150°), (a, 240°), (&, - 30°), and join them by straight lines. What kind of a figure do these lines enclose ? 8. Find the polar coordinates of the vertices of a square whose angular points in rectangular coordinates are (3,-1), (-1, - 1), (- 1, 3), (3, 3). 9. The side of an equilateral triangle is 2a. If one vertex is at the pole, and one side coincides with the initial line, what are the polar coordinates of its vertices ? of the middle points of the sides ? 10. Change " equilateral triangle " to " square " in Ex. 9. 11. Change " equilateral triangle " to " regular hexagon " in Ex. 9. 12. How must p and d vary in order to obtain all points in the plane ? (See § 3.) 13. Show that to each pair of values of p and 6 there corresponds in all the plane one and only one point. 14. Show by plotting the four points, (3,60°), (-3,240°), (3, -300°), ( — 3, —120°), that the converse of Ex. 13 is not true. 15. Show that in general the same point is given by each of the four pairs of polar coordinates, (p,0), (-p,7r + ^), [p, _(27r-^)], [-P, -(tt-^)]. 16. Show that for all integral values of n the same point (p, 6) is also given by (p, e ± 2mr) and [- p, 6 ± (2w + l)7r]. 17. Where does the point (p, (9) lie if ^ = ? if ^ = tt ? if p = 2 ? 18. How can the point (p, 6) move ii 6 = cc? it p = a? where a and a are constants ? 19. What condition must p and 6 satisfy if the point (p, 6) moves along a line perpendicular to the initial line ? parallel to the initial line ? 20. What is the position of the point (p, 0) if p = a cos ^ ? p = a sin d ? 6] COORDINATES Relations between Rectangular and Polar Coordinates 6. Let P be any point whose rectangular coordinates are x and y, and whose polar coordinates, referred to O as pole and OX as initial line, are p and 6. T x' a r S X '/ O y / P / Y^ Draw PQ perpendicular to OX. Then, according to the preceding definitions, Oq = x, QP = y, OP=p, zxop=e. From the right triangle PQO we have 0Q=: OP cos XOP and QP=OPsmXOP. :.sc = p cos 0. 1 y = psme. t (1) These equations (1) express the rectangular coordinates in terms of the polar coordinates. From equations (1) we find the corresponding equations express- ing the polar coordinates in terms of the rectangular coordinates to be ^ ., ^ p = Vic2 + 2/2, e = tan-i^ 3C sine = V cos e = iC . (2) Va;2 + 2,2' ^"'" Vic2 4.yi' By means of formulae (1) and (2) equations in either system of coordinates can be changed into the other system of coordinates. It is seldom necessary, however, to use equations (2). DISTANCES EXAMPLES 1. Change the equation p2 := (fi cos 2 ^ to rectangular coordinates. Multiplying the equation by p^^ and putting cos 2 ^ = cos^ d — sin^ 6 gives p4 = a2 (p2 cos2 ^ - p2 sin2 61) . Whence by substituting equations (1) we have (X2 + 2/2)2 - 052(a;2 _y2). Change to polar coordinates the equations 2. x2 4- y2 — 2 rx. Ans. p = 2 r cos 6. 4. (2 x2 + 2 7/2 - ax)2 = a\x'^ + 2/2). Transform to rectangular coordinates 5. p2 sin 2 ^ = 2 a2. Ans. xy = a^. 6. p^ cos J ^ 3. x2-?/2 Ans. p2 = 0^2 sec 2 ^. Ans. p^ = a^ cos i ^. Ans. ?/2 + 4 ax = 4 a2. Distance between Two Points 7. To j^/i(Z the distance between two points whose rectilinear coordi- nates are given. Let P^ix^j ?/i) and P2(^2? 2/2) be the given points, and let the axes be inclined at an angle w. Draw PiQi and P2Q2 parallel to OY, to meet OX in Q^ and Qg- Draw P^R parallel to OX to meet P^Qi in i?. fY VY Then OQ, = x,, OQ^^x^, QiPi^Vi, ^2^ = 2/2- .-. P2R = ^2^1= OQ, - 0Q2=x, - x„ and liP,= QiPi-Q,R=QiPi-Q2P2 = yi-y2' Also Z P2RP1 = Z OftPi = TT - (o. 7] DISTANCES 9 From the triangle P1RP2 we have, by the law of cosines, P,P^^ = P^P^ + RP,^ _ 2P2R . RP^ cos (tt - io). Whence by substitution, since cos (tt — w) = — cos a>, PiP\ = [(a^i - 352)'^ + (2/1 - 2/2)'^ + 2 (a5i - 052) (Vi - 1/2) cos «]2. (1) When the axes are rectangular, a> = 90° and cos w = 0. Hence for the distance between two points whose rectangular coordinates are given, we have the very useful formula P^Pi = V(a?i- 072)2+ (2^1-2/2)2.* (2) If the plus sign before the radicals in (1) and (2) gives P2P1, the minus sign will give PiPi- It will aid the memory to observe that the meaning of (2) is expressed by writing (Distancey = (EastingY -\- {Northingf. Cor. If P2 coincides with the origin X2 = 2/2 = 0, and equations (1) and (2) give for the distance of a point Pi(a7i, 2/1) from the origin OJ*i = ^ici^ + 2/i2 + 2 xxvx cos CO, for oblique axes, (3) OP\ — ViCi^ _|_ y^2^ for rectangular axes. (4) EXAMPLES 1. Find the distance between (— 5, 3) and (7, — 2). 2. Show that if the axes are inclined at an angle of 60°, the distance between the points (- 3, 3) and (4, - 2) is \/39. 3. Find the distance from the origin to the point (— 2, 4) when the axes are inclined at angle of 120°. 4. Find the lengths of the sides of the triangle whose vertices are (4, 1), (-2, 4), and (1,-2). 6. Show that the four points (2, 4), (1, 7), (- 2, 4), and (- 1, 1) are the angular points of a parallelogram. 6. If the point (a;, y) is 6 units distant from the point (3, 4), then will JC2 + y2 _ 6 X - 8 ?/ = 0. *The student should convince himself of the generality of equations (1) and (2) by constructing other special cases in which the given points lie in different quadrants. He will thus have an illustration of the general principle that formulsB and equations deduced by considering points lying in the first quadrant, where both coordinates are positive, must, from the nature of the analytic method, hold true when the points are situated in any quadrant. 10 DISTANCES [8 8. The distance between two points in terms of their polar coordinates. Let Piipx, 6y) and P2(p2> ^2) be the two given points. Then OPr = p,, OP, = p„ ZXOP, = e,, ZXOP2 = e2, and Z P2OP1 = 0,- O^. Erom the triangle PxOP^^ as in § 7, we have P^Pi = OPi' + OP2' -2 0P,' OP, cos P2OP1, . •• P1P2 = ^Pi'^ + P'2^ - 2 P1P2 cos (61 - 62). Ex. 1. Derive equation (2), § 7, from equation (1), § 8. Expanding the last term and squaring (1), § 8, gives P1P22 = pi2 + p,^2 _ 2(pi cos ^1) (p2 cos 6-2) - 2 (pi sin ^1) (pa sin ^2). Substituting the values given in equations (1), § 6, we have P1P22 = Xi2 + yi^ + x^^ + yo^ -2 xixa - 2 yi^a- (1) . •. P1P2 = V(xo - xi)2 + (?/2 - yxy\ Ex. 2. Show that the distance between the points (4, 90°) and (-3, 30°), is V37. Ex. 3. Find the distance between (2 a, 180°) and (-a, 45°). . 9. To find the coordinates of the point which divides the line join- ing two given points in a given ratio (mi : m<^. Let Pi{x^, 2/1) and P^ix^, 2/2) be the two given points, and let P{x, y) be the required pointo Draw PiQi, PQ, P2Q2 parallel to the ?/-axis, and PR, P^Ri parallel to the ic-axis. Then P^E^ = x — x^, PR = x.2 — x, RiP=y-yx, JKP2=2/2-2/. 9] DISTANCES 11 From the similar triangles PiPIii and PP2R, we have PiP _ PiRi _ RiP _'nh _x — x^ _y — y^ PP., PR RP,, ma x^-x 2/2 y .-. wii (x2 — x) = m^ (x — Xi), and mi (2/2 -y) = m^ (y - 2/1). Solving (1) and (2) for x and y, respectively, we obtain ac miX2 + tn^ooi y = miijQ + ^^22/1 (1) (2) (3) TTxT' ^- i + x • W These equations, (3) or (4), cover all cases, the division being internal or external according as A is positive or negative. It P be the middle point of PiP,, ni^ = mo, and therefore the coordinates of the middle of a line joining two given points are If we let A = mi : mg, equations (3) reduce to the form «! + Xa?2 V\ + X2/.2 05 = » = |(a?i + £C2), 2/ = 2 (2/1 + 2/2)- (5) These formulae, (3), (4), (5), are independent of the angle between the axes, and hold for both rectangular and oblique axes. Ex. 1. Find the points which divide the line joining (2, 5) and (—6, — 2) internally in the ratio 8 : 4, and externally in the ratio 2 : 9. Ex. 2. In what ratio is the line joining the points (2, 1) and (— 8, 6) divided by the point ( - 2, 3) ? by the point (8, - 2) ? 12 AREAS £10 Areas of Polygons 10* To find the area of a triangle in terms of the coordinates of its vertices, the axes being inclined at an angle w. Case I. When one vertex is at the origin. Let Pi(xi,yi), Po(x2,y^ be the other two vertices. Draw PiQi, P^Qo parallel to the ^/-axis, and Q^R perpendicular to ^9^2- Then OQ^ = x^, 0Q2 = x,, Q^P^ = y^, ^2^ = 2/2, RQ^ = Q.2Q1 sin 0) = (x^ — X2) sin w, and A OP1P2 = A OQ2P2 + trap. QsQi^i A* - A OQ^P^, = i[0Q2 • Q2P2 + Q2QM2P2 + QiA) - OQ, . QiPJ sin CO, = i [a;22/2 + (a^i - ^2) (2/1 + 2/2) -^i2/i] sin w, = I (Sx^iUi - ^iVi) sin (0 in the notation of determinants. 2/1 2/2 sin (1) * The area of the trapezoid ABCD, in which the non-parallel sides intersect, is the difference of the areas of the two triangles formed by the diagonal AC. That is, ABCD = ABC -ADC = ABE- CDE. This is expressed analytically by saying that the area is the algebraic sum of the triangles. The base CD is then regarded as changing its direction (and sign) with reference to AB ; for in going along the sides con- secutively in the order ABCD A, the base CD is traversed in the same direction as A B, which is not the case in the ordinary trapezoid. That is, when D is to the left of C, both the base CD and the area of the triangle ACD are positive, say. But as D moves to the right, both CD and the area ACD >B become zero and change sign as D passes through C. 10] AREAS 13 Case II. When the origin is not a vertex of the given triangle. Pa \Y P. Let Pi(xi, ?/i), Po(x2, 2/2)? ^3(^3) y:i) be the vertices of the given triangle. Draw the lines OP^, OP^, OP^. Then by Case I we have ^i, yi A OP1P2 = i(a;i2/2 — X2I/1) sin to=l A OP.Ps = K^22/3 - ^32/2) sin o> = i A OPiPi = i(.T3.?/i - .T12/3) sin o) = ^ •'i'2j 2/2 ^3) 2/3 Sin o). sm o). sin o). .-. A PiP2^3= i[(^*i2/2 - a^iVi) + ('<^-22/3 - ^3?/2) + (^^sZ/i " aTi^s)] sin o> (2) =i ( a^i, 2/1 a^2, 2/2 + ^'2, 2/2 , 3^3,2/3 a-*3, 2/3 a^i, 2/1 ^1, yi, 1 =i ^2, 2/2, 1 sill (U. ^ h, ys, ^ sin (u (3) When the axes are rectangular sin w = 1, and equations (1), (2), (3), respectively, reduce to A O P1P2 = I (i»i?/2 - OC27Jl) = I (4) A P1P2P8 = I i^K^iVi - a^-22/i + ^'iV'i - «82/2 + 3582/1 - ^iVfi) (5) oci, 2/1, 1 a?2, 2/2J 1 = k i»8, 2/8J 1 Xi .Tj, 2/1 3^2 X2 — x^, y^ 2/3 (6) 14 AREAS [11 11* When the origin is within the given triangle, the given triangle includes the three triangles OP^P^, OP^P^, OP^P^ (§ 10) ; hence the expressions ^{x^^ — ^22/i)> ^(^'22/3 — ^32/2)5 and ^{x.^^ — x^y^ must have the same sign. When the origin is outside, the given triangle does not include all of these triangles, and therefore the above expressions can not have the sa^ne sign. Suppose a person to start from and walk consecutively around the triangles OP^P^, OP2P3, OP^P^ in the direction indicated by this order of vertices. This imaginary person would thus walk along each side of the given triangle once in the same direction around the figure, as indicated by P^P^P&y and along each of the lines OP^, OP2, OPsf twice in opposite directions. When the origin is inside the given triangle, he would walk around each of these triangles in such a manner that he would have its area always on his left hand. When the origin is outside, he would go around those triangles which in- clude no part of the given triangle, in such a manner that he would have their area ahcays on his right hand. Thus direction around a triangle may be taken to indicate the sign of its area. (See footnote under § 10.) The expressions for area in § 10 will be found to be positive, if the vertices are numbered so that in passing around in the direction thus indicated the area is always on the left. Let the student show by trial that (x^y^ — x^jy^) is ± according as Z P1OP2 is ±; Z P1OP2 is ± according as the cycle OP^Po is ±. 12.* To express the area of a triangle in terms of the polar coordi- nates of its vertices. Let Pi(pi, ^1), P2(p2) ^2)? ^sfe) ^3) be the three vertices. Then x^ = pi cos 61, X2 = p2 cos 62, x^ = p^ cos 9s, 2/1 = pi sin 61, 2/2 = p2 sin O2, 2/3 = Ps sin ^3. [(1), § 6.] Substituting these values in (5) and (6) of § 10 gives OP1P2 = i pip2 (sin $2 cos 61 — cos 62 sin Oi) = | pipa sin (62 — ^1). (1) PiP2Ps= i [piP2 sin (82 - e,) + pops sin (^3 - O2) + PsPi sin (6, - 6^)^ (2) From (1) it follows that the three terms of (2) represent, re- spectively, the areas of the triangles OP1P2, OP2PS, and OP3P1. 13] AREAS 15 The signs of these terms are the eigns of the angle differences (since p can always be made positive), and we therefore have an independent proof of the statements in § 11. Let the student prove (1) and (2) directly from a figure. 13* To find the area of any polygon when the rectangular coordi- nates of its vertices are known. LetPi(a;i, 2/1), ^^2^ .^2), -^sfe Vs^ A (^'4, 2/4) ••• -^nC^n, 2/«) be the n vertices of the given polygon. Then, we have, from (5) § 10, A OP.Po A OP,P, = i x^, 2/1 X2, 2/2 ^3, 2/3 X,, 2/4 A OP,P, = \ A OP,P, = i. ^2, ?h •^3> 2/3 ^4, 2/4 X3, 2/5 AreaPiPa • ^« A OP.P, = \ a?„, 2/« ^1, 2/1 =H Xi, 2/1 a^2, 2/2 % ^3 a^3, ^A1 2/3 2/4 + ^A, 2/4 + ••• aJ„, y„ a^5, ^5 aji. 2/1 (1) since the area of the polygon is the algebraic sum of the areas of these triangles. This formula is easy to remember, but by expand- ing the determinants and collecting the positive and negative terms it may be written, Area PiPg ••• 1*„ = \ [(0512/2 + ^aVs + ^^V\ + ••• ^nVi) - (yii»2 + 2/2^53 + Vti^i^ + ••• yn«l)]) (2) which gives the following simple rule for finding the area of a polygon when the rectangular coordinates of its vertices are known : (1) Number the vertices consecutively, keeping the area on the left. (2) Multiply each abscissa by the next ordinate. (3) Multiply each ordinate by the next abscissa. (4) From the sum of the first set of products subtract the sum of the second set and take half of the result. If the axes are oblique, the second members of (1) and (2) must be multiplied by the sine of the angle between the axes. The law of the sign of the area is the same as for the triangle. 16 EXAMPLES ON CHAPTER I [13 EXAMPLES ON CHAPTER I Find the area of the polygons the coordinates of whose vertices talcen in order are, respectively, 1. (1,3), (-2, -4), and (3, -1). 2. (2, 5), (-6, -2), and (-1, 5), when w = 60°. 3. (4, 15°), (-5, 45°), and (6, 75°). 4. (3, -30°), (-5, 150°), and (4,210°). 5. (2, 15°), (6, 75°), and (5, 135°). 6. (-a, ^tt), (a, I it), and (-2a, -|7r). 7. (a, b + c), (a, b — c), and (—a, c). * 8. (a, c + a), («, c), and(— a, c — a). 9. (2,3), (-1,4), (-5, -2), and (3, -2). 10. (4,5), (1,4), (-2,6), (-5,3), (-2,-1), (-3,-4), (1,-2), (3, -4), and (2, 1). 11. What are the rectangular coordinates of (4, 30°), (—2, 135°), (-3,1^)? 12. What are the polar coordinates of (3, - 4), (- 5, 12), (1, 3) ? 13. Find the coordinates of the points which trisect the line joining the points (-2, -1) and (3, 2). 14. Find the coordinates of the point which divides the line joining (3, — 2) and ( — 5, 4) internally in the ratio 3 : 4. 15. Find the coordinates of the point which divide:; the line joining (5, 3) and (— 1, 4) externally in the ratio 3:2. 16. Find the length of the sides and medians of the triangle (2, 6), (7, — 6), (—5, — 1). What kind of a triangle is it ? 17. Find the length of the sides and the area of the triangle (3, 4), (—1, 0), (2, - 3). What kind of a triangle is it ? 18. Find the sides and area of the quadrilateral whose vertices taken in order are (5, - 1), (- 1, 2), (- 5, 0), and (1, - 3). What kind of a quad- rilateral is it ? Change to polar coordinates the equations 19. x^ + y^ = r\ 20 21. x^ = y%2a-x). 22, Transform to Cartesian coordinates 23. d = tsin-^ m. 24. 25. p = a sin 2 6. 26. y = xta,u a. (x^^y^)(x-ay = = &2^2. p2 = a2 sec 2 6. pi = ai sin | d. 13] EXAMPLES ON CHAPTER I 17 Prove analytically the following theorems : / 27. The diagonals of a parallelogram bisect each other. 28. The lines joining the middle points of the adjacent sides of any quadri- lateral form a parallelogram. 29. The three medians of a triangle meet in a point, which is one of their points of trisection. 30. The lines joining the middle points of opposite sides of any quadrilateral and the line joining the middle points of its diagonals meet in a point and bisect one another. y 31. The area of the triangle formed by joining the middle points of the sides of a given triangle is equal to one-fourth of the area of the given triangle. 32. If in any triangle a median be drawn from the vertex to the base, the sum of the squares of the other two sides is equal to twice the square of half the base plus twice the square of the median. 33. The sum of the squares of the four sides of any quadrilateral is equal to the sum of the squares of the diagonals plus four times the square of the line joining the middle points of the diagonals. 34. Pi(xi, yi), Fiix^, yz), T-iixz, 2/3), I'^ix^, 2/4) •• • Tr,{x,,, yn) are any n points in a plane. PiPo is bisected at Qi ; QiP^ is divided at Qz in the ratio 1:2; ^2^4 is divided at Q^ in the ratio 1:8; ^jjPs at Q4 in the ratio 1 : 4, and so on till all the points are used. Show that the coordinates of the final point so obtained are a:i + a:2 + a:3 -}- a;4 + ... a:,. ?/i + 2/2 + 2/3 + 2/4 + . . . Vn and n n Show that the result is independent of the order in which the points are taken. [This point is called the Centre of 3Iean Position of the n given points.] CHAPTER II » >o LOCI AND THEIR EQUATIONS 14. It has been shown in § 3 that to each pair of values of x and y there corresponds in all the pl ane one and only one point, and that to each point corresponds one and only one pair of values. Also, if X and y vary independently and unconditionally from — oo to oo, every point in the plane will be obtained. If, on the contrary, one or both of the coordinates cannot take all values, or if all values cannot be independently taken by both, the point cannot move to all positions in the plane. If, for example, ic > 0, the point X (x, y) must lie to the light of the ~ 2^-axis ; ii x<0, the point must lie to the left of the ^/-axis; if x is neither greater nor less than zero, the point can lie neither to the right nor to the left of the ?/-axis ; i.e. if x=0, the point must lie 07i the 2/-axis. 15. If x>a, the point (x, y) must lie to the right of the parallel AB, which is a units to the right of the 2/-axis ; if x-3? x< -S? x= -S? Ex. 2. Where is the point (x, y) if y>&? ?/<6? y = 6? y>-hf y<-b? y=-b? 18 X <0 Y a A X a X 1 B 17] LOCI AND THEIR EQUATIONS 19 16. Draw a circle with centre at the origin and radius equal to a. Then the point P{Xf y) will be outside, inside, or on this circle according as OP>a, OP a% x^ -\-y^ < a-, or oc^ -\- y^ = a-. Ex. 1. Write down the conditions that the point (x, y) shall be outside, inside, or on the circle whose centre is at the origin and radius 3. Ex. 2. What are the conditions that the point (a;, y) shall be outside, inside, or on a circle with centre at (— 3, 1) and radius 4 ? Ex. 3. Draw a circle with centre at (a, 6) and radius r, and write down the conditions that the point (x, y) shall be outside, inside, or on this circle. 17. Let the line AOB bisect the angle XO Y. Y X 1 Y B V y P X < y / y } / y / o ./ / Y' X' Then every point on AB is equidistant from the axes. Hence the point Pix^ y) is above AB, below AB, or on AB, according as y>x, y, <, or = j i.e. according as y — a; is positive, negative, or zero. 20 LOCI AND THEIR EQUATIONS [19 18. Draw CD parallel to ABj cutting the ^/-axis in E, three units above 0. Then every point on CD is three units farther from the a>axis than from the ?/-axis. Therefore the point P{x, y) will be above CD, below CD, or on CD, according as ?/>, <, or =aj + 3; i.e. according as ?/ — a; — 3 is positive, negative, or zero. Y > ' p y^ "" / q. /" / / y / E / / p y y ^/ A. o Y' Ex. 1. Draw a Hne parallel to AB^ cutting the y-axis two units below ; and write down the conditions that the point (ic, y) shall be above, below, or on this line. Ex. 2. What are the conditions that the point (cc, xj) shall be above, below, or on the line through E parallel to the bisector of the angle X'OF? 19. Let CD be the perpendicular bisector of the line joining A{-\,V) and i5(3, -1). Then all points on CD are equidistant from A and B, and all other points are not equally distant from A and B. Hence the point P{x, y) will lie to the rujlii of, to the left of, or on CD, according as ^P >, <, or = BP, or according as AP"- >, <, or = BP'^-, i.e. according as [(2), § 7] (a; + l)2 + (2/-iy>, <,or = (a:-3)2 + (2/ + iy; " whence 2x — y — 2>, <, or = 0. 20] LOCI AND THEIR EQUATIONS Yl p /D 21 Ex. 1. Find the conditions that the point (x, y) shall be above, below, or on the perpendicular bisector of theiine joining (2, 3) and (— 1, — 2). Ex. 2. What is the condition that (x, y) shall be on the perpendicular bisector of the line joining (a, 6) and (c, d) ? \^^ 20. The examples in §§ 14-19 illustrate certain general principles, of which we will here make only a preliminary statement. I. All points whose coordinates satisfy an equation of condition (not an identity) lie on a certain line ; and conversely, if a point lies on a fixed line, its coordinates must satisfy an equation. II. Points whose coordinates satisfy a condition of inequality do not lie on any fixed line. If /(a;, //) be used to represent any expression containing the two variables x and y and certain constants, these principles may be stated more definitely, as follows : I. All points whose coordinates make /(a;, y) = 0, lie on a certain line ; and conversely, the coordinates of all points on this line make /(^, y) = 0. II. If f(oci, ?/i) > and /(ajo, 2/2) < ^j t^© *wo points (ar„ y,) and (x2, y>^ lie on opposite sides of the line the coordinates of whose points make /(a;, y) = 0. Hence every line, as well as the axes of coordinates, is said to have a positive and a negative side. 22 LOCI AND THEIR EQUATIONS [20 Def. The locus of a variable point subject to a given condition is the place, i.e. the totality of positions, where the point may lie and sat- isfy the given condition. Def. The line (or lines) containing all points, and no others, whose coordinates satisfy a given equation is called the Locus of the Equation ; conversely, the equation satisfied by the coordinates of all points on a certain line (or lines) is called the Equation of the Line, or the Equation y^ of the Locus. Def. That part of the plane containing all points, and no others, whose coordinates satisfy a given inequation is the Locus of the Inequatio7i. Thus the Locus of a point in Plane Geometry is not ahvays a line. In the examples of §§ 14-19 only Cartesian coordinates have been used, but the fundamental principles there illustrated, and also the above definitions, hold for all systems of coordinates. Let the student give some similar illustrations with polar co- ordinates. EXAMPLES What is the locus of 1. x2 + ?/2 =0 ? x'-^ + 2/2 > ? x^ -\-y'^<0? 2. X= VX^ + ?/2 ? X > Vx^ + ?/2 ? X < \/x2 -f- ^2 ? i 3. p = a sec ^ ? p > a sec ^ ? p < a sec ^ ? i l )- ■ 4. p = & CSC ? P > & CSC ^ ? p < 6 CSC ^ ? ' J- 5. 4 < x2 + ?/ < 9 ? ^i^^f^ --^ ' 6. 9<(x-2)2+(?/_3)2'<16?/.r^P>' 7. a sec ^ < p < 6 sec ^ ? i-j //^ 8. p = a COS ^ ? p > a cos ^ ? X < a^fffid ? fr^^ * 9. acos0 a sin ^ ? ^^(z sin J^m'*^ — 11. P = a? p>a? p/£__ 12. What is the locus of a poiWmpving so that the sum of its distances from the Unes x = and x = 3 is 1, 2, 3, 4 ? 21] LOCI AND THEIR EQUATIONS 23 To Find the Locus of a Given Equation 21. If the locus of an equation is a straight line, the locus is easily drawn; it is only necessary to locate two points on it (preferably the intersections* with the axes) and draw a straight line through these points. Likewise, if the locus-is a circle, the complete locus can be drawn when the centre and radius are known. It will be shown farther on that straight lines and circles can easily be recognized by the forms of the equations. In general, having given an equation of condition between the coordinates (in any system) of a variable point, we may assign any value we please to one coordinate and find a corresponding! value, or values, of the other. To every such pair of corresponding value's will correspond a definite point of the locus. Since these pairs of values may be as numerous as we please, we can in this way locate as many points of the locus as we please. A smooth curve drawn through these points will be an approximation to the locus of the given equa- tion. The degree of approximation will depend upon the proximity of the points thus located. This method of constructing a locus is applicable to any equation that can be solved for one of the variables, and is called Plotting $ an Equation, or Plotting the Locus of an Equation. The steps of this process are as follows : * Unless both intersections are near the origin, when the line will be inaccurately determined, or both at the origin, when its direction will be quite undetermined. t " Corresponding values " of the variables, x and y say, involved in a given equa- tion are a pair of values of x and y which satisfy the equation. X The logic of the process of plotting is that of induction, and should be so recognized by the ,' ; •'*., \ .'' student. Given the points A, B, C, D, E, F on n curve; then, in the absence of further knowledge, we take as a probable approxi- mation a smooth curve drawn through them like the full curve in the figure. We are not war- ' ' ' ranted in drawing such a curve as the dotted one through the points, because it is unlikely that, taking points at random on such an irregular curve, tlu- jxisititui of these points should fail to disclose any of the irregularity. The student should also be warned that sudden changes of slope or curvature areas unlikely as sudden changes in the value of an ordinate. 24 LOCI AND THEIR EQUATIONS [22 \ T / > L / \ / L P) V / \ \ / p. X > X O y p. p. r. ' X = ^8 -6 -4 — 2 y = 6.8 3.4 .8 - 1 X = 2 4 6 8 y = -2.2 -1.6 .2 2 (1) Solve the equation with respect to one of the coordinates. (2) Assign to the other coordinate a series of values differing but little from each other. (3) Find each corresponding value, or values, of the Jirst coor- dinate. (4) Locate the point corresponding to each pair of corresponding values thus found. (5) Join these points in order by a smooth curve, and this curve will be approximately the required locus. If there be doubt how to fill up any of the intervening spaces, interpolate more points. 22. Illustrative Examples. Ex. 1. Plot the locus of the equation lOy = a;^ — 3x — 20. Assigning to x values from — 8 to + 10, differing by two units, we iind the following pairs of values of x and y to satisfy the equation : -2 10 5 Plotting the corresponding points Pi, Pa, Pa, etc., and drawing a smooth curve through them in the order of the increasing values of x, we find the locus to be approximately the curve drawn in the figure. Ex. 2. Plot the locus of the equation y'^=\.x. Solving for y gives y = ± 2 ^ x. When X = 0, 1, 4, 9, ... to od, y = 0, ±2, ±4, ±6 . . . to ± 00, The corresponding points of the locus are 0(0, 0), Pi(l, -2), P2(l, 2), P3(4, -4), P4(4, 4), PcCO, - 6), and PeCO, 6). . . . When X is negative, y is imaginary. There- fore no points of the locus lie to the left of the ?/-axis. For every positive value of x there are two values of y numerically equal but opposite in sign. Hence the two correspond- ing points of the locus are equidistant from the X-axis. As x increases, both values of y increase numerically. Y ^ P» jpj/. [ x_ o _ ^***«. ^ p." 22] LOCI AND THEIR EQUATIONS 25 Therefore the locus cannot be such a curve as that represented by the dotted line, but must be approximately that indicated by the full line. Ex. 3. Plot the locus of the equation 25(x - 1)2 + \Q{y - 3)2 = 400. Solving for y gives y = 3 ± fVlG — (x-l)2. This form of the equation shows that y is imaginary when a; < —3, or a: > 5, since 16 — (oj— 1)2 is then negative ; and when x is neither less than — 3 nor greater than 5 there are two real unequal values of y, one found by using the + sign before the radical, the other by using the — sign. Hence the locus lies between the two parallel lines a; = — 3 and x = 5. The equation is satisfied by the follow- ing pairs of values of x and y : -3 -2 -1 3 6.3 7.3 7.8 3 - .3 -1.3 -1.8 X y y X = y = y = 2 7.8 1.8 3 7.3 1.3 4 6.3 - .3 7 V l--\ P The corresponding points are P(— 3, 3), Pi(-2, 6.3), P2(-2, - .3), etc., and the locus Is the curve shown in the figure. Ex. 4. Plot the locus of the equation, p = 2a sin d. Here p has its greatest value when sin d has its greatest value, i.e. when d = \ir. As d increases from to J tt, sin ^ in- creases from to 1, and p increases from to 2a ; as ^ increases from ^tt to tt, sin decreases from 1 to 0, and p decreases from 2a to 0. Hence the locus starts from the origin and returns to the origin as 6 is made to vary from to t. Assigning to 6 values from to 180°, differing by 30*^ we find the following points are on the locus : 0(0, 0), A{a, 30°), P(aV3, 60°), C(2a, 90°), D{a^y 120^"^), E{a, 150°), and 0(0, 180°). The complete locus is the curve shown in the figure. Ex. a. Show that the points A, B, . . . all lie on a circle tangent to OX at O and whose radius is a. Show also that every point on this circle satisfies the given equation. 26 LOCI AND THEIR EQUATIONS [22 Ex. 6. Show that the same circle will be described as 6 varies from 180° to 360° ; also as 6 varies from any value a to a -h v. We have in this example an illustration of a characteristic property of equa- tions in polar coordinates containing a periodic function of 6. In such equations p takes all possible values as 6 varies through a limited range of values called the period of the function. The complete locus is described at least once as 6 varies through this period, and is repeated as 6 varies through any other equal period. The period of sin is 2 tt ; hence p takes all possible values from —2a to + 2 a as ^ varies from to 2 tt. The whole circle is described tivice as 6 varies through this period, once as 6 varies from to tt with p positive, and once as 6 varies from tt to 2 tt with p negative. Also the whole circle is described twice if 6 starts from any value and varies through 2 tt in either direction. Ex. 5. Plot the locus of the equation p = sin 2 6. This equation I& satisfied by the following pairs of values of p and 6 : e = 45°, 225°, p=l. e = 135°, 315°, p= -\. e = 30°, 60°, 210°, 240°, P = ly/S. 6 = 120°, 150°, 300°, 330°, P = - I V3. e = 15°, 75°, 195°, 255°, e = 105°, 165°, 285°, 345°, P=-\- e = 0°, 90°, 180°, 270°, 360°, P = 0. The corresponding points are found by drawing three circles with centres at O and radii i, | V'^, and 1, and then drawing radii dividing these circles into arcs of 15°. The locus is the four-leaf curve shown in the figure. As e varies from to 2 tt, the four leaves are described in the order 1, 2, 3, 4, and in the direction indicated by the arrow heads. EXAMPLES Plot the loci of the following equations : * (2x-Sij - 6 = 0. W / 1. \4x-6y- 6 = 0.\ 2. [Qx-dy + 27 =0. 2x-\-Sy + 5 = 0. 3x-2?/-12=0. 5x4-2?/- 4=0. J * For convenience in plotting loci the student should be supplied with " coordinate paper," both " rectangular" and "polar." t Loci grouped under the same number should be plotted on the same diagram. 22] LOCI AND THEIR EQUATIONS 27 r r 2 a; + 9 y + 13 = 0. 3. y = lx-^. 2y-x = 2. 7. 6 a;2 + 5 a;?/ - 6 1/2 = 0. 8. [x2-2/2 = 4. J ' a;?/ = 2. x?/ = - 2. 11. r4(x+i)=i 2)^ 10y=(x+l)2. 4. (a;-4)(y + 3)=0. 6. (x2-4)(y-2) =0. 6. x2- 2/2 = 0. 4x2-«/2_o. f a:2 4. y2 _ 25. 10. (a;-8)2+(?/-4)2 = 25. I (a; -4)-' +(2,-2)2 = 5. 12. «/ = x3-4x2-4x+16. MS. ( !=(,^:-!r- 14. [2/2=(x2-4)2. J y = a:4 _ 20 a;2 + 64. 1 x4 - 20 a;2 + 64. J ^5. (x2 + y2)2 = a2(x2-y2). 16. // = x, x2, x3, x*, x^, ... x«. x = 2/, 2^2^ 2/8, 2/*, y^,-- y". Note the effect of interchanging x and y ; e.gr. the locus of x = ?/* is obtained from the locus of y = x^ by revolving the plane through 180° around the line 17. y = (x-l), (x-l)2, (x-l)3. y. 18. 2/ = x3, x3-x, x8 + x. - f /" 19. 2/^ = X, x2, x^, x*. 20. 2/ = sin x, cos x, sin-i x, cos"* x. 21. 2/ = tan X, cot x, tan-i x, cot-i x. ^2. 2/ = sec x, esc x, sec-^ x, csc-i x. -f • 23. 2/=sin 2 x, sin ^- , | sin 2 x, 2 sin ^ . 24. 2/ = 6 sin -, 6 sin x + c 25. p = sin ^, cos 0, sec 0, esc ^. 27. p = cos 2 e, cos 3 0, cos 4 ^. / 29. p = sin \ e, cos ^ 6. ^31. p=:acos^ + ?). ^^33. 2/ = 2^ log2X. fZb. y = a^, logaX. (a>, =, <1.) 2 a a f 26. p = sin3^, 8in4tf. 28. p = tan ^, cot d. 30. p= ^ 6 1 — cos d 3 — 2 cos ^ 32. p2 = sec 2 ^, CSC 2 ^. (Cf. No. 9.) 34. 2/ = 10^, logio X. i 36. 2/ = 2*, 2-^ i(2' + 2-'). 37. 2/ = e«, e «, ^ (e« + e «). Catenary, if e = 2.7 +. 38.2/ = ^^, (x-l)(x-2) , x-3 x-3 X 40. y = ^+2,(^-l)(^-3). '^ ^ x + 3 x-2 42 y^ (a;-l)(x.-3)(x-5) (x-2)(x-4)(x-6) 44.y- (a;-l)(a;-3)(x-5) (x-2)(x-4) 89 2/= (^-^)(^-^) , (^-^)(»-3). • ^ (x-3)(x-4) (x-2)(x-4) J ,^ (x+l)(x-2) (x + 2)(x-4) (x + 3)(x-4) (x-l)(x-3) 48 ^^ (a;-H)(x-4)(x-6) (x-l)(x + 2)(x-3) 46. (x-l)(x + .3)(x-6), (x-2)(x-4) 28 LOCI AND THEIR EQUATIONS [23 ^ 46. , = ^^. 47. , = — ^ -e 48. y=j^,^y «• y = (^^rry/ ^«- ^ = (x--2)2 • 5^- ^ = (x-2)(x-3y ' 52. Are the points (3, 60°) (f , — 90°) on the same or opposite sides of the loci of Ex. 30 ? , ^ 53. Which of -the following loci pass throug^the origin ?^yuiu&JO ^-^^^^^"^^ ^ 4. (1 ) 2 X + 3 y = O.i- (4) 2/2 - a2a;2>^5?^7) yi = ^a^\\ Lv<^^) iK^ + 2/2 = 1. (5) ax + &?/ + c = 0. (8) ^2 ^ 4a (x + a).)^ iT f_(3) 2/ = 3x2-x. (6) ax2 + &?/2 = 1. (9) (x - a)2 + (y - 6)2 = a2 + 62. . kv'^^^^^ What is the necessary and sufficient condition that the locus of an equation N\ in Cartesian coordinates shall pass through the origin ? The Use of Graphic Methods 23. It has been shown in §§ 14-20 that whenever the relation between two quantities, whose values depend upon one another, can be definitely expressed by an equation, then the geometric or graphic representation of this relation is given by means of a curve. Such a curve often gives at a glance information which otherwise could be obtained only by considerable computation ; and in many cases reveals facts of peculiar interest and importance which might other- wise escape notice. The use of graphic methods in the study of physics, analytical mechanics, engineering, and many other branches of scientific inves- tigation, is already extensive and is rapidly increasing. Graphic methods can be used, however, not only in examples where the equation connecting the two variable quantities is known, such as those already given, but also in examples where no such relation can be found ; in these latter cases the graphic method furnishes almost the only practical means of studying the relations involved. Comparative statistics, and results of experiments and direct observations, can frequently be more concisely and forcibly repre- sented graphically than by tabulating numerical values. The fol- lowing are simple examples of this kind: 1. The following table shows the net gold (to the nearest million of dollars) in the U. S. Treasury at intervals of one month, from Jan. 10, 1893, to Oct. 31, 1894 (Report of the Sec. of the Treas., 1894, p. 8) : >3J LOCI AND THEIR EQUATIONS 29 1898 Millions of Dollars. 1898 Millions of Dollars. 1894 liiUions of Dollars. 1894 MiUiuus of Dollars. Jan. 10 Feb. 10 Mar. 10 Apr. 10 May 10 June 10 120 112 102 106 99 91 July 10 Aug. 10 Sept. 9 Oct. 10 Nov. 10 Dec. 9 97 104 98 87 85 84 Jan. 10 Feb. 10 Mar. 10 Apr. 10 May 10 June 9 74 104 107 106 92 69 July 10 Aug. 10 Sept. 10 Oct. 10 Oct. 81 65 62i 56 60 61 Using time (in months) as abscissas, and dollars (1,000,000 per unit) as ordinates, the separate points represented by the table have been plotted (Fig. 1) and then joined by a smooth curve. 100 50 ■^ ;5 123456789 10 11 2 3 4 5 6 7 1894 8 9 10 11 12 Fig. 1. In this example the curve gives no new information, but it presents in a much more concise form the information given by the tabulated numbers. Observe also that if the points are inaccurately located, the diagram becomes not only worthless, but misleading. 2. An excellent example of the use and advantages of the graphic method of representing comparative statistics is found in the large engraved plate placed under the front cover of the Annual Report of the Secretary of the Treasury for 1894. This plate presents on a single sheet information that covers several pages when expressed in tabulated numbers. All of the curves given on this plate, except one, are shown (on a smaller scale) in Fig. 2. This figure should be carefully studied, and if possible the original plate should be consulted. 3. The curves in figures 1 and 2 were constructed by locating separate points and then drawing a smooth curve through these points. Such curves give no new information, but represent graphically information already ascertained. In some cases, however, curves can be drawn mechanically. When this is possible the curve is constructed, not for the purpose of exhibiting facts previously known, but for the purpose of obtaining new information. For instance, in the stations of the U. S. Weather Bureau an instrument called ■■:T:T'"""""""""7"^r ""■"T"-- t -.-lt._£ :3:l 1- 1 JL J ^1 : --3 - •: _- - ---'T^ V \ § i^ \ r"' K k ) 2 .__ 5. 1 __^-'i__ _.% ^ •k. A. Is f - ^s s. \T I I - 2 ^ \ *»^ l\ ^ \ - ^-•" ^ "^^ 3Z 2 :__>Jv--:---^- - ^T ::: rT : --. :: Si > ^^ s T ■ I-'* -- f'^s^^ 2 N^U t.-vL. WJ ^v ^ 'Xu X \ . / f 00 - v^^- 5 ^>-- \ ' ..^ c ^ J L w t - ^;^ - t I 2 _^L 52> _-- 7^ 1 •...: \ , II ":^ ""s. *• i i_ § - _ _ ^_ - ^\-i"'--i " j '^ 1 00 •^ \ '... /[ 1 ^ _ _ _ ^ — ^_ VT" III I I V I It I^^ V "~ 00 r ^ ^^ f ^ r-l _ ^^u. 4:i\ :± • - - _ _ --. -I _ 3 i ,'^ s ' ^2 _^ > J 1 ^ i ^" « \^~ la ^ " ^v ^ ^ ~ i 2 _ _ \ _._ ^: 2 1 ^ _ __ 1^ _^ 1 \ • -l 00 Z' ^^ 1 ' 2 _ /- i ; 1 '^ - \ « 1 ^ ^ "^'^ SIS ^ ^ I 1 2 >. *» _:i :l- l_ i 3 -^ - i •. L j ^ 2 i i^r \ .'7 1 CO c "3 1 si ;S ^__ . "Vt T « "^ Sk M .»j <« 3 C 1*-. y ^ 1 1 ^ " 1 2 5 - "2 t * »■ 5" ' s S 2 •- G H 1 - - / V -i-i- ' ^ h ^ 2 c c ° ^- = > 3 ■;; "c - **'^ jC !!• ' ^ w i « -2 Z J/ V \ "^ m ^I ^. 11 2 O^UQCQg ,*"" N U 2 1 ? ^ ^ 2 I "^"' -1 ^^ «s 1 s ^ SlE 2 - axis, put y = in the equation of the curve and solve the resulting equation for X. The roots of this equation in x represent the distances from the origin to the points of intersection ; and these distances are called the x-intercepts of the given curve. Similarly, to find the y-intercepts, put a; = in the given equation and solve the resulting equation for y. Ex. 1. How many x-intercepts may a curve of the nth degree have ? <^Vv^ ^ > i^-w^ Ex. 2. What does it mean when in an equation in polar coordinates we put ^ = 0? p = 0? 26. A line may be defined as the path of the moving point. Then, if (Xf y) be the moving point, both x and y are variable quantities, and are called the variable or current coordinates of the moving point. The path of the moving point is then determined by the condition that its coordinates must vary only in such a manner as always to satisfy a given equation ; i.e. although both coordinates vary the relation between them remains fixed. EXAMPLES Find the intercepts and the points of intersection of the following loci : 1. 2 a; + 3 y = 12, 4 a; - y = 5. 2. 3x + 5y = l, x-3y+7 = 0. 8. 5x-22/-f-4 = 0, x-2y = A. 4. X + 3 y = 15, x2 + y2 = 25. 6. 3 X - 4 y = 20, x^ + y2 _ 10 x - 10 y + 25 = 0. 6. 5x+4y = 20, x2 + ?/2 = 4. 11. Find the points of intersection of the loci of Nos. 1, 2, 3, 9, 15, 17, 18, 19, 20, 21, 26 in the last preceding set of examples. 12. Find the intercepts of the loci of Nos. 7, 9, 10, 11, 12, 13, 14, 18, 19, 20 of the same set and check tlie results by the plots already made. 13. Find the area of the triangle whose sides arex — 3y + 5 = 0, 3x-|-4y=ll, *? X + 7 y = 3. 7. x-3y = 0, x2 + y2 + 20 y = 0. 8. y'' = 4.ax, 2xy = a2. 9. y2 = 4ax, y2 _ x2 = a2. 10. y^ = iax, x2 = 4 ay. 34 v> .^ LOCI AND THEIR EQUATIONS Symmetry of Loci [27 27. The process of constructing a locus explained in § 21 is long and tedious. It may often be shortened by an examination of the peculiarities of the given equation, such as the limiting values of the variables for which both are real (see Ex. 3, § 22), symmetry, etc. Such considerations will often reveal the general form and limits of the curve and give all the information desired with little labor. The intercepts (§ 25) are almost always useful for this purpose. Definitions. Two points A and B are said to be symmetrical with respect to a ceiitre when the line AB is bisected by 0. Two points C and D are said to be syminetrical with respect to an axis when the line CD is bisected at right angles by the axis. The two points (x, y) and (— x, — y) are symmetrical with respect to the origin ; (x, y) and (x, — y) with respect to the avaxis. A curve is said to be sym- metrical with respect to a centre O when all lines passing through meet the curve in a pair, or pairs, of points sym- metrical with respect to 0. A curve is said to be sym- metrical with respect to an axis when all lines perpendicular to the axis meet the curve in a pair, or pairs, of points sym- metrical with respect to the axis. Or, in other words, a curve is symmetrical with respect to an axis, if the figure appears the same when a plane mirror is placed on the axis perpendicular to the plane of the curve. The curve PQ is symmetrical with respect to the origin, and RS is symmetrical with respect to the y-axis. The principal kinds of symmetry arising from the form of the equation are as follows : 28] LOCI AND THEIR EQUATIONS 35 28. Equations in Cartesian Coordinates. (1) If f{Xy y) =/(«, — 2/)>* i^^ locus of the equation fix, y) =0 is symmUrical with respect to the x-axis; i.e. If an equation is not altered when the sign of y is changed, its locus is symmetrical with respect to the x-axis. Let (x', y') be any point on the locus f(x, y) = 0. Then, since f(x, y) =f{x, - y), by hypothesis, f(x',y',)=f(x',-y') = 0. That is, the point (x', — y') is also on the locus. Therefore, since the line x = x' meets the locus in any point (x', y'), it will also meet the locus in the symmetrical point (x', — y'), and the curve is symmetrical with respect to the a;-axis, Ex. Let/(aj, y)=y^-4x, thenf(x, - y) = (- yy - 4 x = y^ - i x. Therefore /(x, y,) =/(x, — y) and the curve ?/2 — 4 x = is symmetrical with respect to the x-axis. (See Ex. 2, § 22. ) (2) Similarly, if fix, y)=f(^ — x, y), the locus of f{x, y) = is symmetrical with respect to the y-axis. Ex. y — cos x = y — cos ( — x) . Therefore the locus ot y = cos x is symmetrical with respect to the y-ajoa. (3) If f(x, y) = ±f(- x, - y), the locus of fix, y) = is sym- metrical ivith respect to the origin. Let (x', ?/') be any point on the locus f(x, y) = 0. Then, since /(», y) ~ ±f( — x, — y) by hypothesis, /(a^',2/')=/(-a^', -.^/)=0. Hence the straight line through the origin and the point (x', y') meets the locus again in the symmetrical point ( — x', — y'). Therefore the curve is symmetrical with respect to the origin. a^ b-^ ~a^ 62 a^ 6'"* = i-^)\{-yy _i, ~ a^ b^ * The sign " = " means " identical with," i.e. the same for all values of x and y, and therefore that the two expressions vanish for the same values of x and y. E.g. (x + y)2=x2+2x2/-f-y2, eosx = cos (-x). 36 LOCI AND THEIR EQUATIONS [28 Therefore the curve — + ^ = 1 is symmetrical with respect to both axes or Jr- and the origin. (See figure, § 34. ) (4) If f(x, y) f= {y, x) the locus off(x, y) = is symmetrical with respect to the line y = x. E.g. a^ -f- ?/^ = 1. (5) If f{x,y)=f{ — y, — x) the locus of f{x, y) = is sym- metrical with respect to the line y = — x. E.g. xy — ±1. Let the student prove propositions (4) and (5). The foregoing conditions of symmetry are both 7iecessary and sufficient; i.e. if either one of the conditions (3), for example, is satisfied, the locus is symmetrical with respect to the origin, other- wise not. The student, however, should examine the opposite propositions independently. The following conditions, (6), (7), (8), are sufficient, but not necessary ; i.e. the opposite propositions are not necessarily true. (6) If an equation contains only even powers of y, its locus is sym- metrical with respect to the x-axis. [From (1).] (7) If an equation contains only even powers of x, its locus is sym- metrical with respect to the y-axis. [From (2).] (8) If an equation contains only even powers of both x and y, its locus is symmetrical with respect to both axes and also with respect to the origin. [From (3).] In an algebraic * equation either one of the following conditions is sufficient, and one or the other is necessary. (9) If all the terms of an algebraic equation are of even degree, or if all the terms are of odd degree, its locus is symmetrical with respect to the origin. [From (3).] Show that (6), (7), (8), and (9) follow from (1), (2), and (3). Show that (6), (7), (8) are necessary conditions of symmetry if the equation * A function in which the variables are involved in no other way than by addition, subtraction, multiplication, division, and root extraction is called an Algebraic Func- tion. All others are called Transcendental Functions. E.g. 3a;2_2a; + 4, x2— aa;?/ + 6r/2, ^,J +n Vxy. are algebraic functions; while a*, sin x, sec-i y, log (x^-j-y) are transcendental functions. (jHj^/l7 29] LOCI AND THEIR EQUATIONS 37 29. Equations in Polar Coordinates. The best way to determine the symmetric properties of loci in polar coordinates is to transform their equations to rectangular co- ordinates, and then apply the tests given in § 28. The following conditions, however, are useful in simple cases. They are sufficient but not necessary, conditions of symmetry. (1) If fid) =f{- 0), or, iffiO) = -/(tt - 6), the locus of p = f{e) is symmetrical with respect to OX. (2) Similarly, if f(e)=f(-7r- 6), or, if f(e) = -f(-0), the locns of p =f{0) is symmetrical with respect to O Y. (3) If f{6) =/(7r + 6), the locus of p =f{6) is symmetrical with respect to O. EXAMPLES In what respects are the loci of the following equations symmetrical ? Af* 4 1. y = x2.V>^ 2. y^x^. y 3. t = ^- "^4. y^ = x.Y 5. y = 7^. 4 6. x^r=:yx^\ 7. y2^x3. ^ 8. y^^x^. 9. 2/2 = a;2. -f 10. y = x\ «^ ^ 11. y'^ = x*. 12. ^ = y^- 13. y'^ = xfi. -^ 14. f = x^. ^' 15. y^ = x^. 16. y"' = x*. 18. y=x^-x'^. Ad. y = x*-x^.20. y = x*-3fi. y — x^ — X. 21. xy = a. 22. x'^y = a. ^^S. ax^ + hy'^ = \.\^ Vr>^«w^ 24. ax2 ^2bxy + cy^ = 1. 25. ax^ + 2bxy-^ ay^ = 1. ^ 26. xy - 2 (X + y) = 1. /^7. x8 + y3 = 1. \> «tMWv ^^28. X* 4- y* = l.-l(]^ \sr^H^ 29. x< = y2 (4 a^ - x2). M 30. x(y + x)2 + a^y = 0S> 31. xV = rt2(x2 + 2/2). yZ2. x^ + y-' = a^. \j^ ^m^A^ 33. x^ -\- y^ = a^. 34. (a - X) 2/2 = (a + x)x2. ^ 35. (a - x) y2 + x^ = 0. 36. ?/ = K2' + 2-*). 37. y=i(2*-2-^). 38. p2_cos2^. 39. p2_sin2^. 40. Point out the symmetric properties of the loci in the last two preceding sets of examples, especially those given in polar coordinates. 41. Show that if an equation is not altered when — x is written in the place of y. and y in the place of x, its locus will show no change in position when the curve is turned about the origin through a right angle in its plane. For an example see No. 7, p. 27 ; also 2 x2 — 3 xy — 2 ?/2 = ?. The locus of x* + a'^xy - y* = is also such a curve. 38 LOCI AND THEIR EQUATIONS [30 To Find the Equation of a Locus, having given its Geo- metric Definition 30. It should be borne in mind that to find the equation of a locus we have merely to find an equation that is satisfied by the coordi- nates of every point on the locus, and not satisfied by the coordinates of any other point. It is not easy to give specific directions which can be applied in all cases, but the following plan will be useful to the beginner, at least in the simpler cases : (1) Choose the system of coordinates best adapted to the locus under consideration, and select a convenient set of axes. (2) Write down the geometric equation which expresses the given geometric definition, or any known geometric property of the locus. (3) Express this geometric equation in terms of the chosen system of coordinates, and simplify the result. The following examples will give a better idea of the method of procedure than any formal rules ; they should be carefully studied : 31. To find the equation of any straight line. Y Let ABC be any straight line meeting the axes in A and B. Let OB = &, let tan XAO= m. Let P(xj y) be any point on the line. Draw PQ parallel to OF, and BE parallel to OX 31] LOCI AND THEIR EQUATIONS 39 Then for the geometric equation we have QP=QR + RP=OB+BR tan PBR. But QP = yy OB = h, BR = x, tan PBR = m. .'. y = mx + b, (1) which is the required equation. For any particular straight line the quantities m and b remain the same, and are therefore called constants. Of these, m, tlie tangent of the angle between the line and the »-axis, is called the Slope of the line, while b is the ^/-intercept. By giving suitable values to the constants 7ii and 6, (1) may be made to represent any straight line whatever, e.g. If 6 = 0, we have ^^^^^^ ^2) for the equation of any line through the origin. Quantities entering into an equation, such as m and //, which remain constant so long as we consider any particular curve, but whose variation causes a change in the position, size, or shai)e of the curve, are called Parameters of the curve.* Moreover, any equation that can be put in the form (1), i.e. y equals some multiple ofx plus a constant, represents a straight line. The general equation of the first degree Ax-\-By + C=0 (3) may be written y=— — x— — , and therefore (3) represents a straight line whose slope is — — C ^ and whose y-intercept is . (See § 43.) Ex. 1. If b varies in (1) while m remains constant, how will the line change position ? If m varies while b remains constant ? If m varies in (2)? Ex. 2. What will be true of the signs of m and b when the line crosses the various quadrants ? * The difference between parameters and coordinates shonld be carefully noted ; also tb(! diffi^roiice in the effect of a vsiriation of the parameters of an eqnation and the variation of the current coordinates. (See § 26.) 40 LOCI AND THEIR EQUATIONS [32 32. To find the equation of a circle referred to any rectangular axes. Let r — radius, and let C(a, h) be the centre. Let P(xj y) be any point on the circle. Then CP = r. [Geometric equation.] But CP'= {x - af + (2/ - hy. [(2), § 7.] ... (pc-a)^^{y-l>)^ = r^ (1) is the required equation. If a = r and 6 = 0, (1) reduces to aj2 + 2/2-2/'ic=0. (2) If a = — r and 6 = 0, (1) becomes x^^y^ + 2rx = Q. (3) y The circle at the right in the figure is the locus of equation (2) ; the circle at the left is the locus of equa- tion (3). When the centre is at the origin, a = 6 = 0, and we have for the simplest equation of the circle in Cartesian coordinates the standard form (§ 16), aj2 4. 2,2 = rK W 53] LOCI AND THEIR EQUATIONS 41 Moreover, any equation of the second degree in which the term in xy is wanting and the coefficients of a^ and tf are equal, can be written in the form of equation (1), and therefore will represent a circle, real or imaginary. For example, the equation a;2 + /-4a; + 62/-3 = may be written in the form (x-2f + {y + ^f=U, which shows that its locus is a circle whose centre is at the point (2, — 3), and whose radius is 4. EXAMPLES 1. What is the form of the equation and the position of the circle, if 6 = i r and a = ? 2. What are the parameters in these equations ? Discuss the effect produced by their variation. Find the centres and radii of the following circles : 5. a;2^ i/2^2x-4i/ = 0. 6. a:2 + 2/2-3x + 5i/ = 0. 7. a;2 + y2_,.6^_4y ^9^0.*^ - 8. 4(a;2 -f 2/2)_12 x + 8y - 23 = 0. 9. x2 + ?/2-|. (3a; + 8«/- 11 =0. 10. 4(x2 + ^z^)- 20x - 32 ?/ + 25 - 0. 11. Find tue general equation of a circle which touches both axes. 33. Polar equations of the circle. It follows from (1), § 8, that the polar equation of the circle whose centre is at the point (a, a) and whose radius is r, is p2 -2 ap cos ((9 - a) + a' - /-^ = 0. (1) If the pole is on the circle, the equation is p = 2rcos(^-«); (2) if the centre is also on the initial line, the equation is p = 2rcos^; (3) if the circle is above the initial and tangent to it at the pole, its equation is p = 2 r sin 0. (4) Ex. 1. Why is (1) of the second degree in p while (2), (3), and (4) are of the first degree ? When is the pole outside, and when inside the circle ? Discuss the effect of the variation of the parameters in these polar equations. Ex. 2. Transform equations (1), (2), (3), (4) to rectangular coordinates. 42 LOCI AND THEIR EQUATIONS [34 34. The Ellipse. The ellipse is the locus of a point which moves so that the sum of its distarices from two fixed points, called foci, is con- stant. Take the line through the foci as the a>axis, and the point midway between the foci as origin. Let 2 a = the sum of the distances from any point on the ellipse to the foci. Let F{cj 0) and F\—c, 0) be the two foci. Let P{x, y) be any point on the locus. Then FP + F^P= 2 a. [Geometric equation.] But FP = ^ (x-cy + y\ and FP = -\/ {x-^cf + y\ Transposing the first radical and squaring [(2), § 7.] 2 a. (1) (oj + cf + 2/' = 4a2+(a;-c)24.2/2-4a V(a;-c)2 + /, or a V (a; — c)^ + y- == a^ — ex. Squaring and transposing again {a^ _ c2) a^ + ay = a^a" - c"). If we put a^ — c^—b^, we get the equation of the ellipse in the standard form, a* 62 (2) 35] LOCI AND THEIR EQUATIONS 43 35. An examination of this equation (2) as to symmetry, limiting values of the variables and intercepts, will give the general form and limits of the curve. (1) Only the square of the variables x and y appear in this equation. Therefore the ellipse is symmetrical with respect to both axes, and also with respect to the origin. [(8), § 28.] Hence every chord passing through O is bisected by 0. For this reason, the point O is called the Centre of the ellipse. Likewise the lines AA^ and BB^ are called the Major Axis and Minor Axis, respectively. (2) When yz=0,x=±a, a>-intercepts. When a; = 0, 2/ = ± 6, ^/-intercepts. Therefore the curve cuts the «-axis a units to the right and a units to the left, the ?/-axis b units above and h units below the origin. (3) Solving the equation (2) for y and x respectively we find h , .„ a r^^ Hence y is imaginary when a; > a, or a; < — a ; and x is imaginary when 2/ > 6, or 2/ < — &. Therefore the curve lies wholly within the rectangle formed by the lines x= ± a. and y= ±b. Also, as either variable increases, the other diminishes. The form of the curve is shown in the figure. Such an examination of an equation is called A Discussion of the Equation. Ex. 1. Transform equation (2), § 34, to polar coordinates and show that p is finite for all values of 6. Ex. 2. Where is the point (h, k) if - + ^-l>0? < ? Ex. 3. Show the relation of the ellipse ^^ + ^ = 1 to the circles x^ + y^ = a* and x'^ + y^ = 62. « ^ / X. 4. Find the axes, coordinates of the foci, and plot the ellipses 44 LOCI AND THEIR EQUATIONS [37 36. The Hyperbola. The Jiyperbola is the locus of a point which moves so that the difference of its distances from two fixed points {foci) is constant. Choose axes as in the case of the ellipse, let 2 a be the constant difference, and show that when b^= c^— a^ the equation of the hyper- bola reduces to the standard form. [See Fig. § 90.] Ex. 1. Discuss equation (1). Ex. 2. Show that the hyperbola (1) lies wholly between the two straight lines ay = ±i bx, and that as x becomes infinite the ordinates of the lines become equal to the ordinates of the hyperbola. These lines are called the Asymptotes of the hyperbola. [See Fig. § 110.] C-- r 4. ': v Ex. 3. Transform equation (1) to polar coordinates, and find the value of p b when ^ = ± tan-i -• Ex. 4. Find the foci, equations of the asymptotes, and trace the curves ^ ^ 16 9 ^ ^ 16 25 ^ ^ 4 16 ^ (4) a;2 - if = a2. (5) tf - x'^ = h\ (6) 4 x2 - ^/2 = 4. 37. The Parabola. Tlie parabola is the locus of a point whose distance from a fixed straight line is equal to its distance from a fixed point. The fixed point is called the Focus; the fixed line is called the Directrix. Take the line through the focus perpendicular to the directrix as the a^axis, and the origin midway between the focus and the direc- trix ; let 2 a denote the distance from the focus to the directrix. [See Fig. § 88.] Then show that the equation of the parabola is 2/2 = 4 ax. (1) Ex. 1. Discuss this equation (1), also 2/2 = — 4 ax and x!^ = ±4 ay. Find the foci, equations of'the directrices, and draw the parabolas (2) 2/2 ^ 4 a;. . (3) ^2 ^ _ 8 X. (4) y^ = Qx. (5) a;2 = 8 2/. (6) x2 = - 10 j/. (7) x2 = - 12 2/. 37] LOCI AND THEIR EQUATIONS 45 EXAMPLES I 1. A moving point is always four times as far from the x-axis as from the i/ -s ^ y-axis. What is the equation of its locus ? A 2. Find the locus of a point which is equidistant from the two points (3, 2) \y^ and ( - 2, 1). Ans. 6 x + 2/ = 4. Y- 3. Find the locus of a point which is equidistant from the points (a, 6) ^ and (c, d). 4. A point moves so that its distance from the point (3, — 4) is always 5. ^^"^ Find the equation of its locus. Does the locus pass through the origin ? Why? Ans. a;2 ^ y2 - 6x -f- 8 y = 0. f- 6. Find the equation of a circle touching both axes and having its centre at that point ( — 3, 3). 6. Find the equation of a circle touching both axes and having a radius equal to 4. "f 7. A point P is two units from a circle with radius 4 and centre at (2, — 6). What is the locus of P? 8. A point moves so that its distance from the origin is twice its distance from the x-axis. What is the equation of its locus ? Ans. x^ — 3 y2 _ q. -r 9. A point moves so that its distance from the x-axis is equal to its dis- tance from the point (2, — 3). Show that the equation of its locus is x2-4x + 6y + 13 = 0. 10. A point P moves so that its distances from the points A (2, 2) and 5 ( — 2, — 2) satisfy the condition AF + PP = 8. Show that the equation of its locus is 3 x2 - 2 x?/ + 3 ?/2 = 32. 11. What is the locus of a point which moves so that (1) the sum, (2) the difference, (3) the product, (4) the quotient of its distances from the axes is constant (a) ? 12. What is the locus of a point which moves so that (1) the sum, (2) the difference, (3) the product, (4) the quotient of the squares of its distances from the axes is constant (a^) ? 13. Find the locus of a point which moves so that the sum of the squares \^ of its distances from the points (a, 0) and (— a, 0) is constant (2 c^). 14. Find the locus of a point which moves so that the sum of the squares of its distances from the three points (5, - 1), (3, 4), (-2, - 3) is always 64. 16. Find the locus of a point which moves so that the difference of the squares of its distances from (a, 0) and (— a, 0) is the constant 2 c^. "^ 16. Find the locus of a point such that the sum of the squares of its distances ^ from the sides of a square is constant. CHAPTER III THE STRAIGHT LINE 38. It was shown in § 31 that the equation of any straight line when expressed in terms of its slope m and ^/-intercept h is an equatiion of the first degree, y = mx + 6 ; and also that the general equation of the first degree, Ax-^By-\-C = % represents a straight line. It is sometimes more convenient, how- ever, to write the equation of the straight line in other forms ; i.e. to express it in terms of some other pair of parameters. 39. To find the equation of the straight line in terms of its inter- cepts on the axes. Y Let A and B be the points in which the straight line meets the axes ; let OA = a, and OB = h. Let P (x, y) be any point on the line. Draw PQ parallel to the ?/-axis, and join and P. Then A OAP + A OBP = A OAB. Hence " bx-\-ay = ab. or a o (1) 40] THE STRAIGHT LINE 47 If Z = - and m = j, the equation may be written lx-{-my = 1. (2) 40. To find the equation of a straight line in terms of the length of the perpendicular from the origin iqyon the line and the angle which that perpendicular makes with the x-axis. Let ON be perpendicular to the straight line AB, and intersect it in R. Let OR =p, and angle XON= a. Let P(x, y) be any point on the line. Then since OQPR is a closed polygon, OR is equal to the sum of the projections of OQ, QP, and PR upon OR. That is, 022 = proj. of OQ + proj. of QP+proj. of PR = OQ cos a + QP sin a + 0. .*. occo^a + ymna=p^ (1) which is the required equation. Let Z X AP = y = 90° -^ a. Then cos « = sin y, sin a = — cos y, and, by substituting in (1), the equation of the line becomes a5 sin -y - 2/ cos -y = p, (2) Since equations (1) and (2) involve the trigonometric functions, sin and cos, ON and AB must be regarded as directed Hues. As in Trigonometry, we will consider the directions of the terminal lines of a and y as the positive directions of these lines. If y = 90° 4- «, as assumed above, then standing at R facing the positive direction of ON, the positive direction of AB is to the lefl; 48 THE STRAIGHT LINE [40 and standing at R facing- the positive direction of ABj the positive direction of ON is /rom AB toivard the right. This will be called the positive side of the line AB. Then in equations (1) and (2) p is positive when taken in the positive direction of ON. Hence when p is positive the origin is on the negative side of the line. U.g. In the equations >y^'^ cos a = sina = — . V2 /. a = 45° and y = 135° for both lines ; but for AB p = 3, for CD p=-S. Hence the two lines are parallel but on opposite sides of 0. Also is on the positive side of CD and on the negative side of AB. Since sin (^ ± 7r) = — sin ^ and cos (^ ± 7r) = — cos 0, if the signs of all the terms in (1), or (2), be changed, the direction of AB, and also of ON, will be changed by ± tt ; and therefore the positive and negative sides of the line will be reversed. That is, the equation of a line may be written so as to make either side of the line positive or negative, just as we choose. E.g. The equation of the line AB, 2 2 may also be written X VSy 2 2 In (1) p : -2, 2. -2, cos ct = sin 7 = -, 2' sni a Vs — cos 7 = . \a = - 60° and y = 30°. 41] THE' STRAIGHT LINE 1 49 In (2) i? = 2, cos a = sin y . •. a = 120° and 7 = 210^ ^, sina = — CO87 =A^. 2 2 Angles and directions corresponding to (1) are denoted by single arrow-heads, those corresponding to (2) by double arrow-heads. The origin is on the positive or negative side of the line according as the equation is written in the form (1) or (2). Ex. Point out the combinations of signs of cos «, sin a, and p when the line crosses the different quadrants. 41. Transformation of the equations of the straight line. In §§ 31, 39, and 40 we have found, by independent methods, the three standard forms of the equation of a straight line involving different pairs of parameters, m and h, a and &, a or y, and p ; viz. : y = mx + b. Slope form, (1) - + ? = 1 , Intercept form, (2) "i • ^.^r.^ Z c Distance, or normal form. (3) i X smy - y cos y = p, ) ' *' ^ ' Any one of these forms of the equation may, however, he deduced from any other. I. From the figure we obtain di- rectly the relations cos a _ b ~ a , sin y m = tan y = — — ^ = cos y sin a and j9 = a cos a = 6 sin a = — 6 cos y = a sin y. Then substituting these values of m in (1), for example, gives cos« sin a x-[-b, and smy ^ cos y ' 50 THE STRAIGHT LINE [41 Whence, since 6 sin a = —h cos y = p, we get a; COS ot H- 2/ sin ct =^, and X sin y — yGOSy=p. Moreover, the general equation of the first degree, Ax + By+C = 0, (4) can be transformed into any one of the three standard forms. II. Solving (4) for y gives (see § 31) A O y= --X--. Slope form. (5) a Jo III. If we transpose and divide by C, (4) may be written — TvH p=l. Intercept form. (6) ^A ~B IV. To reduce the general equation (4) to the distance form. In this case we are to transform (4) so that the sum of the squares of the resulting coefficients of x and y shall be unity. Hence, if we assume the transformed equation to be KAx + KBy + KG = 0, (7) then K^A^ + KV- = cos^ a + sin^ a = 1. j^^ 1 Whence V3M^^ .-. ^ x-^ ^ y= ^ (8) V^^ + 52 VA'-^B' VA'+& is the required equation. Hence, to reduce the general equation (4) to the distance form, trans- pose C and divide by ■\/ A^ + B^. The general equation of the first degree must therefore represent a straight line, since, by transposing .and multiplying by a suitable constant, it can be reduced to any one of the standard forms of the equation of the straight line. {Cf § 31.) 41] THE STRAIGHT LINE 51 V. Values of parameters in terms of A, B, and C. Comparing coefficients in (1) and (5), (2) and (6), (3) and (8), we get a = -^» ^ = -^' m = -^, p= ~^ A' B' B' ^ ^W+B' A B cos a = sin v = — . sin a — — cos v = V^2 + ^ V^- + B' Observe that the values of a and h thus obtained are the same as those found by putting y = 0, then x — in (4) ; also that A b m = — — = , as found above directly from the figure. Then B a sin a, cos a, and p can be found by Trigonometry and the relations obtained from the figure. EXAMPLES 1. When is it impossible to write the equation of a straight line in the intercept form ? in the slope form ? Change the following equations to the standard forms and thus determine their parameters. Also draw the lines : 3. 4?/ = 3 a; + 24. 6. 5x+4«/ = 20. 7. 2x-4y + 9 = 0. 9. 2x + 3y = 0. 11. y = 4. Transform Ax ■]- By ^ C = so that the sum of the three coefficients shall be if; so that the square of the first shall be three times the second ; so that the product of the three shall be twice their sum. 13. Transform 5 a; + 4 y — 20 = so that the sum of the three coefficients shall be 22 ; so that the product of the first and third shall be equal to the second. 14. Transform 3a;-4?/+12 = 0so that the square of the second coefficient shall be equal to twice the third minus four times the first ; so that the product of the three shall be minus three times the last. 16. Transform 5 a; — 2 y — 3 = so that the product of the first and second coefficients minus ten times the third shall be equal to — 40 ; so that the s(iuare of the second plus twice the sum of the first and third shall be equal to 24. 2. x-\-V3y + 10 = 0. 4. y = x-6. 6. 5 X - 12 y = 13. 8. 2x-Sy = i. 10. x-a = 0. 12. Transform Ax + B \}^ 52 THE STRAIGHT LINE [42 42. To jind the polar equation of a straight line. Let N{p, a) be the foot of the perpendicular from upon the given line AB. Let P(p, 6) be any other point on AB. Then Z NOP = {6 - a), and OP cos JV'OP = ON. .•.pcos(e-a)=i>, (1) which is the required equation. EXAMPLES Find the parameters and draw the lines whose equations are 1. p cos {6 - 30°) =2. 2. p cos (d - 60°) = 1. 3. p cos {d + 45°) =3. 4. p cos {6 + 120°) +4 = 0. 5. p cos {e - 120°) +1=0. 6. p cos {6 + 60°) + 5 = 0. < 7. Transform aj cos a + y sin a = j? to polar coordinates. 8. What is the polar equation of a line perpendicular to the initial line ? parallel to the initial line ? •^9. What is the polar equation of any straight line through the pole ? of the initial line ? 10. What locus is represented by sin ^ = ? sin 2 ^ = ? sin 3 ^ = ? •.. sin w5 = ? 11. What is the locus of cos nO = when w = 1, 2, 3, ... ? 12. Find the coordinates of the point of intersection of p cos {6 ± 45°) = 1. J 13. Find the polar equations of the bisectors of the angles between the lines p cos (d - 60°) = 2, and p cos (J9 - 30°) = 2. V V^: ■ 44] THE STRAIGHT LINE 53 43. To find the equation of a straight line passing throufjh a fixed point {xij t/i) in a given direction. Let the line make with the a^axis an angle tan~^ m. Its equation will then be (where b is unknown) _ _ y = mx-\-bj (1) and since the line passes through (x^, ?/i), yi = mxi-\-b. (2) Whence, by subtracting (2) from (1), y-yi = m{ac-i€i). (3) The line given by (3) will pass through the point (xy, y^ for all values of m\ and may be made to represent any line through {xiy ?/i) by giving to m a suitable value. If then we know a line passes through a certain point, we may write its equation in the form (3), and determine the value of m from the other condition the line is made to satisfy. Since m = tan y = ^(§ 40), equation (3) may be written in the form ^^^im^iLzJll^r, (4) cos-y sinY ^ ^ where r is the variable distance from the fixed point (x^, y^ to any point (x, y) on the line. This is a very useful formula. Let the student prove (4) directly from a figure. 44. To find the equation of a straight line ivhich passes through two given points (x^ 2/1) ci'^d (x2, y^. Since the line passes through (ajj, 2/1), its equation will be of the form [(3), § 43] y-y^ = m(x-Xi)', (1) then, since (ajg, ^2) is also on the line, we have y2-yi='m(x2-Xi). (2) Dividing (1) by (2) gives the required equation 2/2 - 2/1 iK2 - «! (3) 54 THE STRAIGHT LINE [45 Equation (3) may also be written X, y, 1 xu 2/1, 1=0. (4) which is obvious, since the area of the triangle formed by (fl?i, 2/1) (x2, 2/2) and any other point (x, y) on the line is zero. EXAMPLES Find the equation of the straight line 1. if 6 = f and 7 = tan-i \. /a. if a = h and p 3. if 7 = 30° and p = 4. 4. if 6 = - 3 and 7 = 150°. 6. if 7 = tan-i 2 and the line passes through (3, — 4). 6. if 7 = tan-i - and the line passes through (—a, h). 7. passing through the pairs of points (2, 3) and (— 6, 1) ; (- 1, 3) and (6, - 7) ; (a, 6) and (a + 6, a- b). 8. Find the equations of the sides of the triangle whose vertices are the points (1, 3), (3, - 5), and (- 1, - 3). 9. Find the equations of the three medians of this triangle, and show that they meet in a point. 10. Find the equation of a line passing through (—1, 4) and having inter- cepts (1) equal in length, (2) equal in length but opposite in sign. ^ 11. What is the equation of the line through (4, — 5) parallel to2x4-32/ = 6? 45. To find the angle between two straight lines whose equations are given. Let AB and A'B' be the given lines. 45] THE STRAIGHT LINE 55 Let be the required angle. Then, using the same notation and the same convention as to direction of the lines as in § 40, = a-a' = y-y'. ~ (i) I. If the equations of the given lines be X cos a-\-y sin a =p and x cos a' -\-y sin a' =i>', cos m, then y' > y and (3) will give tan (— <^) = — tan <^ = tan (tt — <^), instead of tan <^. 46. To find the equations of two lines passing through a given point (xi, yi)j the one parallel, the other perpendicular, to a given line. Let the given line be Ax + By+C = 0. Then the parallel line is Ax-\-By + K==0, [§45, III.] (1) and the perpendicular line is Bx-Ay + K' = 0, [§ 45, III.] (2) where K and K' are constants to be determined. Since both (1) and (2) are to go through (xi, y^), these constants are such that Ax,-^By,-]-K=0 I ^3^ and Bxi — ^?/i + ^ = 0, J i.e. K=-(Ax, + By,) | ,^. and K' = -(Bxi-Ay,).] Therefore, the required equations are, respectively, A(ac-iet) + B{y- y{) = 0, (5) and Bix-xt)-A(y-yi) = 0. (6) If the equation of the given line is in the form y = mx + b, the required equations may be written [(3), § 43, and II, § 45] y-yi = m(i€-xi) (7) and V-yi=^-^(^-oci)» (8) 46] THE STRAIGHT LINE 67 EXAMPLES Find the angles between the following pairs of lines : 1. 3 a; + 4 ?/ = 8 and 7 y - X + 14 = 0. 2. 2a; + 3|/ = 6 and 2y = 3x- 12. ^ 3. x + 4 = 2y and x + Sy = 9. 4. 3 2/ + 12 a; + 16 = and 2 y = 4 a; + 5. ^. 5. --f = 1 and ^-- = 1. ^ a b ah -|L 6. Prove that the points (1, 3), (5, 0), (0, - 4), and (-4, - 1) are the vertices of a parallelogram, and find the angle between its diagonals. Find the equations of the two straight lines 7. passing through the point (2, 3), the one parallel, the other perpendicular, to the line 4 x — 3 y = 6. ^ 8. passing through (4, — 2), the one parallel, the other perpendicular, to the line y = 2 X + 4. J 9. passing through the intersection of4x + y + 6 = and 2x — 3y + 13 = 0, one parallel, the other perpendicular, to the line through the two points (3, 1), and (-1, -2). .^, 10. Find the equation of the perpendicular bisector of the line joining the points (3, -1) and (-2, 1). i 11. Find the equations of the lines perpendicular to the line joining (2, 1) and ( — 3, — 2) at the points which divide it internally and externally in the ratio 2 : 3. 12. What is the equation of a line parallel to 3 x + 4 y = 12 and at a distance 4 from the origin '? 13. Find the point of intersection of two parallel lines. (eP, o^] The vertices of a triangle are (3, 1), (- 2, 3), and (2, - 4): 14. Find the equations of its altitudes and show that they meet in a point. 15. Find the equations of the perpendicular bisectors of its sides, and show that they meet in a point which is equidistant from the three vertices. 16. Find its interior angles. 17. Find the equations of two lines through the origin, each making an angle of 30° with the line 4 x -f ?/ + 4 = 0. / 18. Show that the equations of the two straight lines through a given point (3Ci, J/i) making a given angle with the line y = mx + b are m i- tan , 58 THE STRAIGHT LINE [47 47. To find the perpendicular distance from a given straight line to a given point Px{x^y 2/i)- Let HK be the given line, and let H^K^ be parallel to HK and pass through Pj. Let P^Q be the perpendicular from P^ on HK^ and OR, OE' the perpendiculars from on HKsind H'K'. Let the equation of HK be a; cos a-\-y sin a=p. Then the equation of WK is a; cos a + 2/ sin a = p + RR = p + QPi ; and since this line (2) goes through Pi(x^, y^, fljj cos a + 2/i sin a=p -\- QP^. .'. QP\ = oc\ cos a + 2/1 sin a - p, which is the distance from the line a, p to the point (x^y 2/1). If the equation of the given line be Ax + By-\-C = 0, A . B -0 (1) (2) (3) (4) cos a = V ^^ + B' sina = ■Va' + b' i> = ■Va' + b' [§ 41, v.] and substituting these values in (4) gives Aact + Byi + C which is the distance from line A, B, C to the point (x^, y^. (5) 48] THE STRAIGHT LINE 59 Hence the length of the perpendicular from a given line to a given point is found by substituting the coordinates of the point in the equa- tion of the line reduced to the distance form with all the terms trans- posed to the first member. The expression (5) will be positive or negative accordrng- as Ax^ + Byi -f C is positive or negative (if V^^ + J5^ be positive). If Ax^ + By^ -f C is positive, the point {x^, t/i) is said to be on the positive side of the line Ax + By +(7=0; if Ax^ -f- By^ + C is negative, (^1? 2/i) is said to be on the negative side of the line. If the equation of the line be written so that p is positive, the expression (5) will be found to be positive when Pj and are on opposite sides of the line. {Cf § 40.) Hence the points (x^, y^) and (ajg, 2/2) are on the same side or oppo- site sides of the line Ax-^By-[-C — according as Axi-\- Byi-\-C and Ax2 + By2 + C have the same sign or opposite signs. This proves for the straight line the principles illustrated in §§ 14-20. 48. To find the equations of the bisectors of the angles bettveen the lines Ax -\-By+C=0, or x cos a + y sin « — p = 0, (1) and A'x-\-B'y-\- O = 0, or xcos a' -^-ysina' —2)'=0. (2) Suppose the equations of the lines written so that the origin is on the same side of both lines. Then for any point (x, y) on the bisector of the angle which includes the origin, Dist. from (1) to (x, y) = Dist. from (2) to (x^ y) ; and for any point (a;, y) on the other bisector, Dist. from (1) to {x, y) = — Dist. from (2) to (x, y). Therefore the required equations are [§ 47] V^2 + £2 V^/2 + B/2 • ^ ^ or a;cosa + 2/sina-j» = ± (ajcosa' 4- 2/sina' -^')» (4) Ex. Show that these two lines are perpendicular to each other. [Use (4).] • 60 THE STRAIGHT LINE [49 EXAMPLES Find the following distances : 1. From 3 a; + 4?/ + 10 = to (1, 12), (-3, -9), (3,4). 2. From x-Sy = 7 to (3, 2), (6, 3), (2, -5). 3. From 5a; + 12y = 13 to (3, -2), (-3,2), (4, -7). 4. From ?>(x — «)+ a(y — b) =0 to (—a, — &), (- &, — «), (&, «). 5. From 4(x-3)=3(2/+l) to (6,1), (4, -5), (-7,2). Are the above points on the same or opposite sides of tlie lines ? Find the equations of the bisectors of the angles between the lines 6. Sx + 4y + 12 = and 4x-Sy = 12. 7. 3 a: - 4 ?/ + 5 = and 12 a; + 5 «/ + 14 = 0. 8. 2/ = 2x+5 and x — 2y = 8. ^ 9. y=V3x+3 and ic + V3 y = 9. ,J\ 10. Find the lengths of the altitudes of the triangle whose vertices are (3, 4), (-4,1), and (-1, -5). \ 11. What is the locus of a point which is 3 units distant from the line 2x-42/ = 9? f 49. To find the equation of a straight line passing through the inter- section of two given straight lines. The most obvious method of finding the required equation is to find the coordinates x\ y' of the point of intersection of the two given lines, and then substitute these values in equation (3), § 43. The following method of dealing with this class of problems is, however, sometimes preferable, both on account of its generality and because it saves the labor of solving the two given equations : Let the equations of the two given straight lines be Ax + By-^C=0, (1) and A'x + B'y + C" = 0. (2) The required equation is then written Ax + By + C + \ (A'x + B'y + C) = 0, (3) where A is any constant. V^ 49] THE STRAIGHT LINE 61 Equation (3) is of the first degree, and therefore represents a straight line ; if (x', y') is the point common to (1) and (2), we have Ax' -j-By'+C = and A'x'-}-B'y'-{-C' = 0. /. Ax' + By'+C + X (A'x' + B'y' + C) =0, - - which shows that the point (x', y') is also on (3). Hence (3) is the equation of a straight line passing through the point of intersection of the two given lines. Moreover, equation (3) contains one arbitrary parameter, A, and therefore, by giving a suit- able value to A., the line may be made to satisfy any other given con- dition; it may, for example, be made to pass through any other given point, may be made parallel, or perpendicular to a given line. Hence equation (3) represents, for different values of A., all straight lines through the point of intersection of (1) and (2). The other condition which any particular line is made to satisfy will give an equation for the determination of the value of \. Ex. Find the equation of a straight line passing through the point of inter- section of2a;4-52/-4 = and 4x — 2?/+2 = 0, and perpendicular to the line 2x-4ij = 7. (1) Any line through the intersection is given by 2x-\-5y-4 + \(Ax-2y-\-2)=0, or (2 + 4 \)x + (5 - 2 \) ?/ + (2 X - 4) = 0. (2) Now (2) is perpendicular to (1) if (§ 45, III) 2(2 + 4 X) - 4(5 - 2 X) = ; i.e. if X = 1. .'. 6 X + S y = 2m the required equation. EXAMPLES ''^ 1. Find the equations of the lines joining the points (0,0), (4, 2), (— 1, 3), (_ 3^ — 4) to the point of intersection of the lines 2x + y = 2 and 2 ic — 3y = 6. 2. What is the equation of the straight line passing through the intersection oiix-2y = 4: and 7 a; - 3 y + 21 = 0, and parallel to9x-iy = 0? /^ 3. Find the equations of the two lines passing through the intersection of t — 2y — \ and 2x |-5?/ + 4 = 0, the one parallel, the other perpendicular, to z -f- 2 j^ = 0. 4. Find the equations of the two lines passing through the intersection of 7 x — 5 ?/ = 35 and 8 a; — 3 y + 24 = 0, the one parallel to y = 2 x, the other per- pendicular to 3 a; + 4 2/ = 0. 62 THE STRAIGHT LINE [50 ^ 6. Show that it S = and S' = represent the equations of any two loci with terms all transposed to the first member, and \ denotes an arbitrary constant, then the locus represented by the equation will pass through all the common points of the two given loci. Consider the two cases X = and \ = co . 6. Find the equation of the circle which passes through the origin and the common points of the circles x2 + ?/2 = 25 and a;2 + 2/2 _ 18a: + 20 = 0. V 7. Find the equation of the circle which passes through the common points of a;2 4- ?/2 = 16 and x — y = 4, and (1) passes through the origin, (2) touches the aj-axis. 50. To Jind the equation of a straight line referred to axes inclined at an angle — y, we also have 51] THE STRAIGHT LINE 63 y — b _QIi _ sin QOE _ sin y X OQ sin ORQ sin ( — y) which is the required equation. * Let m= . ^^y =-. ^-^^ (2) Sin (o> — y) sin (t) — cos o> tan y mt- J. *W sin « /ON Then tan v = ,— , (3) ' 1 + wt cos »' ^ ^ and equation (1) becomes y = ma? + b, (4) which in oblique coordinates represents a straight line inclined to the a?-axis at an angle tan-^f ., ^^^^°" ). ^ \l + »*cos«/ 51. Some of the investigations in the preceding sections of this chapter apply to oblique as well as to rectangular axes. Let the student show that this is true of the following equations : - + f=l. [(!),§ 39.] CL y-y, = m{x-x,\ [(3), § 43.] [(3), § 44.] EXAMPLES ON CHAPTER III 1. What are the loci of the following equations ? (1) x'^ + axy = {i. (2) x^-xy'^z=iO. (3) a;3 + ?/8 = o. (4) x^-y^=Q. (5) a2^,2_ 52^2 = 0. (6) a2a;2^.^,2y2 = o. (7) (a;2-l)(y2_4)^o. (8) (ax + 6y)2 = A (9) 2/2-(a;-a)2 = 0. (10) (x- a)2+ (y- 6)2 =0. (11) (x - a)2 - (// - 6)2 = 0. (12) x3-x2y + xy2_y2 = o. (13) /) = asec (^-a). 64 THE STRAIGHT LINE [51 2. Find the equations of the lines which bisect the opposite sides of the quadrilateral (3, 4), (5, 1), (- 3, 4), and (5, - 1). ■^ 3. Find the equations of the lines which go through the origin and trisect that portion of the line Sx — 2y = IS which is intercepted between the axes. 4. Find the equation of the line through (a, b) parallel to the line joining (0, -a) and (fc, 0). 5. Find the equations of the lines which pass through (—2, 1) and cut off equal lengths from the axes. ^ 6. Show that the three lines 2x — y = 4:, x-{-2y = 7, and Sx + y = 11 meet in a point, 7. Show that the three points (1, 3), (—1, 4), and (9, — 1) are on a straight line ; also (3 a, 0), (0, 3 6), and (a, 2 b). / 8. For what value of m will the line y = mx — 4 pass through (4, 2) ? be 2 units distant from the origin ? 9. A line is 3 units distant from and makes an angle of 60° with OX. What is its polar equation ? its rectangular equation ? 10. Find the locus of all points which are equidistant from the two lines Sx-2y = S and Sx-2y + 2 = 0. }/ 11. What is the distance between the parallel lines 3 X + 4 ?/ = 5 and 6x-\-Sy-\-lb = 0? 12. Find the points on the axes which are 4 units from the line x-7y-h21=0. ^ 13. Show that the perpendiculars let fall from any point of 22x — 4y = 15 upon the lines 24:X-\-T y = 20 and 4x — Sy = 2 are equal. Find another line of which this statement is true. 14. Find the perpendicular distance of the point (I, m) from the line through (a, b) perpendicular iolx-{- my = \. * 16. Show that the bisectors of the interior angles of a triangle meet in a point. 16. Find the locus of a point which is equally distant from the lines 5 aj - 3 ?/ = 15 and Zy = 5x^Q. » 17. Show, by the use of (1), § 42, or by transforming (3), § 43, that the polar equation of a line passing through the fixed point (pi, ^i) may be written p cos {d — a) = p\ cos (^1 — a). 18. Show, directly from a figure, or by transforming (3), § 44, that the polar equation of the straight line which passes through the two fixed points Cpi, ^i) and (/02, ^2) is pipo sin (02 — 61) + P2P sin (d — 62) + ppi sin (^1 — 6) =0, 51] THE STRAIGHT LINE 66 ^ 19. Show that the equations n ^ COS ^ + -B sin + - = 0, ^ cot ^ = IT, V P p = kaec{d — a), p = icsc (^- /3), represent straight lines. 20. Show that the equations of the lines passing through (—3, 2) and inclined at an angle of 60° to the line VSy — x = 'S are ic + 3 = and VSy + x + 3 = 2 V3. / 21. Find the equations of the sides of a square of which the points (2, 2) and (—2, 1) are opposite vertices. 22. What are the equations of the sides of a rhombus if two opposite vertices are at the points (— 1, 3) and (5, — 3), and the interior angles at these vertices /e each 60° ? 23. Prove that the equation of the straight line which passes through the point (a cos^ 6, a sin^ d) and is perpendicular to the straight line x sec 6 -{- y esc ^ = a is X cos 6 — y sin 6 = a cos 2 0. 24. Show that the equations of the lines passing through the point (4, 4) and whose distance from the origin is 2 are x{l ± y/1) + 1/(1 T V^) = ^' 25. Find the area of the triangle formed by the lines y + 3a; = 6, y = 2a:-4, ?/ = 4x + 3. 26. Show that the area of the triangle formed by the linea y = m\X + 6i, y = m^x + 62, and « = 0, is 1 (&1 - &2)^ 2 m\ — m2 27. Show that the area of the triangle formed by the lines y = mix + 61, y = m^x + ^2, and y — mspc -f 63 is 1 r(ft.-W + (6.-W' + (fes-ft.)'] . (Use Ex. 26.) 2 L wii — m2 m2 — ms ma — mi J 28. What is the equation of a line passing through the intersection of 3a; — 2?/ + 12 = and x + 4 y = 20, and (a) equally inclined to the axes ? (6) whose slope is — 2 ? 29. The distance of a line from the origin is 6, and it passes through the intersection of 2x + Sy = 6 and 3a; — 6y + 29 = 0. Find its equation. 30. Find the equations of the two lines which pass through the intersection of a; + 2 y = and 2x — y + S = 0, and touch the circle x2 + yi = 9. ee THE STRAIGHT LINE [51 31. Find the equations of the two lines which pass through the intersection of x + Sy + 9 = and 3 x = ?/ + 13, and touch the circle (x + 2)2 +(y- 3)2 = 25. 32. Find the equations of the diagonals of the rectangle whose sides are x-\-2y = 10, x + 2y + 2 = 0, 2x- y = 12, and 2x — y = lQ, without finding the coordinates of its vertices. 33. A circle passes through the common points of a;2 + 2/2 _ 25 and a; - 4 y + 13 = 0, and cuts the avaxis in two coincident points. Find its equation. 34. Show that the locus of a point which moves so that the sum of its dis- tances from the two lines X cos a -{- ysin a=p and x cos cc' + y sin a' = p' is constant and equal to K is the straight line 2 X cos ^ (a + a') +2y sin H« + a') = (p -\- p' -\- K) sec \ (a - a'). Show that the locus is parallel to one of the bisectors of the angles formed by the two given lines. Show also that if the difference of the distances from the two given lines is constant, the locus is a straight line parallel to the other bisector. 35. If p and p' be the perpendiculars from the origin upon the straight lines whose equations are X sec 6 -\- y esc = a and x cos 6 — y sind = a cos 2 0, prove that 4p^ + p'^ = a\ 36. Show that the equation of the line passing through the points (a cos a, 6sin«) and (acosjS, 6sin)3) is bx cos ^ (a + ^) +ay sin ^ (a + j3) = a& cos ^ (a — p). 37. Show that the equation of the line which passes through the points (a sec a, b tan a) and (a sec /3, b tan /3) is bx cos I (a — /3) —ay sin ^ (a + j3) = a& cos | (« + /3). 38. Show that the three straight lines aix + biy + ci = 0, azx + b2y + C2 = 0, a^x -\- bsy + cs = will meet in a point if ai, &i, Ci a2, 62, C2 = 0. 39. Find the determinant expressions for the coordinates of the vertices, and for the area of the triangle formed by the three lines in Ex. 38, and show that the determinant there given is a square factor of the determinant expression for the area of the triangle. y CHAPTER IV TRANSFORMATION OF COORDINATES, OR CHANGE OF AXES 52. The formulae for changing an equation from rectangular to polar coordinates and vice versa have already been found in § 6, and their usefulness amply illustrated. Moreover, the equation of a curve in any system of coordinates is sometimes greatly simplified by referring it to a new set of axes of the same system. Hence, it is also desirable to be able to deduce from the equation of a curve referred to one set of axes its equation referred to another set of axes of the same system. Either of these operations is known as a Transformation of Coordinates, or Change of Axes. The equations, which express the relations between the two sets of coordinates of the same point, and by means of which these opera- tions are performed, are called Formulae of Transformation. 53. To move the origin to the point (7i, k) without changing the direc- tion of the axes. Let OX and F be any pair of axes inclined at an angle w, and let O'X' and O'Y' be a new pair parallel respectively to the old. Let/ P be any point whose coordinates are (x, y) with respect to the original axes, and (x\ y') with respect to the new axes. Then from the figure, OQ=ON-^NQ, and QP=QE-^EP. But OQ=x, NQ=x', ON=^h, QP=y, RP=y', QR^k. .'. oc = gc' -\-h9 y = y'^k. (1) 68 CHANGE OF AXES [54 As these equations are independent of w, they hold for both rec- tangular and oblique coordinates. Hence to find what a given equation becomes when the origin is moved to the point (h, k), the new axes being parallel to the old, substitute x' + h for x and y' -\-k for y. After the substitution is made we can write x and y instead of x' and y' ; so that practically this transformation is effected by simply writing x-{-h in the place of X, and y -{-k in the place of y. 54. To transform from one set of rectangular axes to another, having the same origin. Let {x, y) be the coordinates of any point P referred to the old axes OX and OF; and {x\ y') the coordinates of the same jDoint referred to the new axes OX' and OY'. Let the angle XOX ' = 0. Draw the ordinates MP and J^P, and the lines QJ^ and BN par- allel to OX and OY respectively. Then Z NPQ = 6, OM=x, MP^y, ON=x', NP=7j\ OR=ONGose = x'eose, BN= ONsin e = x' sine, QN= NP sine = y' sinOj QP = NP cos e=y' cos 9. But OM=OR-QN, and MP = EN-j- QP. Therefore and a? = a5'cos6 - 2/'sm0, 1 2/ = iK' sin 6 + y' cos B. j (1) If at the same time the origin he changed to the ptoint (h, k), the re- quired formulm will he a? = a?' cos 6 - y' sin 6 + ^, 2/ =: a?' sin e + y' cos + Tc,] (2) This transformation is clearly obtained by combining the two formulae (1) and (1) of § 53. 54] CHANGE OF AXES 69 EXAMPLES Transform to parallel axes through the point (3, — 2) 1. 2/2_4a; +4y + 16 = 0. 2. 2x^ + 3y^ - 12 x + 12y + 29 = 0. What are the equations of the following loci when referred to parallel axes through the point (a, 6) ? 3. (x - a)2 + (y - 6)2 =z r^. 4. xy - bx - ay -\- ah = a^. 6. 1/2 _ 2 6y + 4 ax = 4 a2 _ 62. e. x^ - y2 _ 2 ax + 2 62/ = 6^ - a^. 7. 62(aj2 _ 2 ax) + a2(?/2 _ 2 &«/) + a2&2 = 0. Transform by turning rectangular axes through an angle of 45°. 8. x2 - 2/2 = flj2. 9. 3x2 - 2 xy + 3 2/2 = 32. (10. p. 45.) 10. 2(y + x) = (y- x)K 11. ax2 + 2 tey + ay^ = 1. 12. x^ + 2/^ = ai 13. (x^ + y'^y = 4a^xY- In 13 change the given equation and the result to polar coordinates. yl!l4. Transform - 4- ^ = 1 by turning the axes through tan-i^ * a b b "f- 15. What does 2x2 — 3x2/ — 2^/2 — 50^2 become when the axes are turned through tan-i — 2 ? 16. If the axes be turned through an angle of 30°, what does the equation 9x2 - 2 V^xy + 112/2 = 4 become ? / 17. Show that the equation 2x2 4-x2/ — 2/^ + 5x — 2/ + 2 = can be reduced to 2 x2 4- xy — y^ = 0, by transforming to parallel axes through a properly chosen point. Through what angle must the axes be turned to cause the term in xy to disap- pear from the following equations ? 18. x2-6xy + y2 = i6. 19. 8x2 + 4xy + 5y2 = 36. 20. (4 2/ - 3 x)2 - 20 X + 110 2/ = 75. 21. ax2 -{.2hxy + by^ = c. 22. Show that the transformation a: = ^, 2/ = x simply changes the scale of the curve, k being the factor of magnification. 28. Compare the curves y = sin x and 2/ = i sin 2 x. 24. Show that the curve y = sin2 x differs only in position and size from y = sin X. CHAPTER V SLOPE, TANGENTS, AND NORMALS 55. It sometimes happens, that the substitution of a particular value for the variable in a fraction causes both numerator and de- nominator to vanish, and the fraction takes the form ^. Thus, z — becomes j- when x = ^. - . 1 — CSC a; 2 The fraction is then said to be indeterminate ; that is, the fraction has no value, or meaning, for this particular value of the variable. Such a fraction, however, usually approaches a defiyiite limit as the variable approaches this particular value as its limit. This limit is the value we then assign to the fraction, because it fits in continu- ously with the other values of the fraction. This definite limit can be found by reducing the given fraction to an equivalent one whose terms do not both vanish when the particular value is substituted for the variable. In all the investigations which follow in this chapter it will be found to be necessary to determine the limit which a ratio approaches when its terms both approach zero. Hence the student should now fix in mind the following definition, viz . : A constant is called the limit of a variable if the difference between the constant and the variable can be made to become and remain as small as ive please. 56. Examples of limiting values of ratios. (1) Let Khe the area of a square whose side is x. Then rlil^L^l = 5. But ^'^^ ^ = i^"^. ^ = Jl% (x) = 0. * The sign " = " in these conditions for a limit should be read " approaches." 70 56] SLOPE, TANGENTS, AND NORMALS 71 (2) Let K be the area of a rectangle with a constant base h and a variab)e altitude x. Then pill^l =5. But ^^!" :^=1'[^^=:6. LliraxJ;,^o ^=^ X ^=^ X (3) Let V be the volume, T the total surface, C the circumference of the base of a right circular cylinder whose altitude is constant and radius variable. Then r^^^^l =^' \^^^^JL^ =2. LlimCJr=o LlimFJr=o , lim T _ lim 2 7rr(r + h) lim 2 (r + /i) 2 /i If S be the convex surface, find ^!^^ — • Llim (x2 - a-2) J^^« 0' lim (x — a)2 _ lim a; — « _a ^-(^x'-a^~^ = o,x + a~ ' Multiplying both numerator and denominator by 1 + y/l — x^ gives lim 1 - Vl - a;-' _ lim x^ lim 1 _1 x = ^2 -x = 0^,^^_^^.^--^^-x = 0Y:^-/==-1' EXAMPLES Find the limits indicated in the following expressions : - lim x^-gg „ lim x^ - a^ , lim (x-a) x = ay^2_a^ x = ay;2_a2 ' x-ax^-ax'^-a'^x + a^ ^ lim 3 a;2 - 6 a; + 3 _ lim x'^ lim 2 x^^ + a; - 1 • x = l2x2-4x + 2' ' ^-^a-VS^^T^^' **• x = co x^^x + 2' lim V4+X-V4-X lim / , , , 9 Hm sin^.j ,Q lim sec x _ ■. ,, lim 1 — cos x _ 1 -g li"i tan x — sin x _/^ ' x-^^°t'dnx~ ' ' x = sin2x ~2* * x = i_cosx ~ ' ,0 lim sinx _ lim tanx _ -, ^m lim sec x — 1 _ 1 ***• X = a; ~ X = y, ~ • ■^*' X = a;2 ~ 2* 16. If V be the volume, T the total surface, ^S* the convex surface, C the circumference of the base of a cone of revolution whose altitude h is constant, oi,«™ *u»* lim T h lim T lim T . lim T « show that .^_ = -, .^ — = Go, .^ — =1, . — = 2. 72 SLOPE, TANGENTS, AND NORMALS [57 57. Definitions. Let two points P and Q be taken on any curve PQMf and let the point Q move along the curve nearer and nearer to P; the limiting position, TT', of the secant PQ when the point Q ap- proaches indefinitely near to P is called the Tangent to the curve at the point P. The straight line PN through the point P, perpendicular to the tangent TT', is called the Normal to the curve at the point P. The Slope, or Gradient, of a curve at any point is the slope of the straight line tangent to the curve at that point. 58. To find the slope of a curve at any point* Y Let P(x,y) and Q(x + Sx, y -h ^y) be two points close together on any curve AB ; then 8x is the difference of the abscissas, Sy the difference of the ordinates of P and Q. Let the secant PQ meet the ic-axis in S, and let the tangent line at P meet the a>axis in T. Draw the ordinates MP, NQ, and draw PR parallel to the a;-axis. Then PR = hx, RQ = Sy. Let the equation of the curve be y^m- (1) Bead Ex. 1, § 59, in connection with this general demonstration. 58] SLOPE, TANGENTS, AND NORMALS 73 Then at the points P and Q we have OM=x, MF = y=f(x), ON=x + ^, NQ = y + 8y=f(x-\-8x). .'.8y=:f(x + Sx)-f(x). _ _ Also tan XSQ = tan RPQ = ^ = ^ . PR &c .•.tanX^Q = |^==£(^±MziiM. (2) oX ox The slope of the tangent TP, which is the slope of the curve at the point P, is the ultimate slope of the secant SPQ when the point Q moves along the curve close up to P; i.e. tan XTP= lim tan XSQ = lim -^ as Q approaches P. ox When the point Q approaches the position of P as a limit, the dif- ferences Bx and 8y simultaneously approach zero as a limit, and the limiting value of the ratio -^ is denoted by -^ ; therefore in the limit ox ax we have The ratio represented by the last member of equation (3) is also a function of x ; and if, x being regarded as fixed, this ratio has a definite limiting value as Sx approaches zero, this limiting value is called the Derived Function, or the Derivative of f(x) with respect to X, and will be denoted hy f'(x), or i)x[/(^)] > i.e. if V=f{ic), then ^| =f\x) = I)^if{x)^. Hence to find the slope at any point of a curve whose equation is in the form y =f(x) we find/'(x), the derivative oi f(x) with respect to Xf and in this substitute the abscissa of the given point. To find the derivative of a function of a;, denoted by /(»), we assign a small increment Sx to ic, producing an increment, denoted by f(x -{- 8x) —/(«), in the function, and then find the limiting value of the ratio Sx 74 SLOPE, TANGENTS, AND NORMALS [59 59. Examples of derivatives and slope of curves. Ex. 1. Find the slope of the curve whose equation is y z=x'^ -\- a. (1) Let P{x, y) and Q{x + 5a;, y + 8y) be any two points close together on the curve ; and let TP be the tangent at P. Then at P, y = x^ -\- a, (2) and at §, y -\- Sy = (x + Sxy + a. (3) Whence (y + 5y)-y ^ (x-j- 8xy + a - (a;^ + a) 5x 5x = tan EPQ. (4) .:^ = 2x-{-8x = tan XSQ. (5) dx When Q approaches P, or as we say, pro- ceeding to the limit 5x = 0, we have (§ 58) dy ^ doc J? X = tan XTP, (6) Hence the slope of the curve at any point is equal to twice the abscissa of the point. At Po, x = Q. .'. PqTq is parallel to the x-axis. At Pi, x = l, .-. tan XTiPi = 1. At Pa, x = l .: tan XP2P2 = 3. At P3, X = — ^, .-. tan XT3P3 = - 1. Ex. 2. Let the equation of the (jiven curve he y In this example we have given /(cc) = x5. Then from the definition of the derivative given in equation (3) of § 58, we have, fir^\- ^i^" /(x + 8x)-f(x) _ lim (x + 8xy — x^ J '-''>- 8x = ^ -Sx = sx = 3^*2 (^ ** + 10 xHx + 10 xHx"^ + 5 x8x^ + 8x^) = 5 a*. That is, the slope of this curve at any point (x, y) on the curve is equal to five times the fourth power of the abscissa of the point. 59] SLOPE, TANGENTS, AND NORMALS 76 Ex. 3. Find the slope of the curve y = — ^ 1 We now have from the definition of a derivative, since f(x) =-, z 1 1 dy _ lim x -\- bx dx~ " lim 5a; = lim a; + 5a; x 8x -1 lim = 5x = 1 a; - (a; + 5a;) ' x(x + 5a;)5x 5a; = a;(a; + 5x) That is, the slope is always negative and varies inversely as the square of the abscissa of the point. Ex. 4. Let y = Vx be the given curve. Then, since f(x) = Vx, we have from the definition dy _ lim f(x + 5x) - /(x) _ lim y/x + 5x — Vx ^^~5x = 5a. -5x = 5a; ' lim 1 ^1 ^^-^v^T5x + VS 2V^ Ex. 5. To find the derivatives of sin x and cos x. * Let 5x = h, for convenience, then will i>,(sin X) = /f (J sin(x + ^) .,-sinx ^ ^iim^ r ^ / ^ |\ sii^i^j i.e. Da.(sina?) =cosa5; (Ex. 13, p. 71.) D.(cos x) = j^% co8(x + A)-cosx ^ Mm ^ ^_ ^.^ / ^ |\ sin|^j 1. e. I>x (cos x) = — sin x. Check the results found in Exs. 2, 3, 4, and 5 by constructing the loci. EXAMPLES Find the slope at the points where x = 0, ± 1, ±2, etc., of the curves whose equations are . / 1. y = x\ * 2. y = x4. 3. x^ = 1. 4. y^ = x^ ^'b. y = x8-4x. / 6. y = x*-20x2 + 64. ^7. y = (^~p . X — £i 8. Find the slope ot y = Va^ + x-^, where x = 0, ± a, oo . 9. Find the slope oi y = Vd^ — x^, where x = 0, ± a, ± la. 10. Find the slope of 10 ?/ = x2 - 3 x - 20, where x = 0, ± 1, ± 4. [§ 22.] V y^ 76 SLOPE, TANGENTS, AND NORMALS [60 General FoRMULiE for Differentiation 60. Tlie derivative of the product, and sum, of two functions. Let ff>{x) and F{x) be any two functions of x. Then (§ 58) ox Introducing <\>{x + 8x)F(x) — {x)F{x)2 = i.e. I>«,[(ie). [(3), § 58.] (3) By an extension of this process it can be shown that I>x[«|>i(a5)<|>2(a5)<|>3(a5) ...] = i'(a5)<|>2(a5)<|)3(cc) ... + 4>2'(iK)<|>i(a5)«|>3(a?) ... + 3'(a?)<|>i(a5)«|>2(a5) ... + .... (4) Or, as a special case of (4), we have, if n is a positive integer, ALX«)]" = D,[(x)(fi(x) -'ton factors] (5) = [(aj)]""V(^) + ••• to n terms] (6) E.g. i>x(sin x)^ = S (sin xyD^isin x) =3 sin^ x cos a;. [Ex. 5, p. 75.] One of the most important results that follows from (7) is I>a,(cc") = nx^-^. (8) In like manner it can easily be shown that I>a.[cf(x)'\ = cf'(x}, where c is a constant, (9) and I>a5[i(a5) + <|>2(a5) + <|>3(a5)+ ...] =«|>i'(a?)+ 3'(ic)+.— (10) Hence, if f(x) is a rational and integral algebraic function of x (§ 63), f'(x) is found by midtiplying the coefficient of each term by the exponent of x in that term and diminishing each exponent by unity. E.g. D^[x* - 2 x3 + 4 x2 - 3 cc + 6] = 4 ^3 - 6x2 + 8 X - 3. 61] SLOPE, TANGENTS, AND NORMALS 77 61. To find the derivative of a function of the type F(x, y) = 0. When we desire to differentiate a function of the type F(Xj y) = 0, we may try first to solve the equation with respect to ?/, so as to put it in the form y = f(x) ; or to solve with respect to x, so as to bring it to the form x=f{y). It is useful, however, to have a rule to meet cases when this process would be inconvenient or impracticable. It will be sufficient for the purpose of this book to illustrate the rule by considering the general equation of the second degree (§ 87). Let F{x, y) = ax' + 2 hxy + hy^ + 2 gx-it-2 fy + c = 0. (1) Let P{x, ?/) and Q{x-{-hx, y-{-hi) be two points close together on the locus of (1) ; then at P and Q, respectively, ax^-\-2hxy + hy^-\-2gx-{-2fy-\-c = 0, (2) a{x + hxY + 2 h(x + 8x) (y + 8y) + b(y + Syf -^2g{x-\-dx)-^2f(y-{.Sy)+c = 0. (3) Subtracting (2) from (3) gives a(2 xBx + Sx2) + 2 h(ySx + xSy + SxSy) + 6(2 ySy + Bf) + 2 g8x-{-2 fhy = 0. (4) Whence ^^ - - ^ aa^ + 2 % + 2^ + a3a^ + /% wnence ^^- 2 hx + 2 by-\-2f+b8y -hhSx ^^A In the limit when &c and Sy approach zero, we have dy__ ax^^hy + g ,„. dic~ hic + hy+f' ^^^ Now apply to (1) the rule deduced in § 60 and differentiate first with respect to x regarding y as constant; then differentiate with respect to y regarding x as constant. Denoting these partial deriva- tives respectively by FJ(xy y) and FJix, y), we thus obtain FJ(x, y)=2{ax + hy + g), (7) and FJ{x, y) = 2 (hx + by +.0- (8) , dy_ _ F^'Juo^ V ) ^ _ ax-{ hy + g .g. * dx~ Fy'ix, y) hx + by^f' ^ ^ It can be proved that this formula (9) expresses the rule for differ- entiating any function of the type F{Xy y) = 0. 78 SLOPE, TANGENTS, AND NORMALS [62 Tangents and Normals 62. To find the equations of the tangent, and the normal at any point (x', y') of a curve. dv' For the tangent, m = /-,. (§ 58.) aoc dx' For the normal, m = — t-,- (§ 57 and § 45.) dy' The primes in ~-^ denote that the coordinates x', y' of the point of contact are to be substituted in the derivative of the equation. Since both lines pass through the point (x', y'), the equation of the tangent is (§ 46) y-y'=^,(^-^')', (1) and the equation of the normal is V-y' = -^, {oc-oc'). (2) CoR. If the axes are oblique, dy sin -y zo ^a \ doc sm C« - 7) ^' ^ Hence equation (1) holds also for oblique axes.* EXAMPLES ON CHAPTER V Find the equations of the tangent and the normal to the curve. 1. x2 + ?/2 Z3 25 at (3, 4). 2. x2 + ?/2 = 169 at (- 12, 5). 3. ?/2 = 8xat(2, 4), (8, 8). 4. 6 ?/ + a;2 = 0, at (6, -6). 5. y = x^-A:X2X (2, 0), (- 1, 3). 6. y^ = a;2, at (- 8, 4). 7. x2 + ?/2 — 4 aj + 6 y = at the points where x = 0. 8. x2 + 1/2 + 4 X — 6 y = 12 at the points where x = 2, x = — 6. 9. x2 4- ^/2 — 8 X — 4 ?/ + 15 = at the points where x = 3, x = 5. 10. x2 + ?/2 _ 16 X - 8 2/ 4- 55 = at the points where x = 3, x = 5. * The theory of this chapter proves what has hitherto been assumed (see note on logic of plotting, § 21), viz., that loci of equations are usually smooth curves without sudden changes in slope or curvature. For, since the slope of a curve f{x, ?/) = at any point (x, y) is a function of x and y, a small change in x and y will ordinarily produce only a small change in the slope. > Ans. x' yi Ans. y' X' • Ans. 3x y ^1 2x' 2y' Ans. 3 X 2y , x' 2/' Ans. ^ + 2^=1. 2x' 2y> Ans. xx' + 2/2/' = 1. 62] SLOPE, TANGENTS, AND NORMALS 79 Find the equation of the tangent to each of the following curves at the point (x', y') : 11. y = x^. ^ 12. 2/2 = X. 13. y = x^. ^^ > 14. y^ = xK 15. xy = l. ^ 16. x2 + 2/2 = 1. 17. x2- 2/2 = 1. ^ 18. x^ + y^ = 1. ^ws. xx'2 + yy'2 = 1. 19. -, + ^ = 1. 20. x« + r = l- \ii!}' 21. What are the equations of the tangents to 16, 17, 18, 20 at the point (1, 0) ; and to 16, 18, 20 at the point (0, 1) ? Find the equation of the tangent to ' 22. y^ = ix-S x2, at the point (1, 1). 23. 10 2/ = (x + 1)2 at the point where a; = 9. (Ex. 11, p. 27.) 24. 4 (x + l) = (y- 2)2 at the point where x = 3. (Ex. 11, p. 27.) 25. (x - 8)2 + (2/ - 2)2 = 25 at the points where x = 4. 26. x(x2 4- 2/2) = a(ic2 — y^) at the point where x = 0, and ± a. 27. Find the equation of the tangent to [-] 4-(-) =2, and show that at the point (a, b) it is the same for all values of n. Cf^'^-JS. Show that the curve a;* + 2/^ = a^ becomes steeper as it approaches the 2/-axis, and is tangent to the axes at the points (± a, 0) and (0, ± a). 29. Let y=f{x) and y = F(x) be two curves intersecting in the point (xi, 2/1) » ^^^ l6t be the angle at which they intersect. Show that tan0= /^CxO-F^(xO, l+/'(Xi).i^'(Xi) What is the condition that the two curves shall meet at right angles? be tangent to each other ? [The angle at which two curves intersect is the angle between their tangents at the point of intersection of the curves.] 80 SLOPE, TANGENTS, AND NORMALS [62 30. Find the angle of intersection between the parabolas y'^ = iax and x^ = 4 ay. 31. Show that the confocal parabolas y^ = 4 a{x -h a) and ^2 _ _ 4 ^(^ — b) intersect at right angles. 32. At what angle do the rectangular hyperbolas y.^ _ yi — gp. aj^(j xy = b intersect ? Draw several sets of these curves by assigning different values to a and b. 33. Find the angle at which the circle x^ + y^ + 2x = 12 intersects the parab- ola y^ = 9x. ^ 34. Find the angle of intersection between x^ -\- y^ = 25 and 4 ?/2 = 9 x. >^ 35. Find the equations of the tangent and the normal to the parabola y^ = 'ix at the point (4, 4). Also find the angle at which the normal meets the curve at its other point of intersection with the curve. 36. Find the derivative of the quotient of two functions. f(x) Xet y = -'^ ^ , and write h in the place of 5x. (1) (p(x) Then ^ = /"i /r/(^±^_/Ml^a. (2) dx h = ^\l(x)f(x + h) -/(x)0(x + h) l ,3^ ^-OL h(f>(x)(/>{x + h) J Introducing {x)f{x) in the numerator gives [x + h)- (x) d^ h — ] dx A • (4) (ic)]2 Find the derivatives of the following functions : o- ^ - « Ans. _2A_. 38. ^^ + ^. Ans. " ^ x + a (x + a)2 x + l (x + 1)2 39. '-±^. 40. -^ ^. ^ns. /^ ■ ^j a-2bx ^2 __^!L_. (a - &x)2 ■ (1 -I- x)« 62] SLOPE, TANGENTS, AND NORMALS 81 43. Show that formula (7), § 60, holds (a) when n is a negative integer, and (6) when « is a rational fraction. [To prove (a) use the formula in Ex. 36 ; for (6) use (9), § 61.] ^ 48. Show that I>a.(taii a?) = sec^ a?. Let tf = tan a; = ^^^. Then from (5), Ex. 36, we get cosx dy _ cos a; • Z>x(sin x) — sin x • Z)a;(cos x) dx cos2 JC But Z>a;(sin x) = cos x, and i)x(cos x) = — sin x. [Ex. 5, p. 75.] . dy cos2 X + sin2 x 1 dx cos2 X Prove the following formulae : = sec2 X. f- 49. l>a.(cot 05)= -csc^o?. ^ 60. I>a,(sec ») = scc'a? tan a?. sj^ 51. l>a. (CSC a?) = — CSC a; cot x, ^ 52. Z>a.(sin oc cos a?) = cos 2 x. Find the derivatives of the following functions : / 63. cos^ X. 54. sin X — 4 sin^ x. Ans. cos^ x. 65. tanx-x. se. 3tanx + tan3x. 67. X sin X + cos x. 53^ ^^^3 ^ _ 3 ^os x. Ans. 3 sin-3 x. 60. (rtx2 -}- 6)3. ^ws. 6 ax{ax'^ + &)2. 59. sec* X — tan2 x. 61. (x2 + a)(x2 + 6) 63. ^2_^2 62. (a4-x3)(6 + 3x2). a^ + a:2* 64. tan2x ^ ^ ^"^^^"^^•- ^ns. 2sec22x. 66. (« + x)V^^. l-2sin2x ^ . g 66. cot2x = Kcotx-tanx). Ans. -2csc-2x. a- 68. (2x3+3)2(1-3x2)3. Vl4-x2 70. sec x + cosx. '0S2 X X3 *^^' (1 + jca)2 * 72. 2 X sin x + (2 - x2) cos x. ^ws. x2 s. x. 73 2 x2 — 1 sin" .X COS"* X xv'l + x2 * cos'"x sin^x V:v,v^' y CHAPTER VI THEORY OF EQUATIONS, QUADRATURE, AND MAXIMA AND MINIMA W\,\ THEORY OF EQUATIONS >J 63. An expression of the form aa;" + &aj"-i 4-ca;'^-2 _| y-kx-^-l, (1) where n is a finite positive integer and the coefficients a, 6, c, • • • A:, I do not contain x^ is called a Rational and Integral Algebraic Function of X of the nth degree ; and ax"" + 6a;"-^+ ca;"-2 H \-'kx-\-l = ^ (2) is called the General Equation of the nth degree. This is the kind of equation we shall consider in this section. If we divide the left side of equation (2) by a, the coefficient of a;", we shall obtain the genefal equation of the nth degree in the standard form, X- -\-p,x^-' -{-p^^-' + ... +p^_^x +p^ = 0, (3) where Pi, P2, " • Pn-i^ Pn do not contain x, but are otherwise unre- stricted. As will be seen hereafter, some of the properties of equations can be stated more concisely when the equation is in the standard form. In this section the symbols /(ic), /i(a;), (l){x), pi 4-^3- (1) Then we wish to calculate the numerical value of /(a) =po«' +p^a^ -\-p2a +ps. (2) This result is most easily obtained as follows : Multiply Pq by a and add to pi, this gives poa -{-pi ; Multiply this by a and add to j72, this gives Pffi^ -\- p^a -\- P2', Multiply this by a and add topg, this gives Poa^ +i>ia^ +i?2« +pt' The process may be arranged in the following way : Po Pi P2 Ps Po« Poa^-\-Pia poa^+pia^+p^ Po i>oa+i>i Poa^-hl)ia-\-p2 Poa^ + Pia^ + p^a -\- ps. 84 THEORY OF EQUATIONS [66 We may proceed in the same way, whatever the degree of /(x). Ex. Find the numerical value of /(3) if /(a;) =2 X*- 7x3 + 13 a^ _ 16. 2 -7 13 -16 6 -3 -9 12 V -1 -3 4 -4 •• /('^) = -4. is process is called Synthetic 1 Substitution. 66. To find the remainder and the quotient when f(x) is divided by X — a, where a is any constant. Divide /(a?) by a; — a until the remainder no longer contains x. Let fj>{x) denote the quotient and R the remainder. We then have the identical equation f{x) = (x) =P(fl^-h (poa -\-p;)x 4- (Poa^ +i>ia +P2), and R = p^a^ -f- p^a"^ 4- p^ci + p^. By comparing these expressions with the results found in § 65 67] THEORY OF EQUATIONS 86 we see that R and the coefficients in <^(cc) are the same as the sums obtained by synthetic substitution. Ex. Find 0(x) and B when 3 x^ — 2 x* — 16 x^ - x + 7 is divided by (x + 2). 3 _2 -10 -1 +7 -^ _ -6+100 0+2 3 -8 U -I +9 Thus 0(x) = 3 X* - 8 x3 - 1, and i? = 9. . •. 3 x5 - 2 x* - 16 x3 - X + 7 = (x + 2) (3 x* - 8 x^ - 1) + 9. This process can be applied to any function of any degree, and is a particular case of Synthetic Division. (See Todhunter's Algebra, Chap. LVIII.) 67. An equation of the nth degree has n roots, real or imaginary. Let the equation be fix) = a;" +piaj"-^ +^2^""^ + ••• + Pn = 0. (1) Let «! be one root* of the equation /(ic) = 0, then/(.c) is divisible by(a;-aO. (^66.) .'. f{x) = {x-a,)f{x), (2) where f{x) is an integral function of x of degree (w — 1). In like manner if ag is a root of /i(a;), then f{x) = {x-a,)f^{x\ (3) where /2(a;) is an integral function of x of degree {n — 2). Proceeding in this way we shall find n factors of the form {x — a^), and we have finally, f(x) = (qc — ay) ipc - at) (a? - as) ••• (» - a«) = 0. (4) It is now clear that ai, as, otg . . . a„ are roots of the equation f{x) =0; and as no other value of x will make fix) vanish, the .equation can have no other roots. The factors of /(ic) need not all be different from one another; thus we may have * We here assume the fundamental theorem that every equation has one root, real or imaginary. Proofs of this theorem have been given by Argand, Cauchy, Clifford, and others, but they are too difficult to be included in this book. The student, however, is already familiar with the fact that every equation of the first degree has one root; that every equation of the second degree has two roots, real or imaginary ; and it will be shown in § 71 that every equation of an odd degree has oive real root. 86 THEORY OF EQUATIOISTS [67 f(Qc) = (Qc-a\)p(pc-a2Yidc-azy-", " (5) where p -\-q + r -\- • • • = 7i. In this case f{x) has p roots each a^, q roots each ag, etc., the whole number of roots being p + q + r -\- ••• =n. Therefore the graph of f{x) will cut the x'-axis in n points, which may be real, coincident, or imagmary: and the real roots are its i^ntercepts. f ( Hence the real roots of a function may be found exactly or approxi- mately by constructing its graph. EXAMPLES 1. Divide 2 a;^ - 6 a:* - 5 ic2 + 10 x + 18 by x - 3. Find the other roots of the following equations : ' 2. Two roots of x* - 12 x^ + 49 x2 - 78 x + 40 rr are 1 and 5. 3. One root of x^ - 16 x2 + 20 x + 112 = is - 2. 4. Two roots of x* + 8 x^ - 22 x2 - 16 x + 40 = are 2 and - 10. 5. Two roots of x* - 12 x^ + 48 x2 - 68 x + 15 = are 5 and 3. 6. Three roots of 6 x^ + 11 x* - 21 x^ + 7 x^ + 15 x - 18 =^ are ± 1 and - 3. Find graphically the exact or approximate roots of 7. x3-2x2-llx+ 12 = 0. 8. x4 - 8 x3 + 14 x2 + 8 X - 15 = 0. 9. x* - 2 x3 - 13 x2 - 14 X + 24 r= 0. 10. x3 - 8 x2 - 28 X + 80 = 0. 11 . 6 x3 - 13 x2 - 21 X + 18 = 0. 12. 8x^-18 x2- 71 x + 60 = 0. 13. x*-6x3-5x2 + 56x-30 = 0. Form the equations whose roots are 14. 1,3,-5. 15. -2, 3, -4, 6. 16. h - I, f. 17. ± 1, ± 4. 18. 0, 1, -4, 5. 19. ± V2, ± V3. 20. 0, - 2, ± V^^. 21. 3, 5 ± V^- 22. 4 ± V3, - 1 ± v/6- 23. 1, - 2, 3, - 4, 5. 24. 0, 2 ± V^^, - 3 ± y/Q. 25. 0, 0, \, - f, 1 ± V2. 26. 1 ± V^=^, - 2 ± V^. 27. - 3, 2 ± V-Z, - 3 ± V^^. 68] THEORY OF EQUATIONS 87 68. Relations betiveen the roots and the coefficients of an equation. If there are two roots, a^ and ag, we have (§ 67) x^-hPiX-\-p2 =(x- ai)(x — a^ = x^— (ai + a^^x + aya^. ~~ (1) .-. «! 4- ttg = — pi, a^a^ =p2. If there are three roots a^, a^^ and a^, we have yf' + pior H- p^ +p^={x— ay){x — a^{x — a^) = a^ — (tti + tta 4- ag)^.-^ 4- (aiCtg + «2 «3 + cisai)x — a^a^a.^. (2) .-. Oi 4- ttg + ttg = — j9i, aia2 + «2«3 4- ^s*^! = i^2j «i«2a3 = — i>3- In like manner if the equation is of the nth degree and therefore has n roots a^, ag ••• a,. ••• a„, then a;" 4-i)i^"-' +i>2^"-' 4- ••• H-p.a;''-'- 4- ••• +i)„ = {x — ay){x — as) ••• {x — a,) ••• (« — a„) (3) = a;** - S,x^-^ 4- /Saa)"-^ + ( - 1)'->S'X"*" ±-+(-ir^n, (4) where S,. is the sum of all the products of aj, a2, •" a^"- a^ taken r together. Equating the coefficients of the same powers of x on the two sides of the identity (4) gives Sl = -Pu S2-=P2, Sr=(-lVPr9 Sn = i- ^)^Pn = «l«2 ••• «r •.• «n. //* Pn = 0, one rooi is zero ; if p^ = p„_i = 0, tivo roots are zero ; if Pn = Pn-i = • • • Pn-r = 0, r + 1 roots are zero. EXAMPLES Find the other roots of the following equations : 1. Two roots of a;3 + a;2 — 4 X — 4 =0 are 2 and — 1. 2. Two roots of a;^ _ 4 a;2 _ 3 a; + 12 = are 4 and ^3. 3. Two roots of ic^ - 13 a; + 12 = are 1 and 3. 4. Three roots of x* - 10 x* + 35 a;^ - 50 x 4 24 = are 1 , 2, and 3. 5. One root of x» - 6 a:^ ^ 12 x^ = is 3 - V - 3. 6. Two roots of 6 ar* - 7 a:^ _ 14 ^2 .{- 15 ^ = are 1 and f . 7. Two roots of 4 xf^ - 5 x* + 2 x^ 4 6 x2 = are 1 ± V^HT 88 THEORY OF EQUATIONS [69 69. Tlie first term of f (x) can he made to exceed the sum of all the other terms by giving to x a value sufficiently great. Let fix) = ii^x" +pi.c'*-^ +^2^*'"^ H VPn, and let k be the greatest of the coefficients ; then \i^ _ p^x'Xx - 1) p^\x - 1) ^ Po . . . ~ k{x''-l) -^ kx'' k^ ^' Now %^(x — l) can be made as great as we please by sufficiently k increasing x, which gives the proposition. 70. An even number, or an odd number, of real roots of f(x) = lie between a and b according as f(a) and f(b) have the same sign, or opposite signs. The two points A[a, /(a)] and B[b, /(6)] are on the same side, or on opposite sides, of the a>axis according as /(a) and f(b) have the same sign, or opposite signs. Therefore, since the graph of f{x) is a continuous curve (§ 64), in passing from ^ to B along the graph the ic-axis will be crossed an even number, or an odd number, of times according as f(a) and f(b) have the same sign, or opposite signs. This proves the proposition. (An even number includes the case of ?io roots.) ■E.g. If /(x) =x^-Zx + \, then /(I) = - 1 and/(2) = 3. .-. At least one real root ofa;^ — 3x + l=:0 hes between 1 and 2. 71. An equation of an odd degree has at least one real root. Let the given equation be f{x) = x^--^^ +Pia^" 4-JP2aj'"-' + • • • +i^2n+i = 0. Let a be a positive value of x sufficiently large to make the first term of f{a) greater than the sum of all the other terms (§ 69). Then the sign of f(a) will be the same as the sign of a-"+^, i.e. the same as the sign of a. Hence if a be sufficiently great, /(a) is positive, /(O) =_P2n+ij and f( — a) is negative. Therefore in all cases there is one real root, which is positive or negative according as p2n+i is negative oi positive (§ 70). 78] THEORY OF EQUATIONS 89 Hence the graph of a fiinctioii of an odd degree in the standard form extends to infinity in the first and third quadrants. 72. An equation of an even degree in the standard form with the last term negative has at least tivo real roots with opposite signs. Let the given equation be ~ f(x) = x'- +i>iar'"-' +i>2a^-'"-' + ••• + P'm = 0. If a is taken sufficiently great, f(a) will have the same sign as a'^"(§ 69), which is positive for both positive and negative values of a; that is, /(a) and/(— a) will both be positive, while /(O) = 2>2n; which by hypothesis is negative. Therefore there is at least one real root between and a, and another between and — a (§ 70). Tlie graph of a function of an even degree in the standard form ex- tends to infinity in the first and second quadrants. 73. To find approximately the real roots of f{x) = (). Plot the graph of f{x) and thus find the pairs of numbers, usually consecutive integers, between each of which one root lies. Suppose /(a) = CA, a positive number ; and f{a + 1) = DB, a negative number. Then there is at least one real root (§ 70) between a and a -h 1. Draw the chord AB cutting the a;-axis in E ; draw BF parallel to the a?-axis meeting AC produced in F. 90 TIIP:0RY of equations [74 Then, if there is only 07ie root between a and a + 1, it is approxi- mately equal to OE ; if the graph were a straight line, it would be exactly equal to OE. Since the triangles ACE and AFB are similar, and FB = 1, ^^^ FB^CA ^ CA ^ f(a) .^. FA CA + BD f(a) - f(a + 1) If we use numerical values of f{d) and f(a -f 1), we shall then have for all cases OE = a + f~^^ * (2) Ex. Find the roots of x^ - 29 a; + 42 = 0. Here / (4) = — 10 and / (5) = 22. Hence there is a root between 4 and 5. Substituting in (2) gives OE = 4 + — — — = 4.4 -. " ^ ^ ^ 10 + 22 Then /(4.4) = - .416 and /(4.5) = 2.625. Hence the root lies between 4.4 and 4.5. When the root is greater than OE, as in the diagram and also in this example, it is better to try the figure next greater than that given by the quotient. The next figure of the root may now be approximated in the same way. Thus f(iA)x.l ^-OilG^ Q^ gj ^^-^ ^ /(4.4)+/(4.5) 3.041 .*. The approximate root is 4.41. The exact root is (3 + ^^2). EXAMPLES Calculate to two places of decimals the real roots of the equations 1. a;3 - 3 a: - 1 = 0. 6. x* - 12 x + 7 = 0. 2. ccS - 7 ic + 7 = 0. 7. a:* - 5 a;3 + 2 x2 - 13 a; + 55 = 0. Z. x^ + 2x'^ -Sx-9 = 0. 8. a;3-3x2-2a; + 5 = 0. 4. x3 + 2 x2 - 4 a; - 43 = 0. 9. a;^ - 81 a; + 40 zir 0. 5. a:3 - 15 X + 21 = 0. 10. x^ - 55 x2 - 30 x + 400 = 0. 74. In any equation with real coefficients imaginary roots occur in pairs. I. Let f(x) = be an equation with real coefficients having r real roots and the other roots imaginary. Then f{x) = (x- a,) (x - a,) ... (x - a,)(x) = 0, (§ 67) (1) * The student should compare this method with Horner's Method of Approximation found in almost any complete algebra. 74] THEORY OF EQUATIONS 91 where (x) is a function with real coefficients whose roots are all the imaginary roots of f{x), and no others. Hence axis in Q, and the curves in P and P\ We will call P and P' corresponding points. ~ ~ Then, if OQ = a, we have by § 58 qp =f(a) = slope ofLMat P. Hence the D. C. is a curve such that its ordinate at any point is the slope of the integral curve at the corresponding point. Let A, B, C, D be the points on LM where the slope, i.e. f\x), is zero ; then the ordinates of the corresponding points A\ B', C, D' on L'M' are zero. Hence A', B', C, D' are the intersections of V3r with the cc-Sixis. Between A and B the slope of LM is positive, between B and C negative, etc. Therefore, between A' and B' the curve L'M' is above the a>axis between B' and C below, etc. It will be convenient to call such points as A, B, C, D, Elbows of the curve. Then the abscissas of the elbows of the graph of f(x) are the roots of /'(«), and may therefore be found by plotting the D. C. or by solving the equation f'(x) = 0. Since / (x) is of degree {n — 1), (§ 78) the graph of f(x) cannot have more than (n — 1) elbows. If f(x) is of an odd degree, its graph will have an even number of elbows (including no elbows), and therefore f(x) will have at least one real root. (Cy. § 71.) If the roots of /' (x) are all imaginary, the graph of f(x) will have no elbows. If two roots of f'(x) are equal, its graph will touch the a^axis, as at D', and the two corresponding elbows of the integral curve will coincide as shown at D. Hence the slope of LM has the same sign on both sides of D. The integral curve therefore changes the direc- tion of its curvature at D, and crosses its own tangent, which it cuts in three coincident points. Such a point is called a Point of Inflection. Ex. Find the coordinates of the elbows of the following loci : 1. y = x^-12x. 2. 2/ = 2x8 -16x2 + 24x4-5. 3. ?/ = x3 - 6 x'-^ + 32. 4. y = 3x*- 20x8+ 18x2+ 108x. 6. y = 3x6 - 20 x^^ + 10. 6. y = 3 X* - 8 x8 - 66 x2 + 144x. 98 THEORY OF EQUATIONS [80 Equal Roots 80. Rolle's Theorem. At least one real root of the equation f(x)=0 (1) lies between any two consecutive real roots of f(x) = 0. ■ (2) For there is at least one elbow of the integral curve, LM (§ 79), between any two consecutive intersections of it with the a;-axis. Conversely, LM cannot meet the ic-axis more than once between any two of its consecutive elbows. Therefore, at most one real root of (2) lies between any two con- secutive real roots of (1). That is, the real roots of (1) separate those of (2). If by a continuous modification of the form oif{x) — for example, by the addition or subtraction of a constant (§ 74) — two roots are made equal, the root of f{x) lying between them must approach the same value. Hence a double root of (2) is also a root of (1). In general, if f{x) has an r-fold root, such a root being regarded as due to the coalescence of r distinct roots, then will /' {x) have an (r — l)-fold root due to the coalescence of the (r — 1) intervening roots. That is, if f{x) has r roots each equal to o, f{x) will have (r — 1) roots each equal to a. Then, by the application of Rolle's theorem to f'(x) and f'\x), f"{x) and f"'(x), and so on, if f(x) = {x- aycf>(x), we have . f'(x) = (x — a)''~Vi(^)j f"(x)=^(x-ay-',(x), (3) j(r-i)(^) = (a: - a)cf>,_,(x). Conversely, if r roots of /'(a?) coalesce and become equal to a, the corresponding r elbows of the integral curve LM will coalesce ; then, if a is a root of f(x), this r-fold elbow will rest on the «-axis and give an (r -j- l)-f old root of f(x). Hence, if /-='(«) =/"-''(«) =/"-''(«) = •••/;(«) =/(a) = 0, 80] THEORY OF EQUATIONS 99 and a is a single root of p''~\x), then a is a double root of P''~^\x), a triple root oi p''~^\x)y ••• an (r — l)-fold root of /'(aj), and an r-fold root oif{x). This suggests an easy method of finding real multiple roots of an equation, when the roots are all equal except one or two. ^__ZI E.g. it /(ic) = x5-5a^ + 40a;2-80x + 48 = 0, we have f'{x) = 5 a;* - 20 x^ + 80 a: - 80, /"(a;) = 20x3 -60x2 4- 80, /'"(x) = 60x2- 120 X. The roots of 60 x^ - 120 x = are and 2. Since/'"(2) = /"(2) = /'(2)=/(2)= 0, 2 isafourfold root of/(x) = 0. Hence all its roots are 2, 2, 2, 2, — 3. Moreover, equations (3) are true whether a is real or imaginary. For suppose f{x) has an r-fold root equal to a, then, whether a is real or imaginary, we have {%^^ and § 67) f{x) = (x-aY{x). . (4) In this case the given function f(x) is expressed as the product of two distinct functions of x, viz. {x — aj and (x) • D,{x - ay. (5) But D,{x-ay=r{x-ay-^ • DXx-a) = r{x-ay-^ . [(7), § 60] (6) .-. fix) = {x- ay(x) (7) = (a, _ ay-'[{x - a)4>\x) + r<^(x)] (8) = {x-ay-',{x). (9) That is, if a is an r-fold root oi f{x), then it is also an (r— l)-fold root of fix), whether a is real or imaginary. In like manner if f{x) also has a g-fold root equal to h, and an s-fold root equal to c, and so on, then f{x) = {x-ay{x-hyix-cy ... <^(^); (10) and f{x) = {x-ay-\x-hy-\x-cy'^ ... 4>,{x). (11) ... (^x-ay-\x-hy-\x-cy-^'^* isthe G. C. D of /{x) and/'(x). 100 THEORY OF EQUATION'S [80 Hence the multiple roots of an equation f{x) = 0, if there are any, can be detected by finding the G. C. D. of f(x) and f'(x) by the usual algebraic process. Likewise the common roots of any two functions can be obtained by finding the G. C. D. of the two functions, and then finding the roots of this G. C. D. Ex. If f(x) = x^ + x^-lSx^-x'^-\-^8x-S6=0, then /'(x) = 5x* + 4x3- 39x2-2£c + 48. The G. C. D. oif(x) and/'(ic) will be found to be x^ + X - 6 = (x - 2)(x + S). .'. f{x)={x - 2)2(x + 3)2(x -1) = 0, and the roots are 2, 2, — 3, — 3, 1. EXAMPLES Solve the following equations by testing for equal roots : 1. x^-\- Ilx2 + 24cc-36zr0. 2. ic3-2ic2-15x + 36r=0. 3. a;*-7a;3 + 9x2 + 27^-54 = 0. 4. X*- 11x3 + 44x2- 76x + 48 = 0. 6. x* - 5x3 - 9x2 + 81 X - 108 = 0. 6. x5 - 15x3 + 10x2 + 60x - 72 = 0. 7. x» - x* - 5 x3 + x2 + 8x + 4 = 0. 8. x* - 2x8 - 11x2 +12X + 36 = 0. 9. x5- 10x2 + 15x- 6 = 0. 10. X* -3x3 -6x2 + 28x -24 = 0. 11. x5- 10x3 + 20x2- 15x + 4=0. 12. X* + 10x3 + 24x2 -32x- 128 = 0. 13. x5 + 19x4 + 130x3 + 350x2 + 125x - 625 = 0. 14. x6 - 5x5 + 5x* + 9x3 - 14x2- 4x + 8 = 0. 15. x5-2x4-6x3 + 8x2 + 9x+ 2 = 0. 16. x6 + 7 x5 + 4x4 - 58x3 - 115x2 - 49x -6 = 0. 17. x^^ - 8x3 + 24x2 -28x+ 16 = 0. 18. x5 - 6x3 - 28x2 -39x- 36 = 0. 19. What is the condition that the cubic equation x3 + ^x + r = shall have a double root ? 20. Show that in any cubic equation with rational coefficients a multiple root must be rational. 81] QUADRATURE 101 Quadrature 81. Let y = f(x) and yz=f'(x) be the equations of the continuous curves LM and L'M' respectively. It is required to find the area included between the curve L'M', the avaxis, and the ordinates corresponding to x — a = OQ, and x = b= OR, where b>a. Let K denote the area QA'B'R. Divide the distance QR into (n + 1) equal parts, each equal to h = Sx. Then (n-\-l)h = b — a. Draw ordinates at the points of division and construct rectangles as shown in the figure. Let Xi = a-\-h= OQi, X2 = a + 2h,'-'Xn = a-\-nh= OQn- Then QA'=f'{a), Q, P/=/'(a^,)r- QnPJ = f'(^nl and the sum of the areas of the (m + 1) rectangles is hf'(a) + hfXx,)+hf'ix,) + - hf'(x„). Now f(x) = ^f/J^±R:iJM. (§68.) Let /-(.)+p = -^^^±4=M, where p is a quantity that approaches zero when A = 0. Then lif{x) + lip = f{* + h)-f{x). (1) (2) (3) (4) 102 QUADRATURE [81 Hence we may put /i/'(a) + hpQ = f(xi) —/(a), ¥'(^2) + hp2 = f(x^ - /(a-s), ¥'(^n-l) + JiPn-1 =f(Xn) - f(.^n-l), From these equations we have by addition 2hf(x) + ^hp=f{b)-f(a). (5) The second member of (5) is independent of 71, %hf'(x) represents the sum of the areas of the (n + 1) rectangles however great their number, and '^hp = when h = 0, i.e. when n becomes infinite. For "^hp < (n -[- l)hp' = (b — a)p', where p' is the greatest of the quanti- ties pi, p2"-pn} and p'= when k = 0. .•.^=,/r^ :^f'(x)8x=f(b)-f(a) = EB-QA. (6) The notation used to express this is K=rfix)dx = f{b)-f(a), (7) where the symbol J stands for ^Hhe limit of the sum," in this case, the limit of the smn of an infinite number of infinitesimal rectangles. Therefore, in order to find the required area, we must first obtain a function which when differentiated will give f'(x) ; then substitute in this new function /(x) the abscissas of the bounding ordinates and take the difference of the results. Hence equation (7) may be written In applying the formula we must first find /(a;) from /'(a;), i.e. we must reverse the operation of differentiation. In this sense the sym- bol r denotes an operation which is the inverse of differ eyitiation. This inverse process is called Integration. If then the symbol D be used to denote differentiation, the two symbols f and D neutralize each other, i.e. \ Df(x) =f{x). 81] QUADRATURE 103 E.g. if Df{x) = f {x)dx = (4 x^ - 3 x2 + 4 x - 6) dx, . then f -0/(35) = {f'{^)dx = f(x) = x* - x^ + 2 x^ - 6 x + c. Hence, to integrate an integral function of x, increase tlie exponent of each power of X by unity and divide the coefficient by the increased exponent. Thus, I x^cZx = , provided n ^—1. J n-\-l If /'(it*) is the derivative of f(x), then f(x) is called the Integral of f'(x). The curve LM may be called the Integral Curve with respect to L'M'. Then we may say that the area bounded by the D. C, the a>axis, and two ordinates is numerically equal to the dif- ference of the two corresponding ordinates of the I. C. If L'M' lies below the ic-axis between A' and B', the slope of LM between A and B will be negative (§ 79). Hence BB < QA, i.e. f(h) axis between A' and B', the formula gives the excess (positive or negative) of the area which lies to the right over that which lies to the left. Ex. 1. Find the area of the segment of the parabola y^ = 4ax cut off by the double ordinate through P(x', y'). Here y = 2 Vax"^ =f'(oc)- .-. Area ONP = P 2 Vax^dx = 2 y/aC x^dx 2V~a A^^'l- 2 Va-|x /f = 1 x'.2\/ax'^ = |xy = f rectangle OBPN. .'. Area OPQ = f rectangle ABPQ. .*. Area between AB and the curve is equal to ^ ABPQ. That is, the parabola trisects the rec- tanffle. B Y P ^^^ x' y' O A N c ^^^^ 104 QUADRATURE [82 Ex. 2. The curve y = x^ — Sx^ -\- 2x cuts the a-axis in the points (0, 0), B(h 0), i>(2, 0). p We now have f'{x) =x^ -Sx^ + 2x. : OAB= Cf'(x)dx = Cix^ -Sx'^-\-2x)dx = r J - a:3 + x2 + cT = 1. BCD =z r^ - cc3 + x2 + cT = (4-8 + 4 + c)-a + c)=-i. |3 -r-^ DEF = [j - x^ -{- x^ -{- cY DEF = ( 8^ - 27 + 9 + c) - (4 - 8 + 4 + c) = 2f EXAMPLES 1. Find the area included between the curve y = x^ — Qx"^ ■\-2Zx — \b^ the X-axis, and the lines a; = 1, x = 3 ; also x = 3, x = 5; a: = l, 5C = 5. 2. Find the area included between the curve ^ = ^2 — 2x— 8, the x-axis, and the lines X = — 2, x = 4 ; also between the curve y = x'^ — 2x + \ and the same lines. Find the area between the x-axis and the curve 3. y n: x3 - 3 x2 - 9x + 27. 4. ?/ = x^ + ax^. 5. ?/ = x* - 4 x3 - 2 x2 + 12 X + 9. Find the area between the curves 6. ?/2 =: 4 ax and x^ = 4 ay. 9. ?/"» = x" and y" = x"*. 7. ?/2 = 4 X and ?/2 = x^. 10. y = x^ — x and y = x. 8. ?/3 = x2 and 2/2 = x^. 11. y = x* — x and y2 =: ^^^2. 12 ylws. Wl — 71 12. y^ = 4 ax and y = 2 x — 4 a. -4ns. «'(^1 13. x2?/ = a^, x = b, X = c, and ?/ = 0. 14. y = x2 — 5 X + 4 and x + y = 4. 16. y = x^ and ?/8 = x. 16. Show that the area included between the curve y = Ax**, the x-axis and the line x = a is ah w + 1 Show that the parabola is a particular case where & is the ordinate corresponding to x = a. b2] MAXIMA AND MINIMA 106 Maxima and Minima 82. Let the curves ML^ L'M\ and L"M" be the loci, respectively, of the equations (1) y=f(x), (2) y=f'(x), (3) y=f\x)r ~ It is assumed in this investigation that the functions f(x), f'(x), f\x) are finite and continuous for all finite values of x. Then 7>"Jf " is the Second Derivative Curve. Since f\x) is the first derivative of f'ipi), the ordinate of L"M" at any point represents the slope of L'M' at the corresponding point; and the intersections E", F'\ G" of X"J/" with the a?-axis correspond to the elbows E', F', G' of L'M' (§ 79). Let the line x = a meet the curves in the corresponding points P, P'j P"f and the x-axis in Q. Then QP=:f(a), QF =f{a\ QP'^ =f"(a). That is, QP' is the slope of LM at P, and QP" is the slope of L'M' at P. Suppose the point P to move along the curve LM toward the rigJU. As P approaches the elbow B, the ordinate QB increases ; but as P passes through B, the ordinate ceases to increase and begins to 106 MAXIMA AND MINIMA [82 decrease. At such a point the ordinate, i.e. f(x), is said to have a Maximum Value, or to be a Maximum. In like manner as P approaches the elbow A, or C, the ordinate QP decreases ; but as P passes through A, or C, the ordinate ceases to decrease and begins to increase. At such points QP, i.e. f{x), is said to have a Minimum Value, or to be a Minimum. That is, a function, fix), is said to have a maximum value when x = a, if /(a) >/(a ± h) ; and a minimum value, if /(a) If a is an r-fold root of f'(x) = 0, then f"(a) = when r>l (§ 80) and the conditions (5) fail to disclose the nature of the corre- sponding ordinate. If r is an odd number, the curve L'M' will cut the avaxis in an odd number of coincident points, and hence will cross the a?-axis at the point (a, 0). Therefore the sign of /'(a;), will change from -f to — for a maximum, and from — to -f for a minimum. In this case we must use conditions (4) to determine the nature of /(a). If r is an even number, L'M' will not cross the aj-axis at the point (a, 0), as at D'. Hence f'(x) will not change sign, and therefore /(a) is neither a maximum nor a minimum. The maximum and minimum ordinates of L'M' can be determined in the same manner. The points E, F, G, D on IjM corresponding to the maximum and minimum ordinates of L'M' are therefore, respectively, the points of maximum and minimum slope of LM. At the points where the slope of a curve ceases to increase and begins to decrease, or vice versa, the curve changes the direction of its curvature. Therefore E, F, G, D are the points of inflection of LM (§ 79). Hence the position of the points of inflection of a curve are obtained by finding the position of the maximum and minimum ordinates of the p. a 108 MAXIMA AND MINIMA [83 83. Illustrative Examples. Ex. 1. The curves y = s'm x and y = cos x are good examples of the relations and principles explained in § 81 and § 82. Let / (x) = sin x. Then fix) = ^% ^n(x^h)-sinx ^ ^lim^ ^^^^ ^^ ^ , ^^ sin|_^J ^^^ = cos x. (Ex. 13, p. 71.) (2) Similarly it has been shown that Dx(,cosx) = — sin x. (p. 75.) Let and y =f (x) = sin x, equation of LM, y =fi(x) = cos X, equation of L'M', y =f"(x) = — sin x, equation of L"M". Y A p. -. ^"^^^^""^^ /P X ^ \./-' / \/\ O Q \ /\ \ \ \,- ^^ y / '-- \ .c..--\ - L' /\^__^ "■--»?:---' 2 " > Z "1 ^ "•) Then /'(^) = cos x = 0, when x = ^ tt, %ir, and /"(I tt) = — sin ^ TT = — 1. .-. sin ^ tt = 1 is a max. /"(I tt) = — sin f TT = 1. .-. sin f ir = — 1 is a min., etc. Also f"(^) = — sin X = 0, when x = 0, tt, 2 tt, 3 tt, etc. These values of x make cos x alternately a maximum and a minimum, and hence give the points of inflection of LM. That is, the sine curve changes the direction of its curvature as it crosses the x-axis. Let X = 0^ be any line parallel to the y-axis. Then f '(x) = cos x=: QF' = slope of L3I at P. Moreover, by § 81 we have Area OAP' Q= ( ''f'(x)dx = T ""cos xdx = fsin xT = sin x = QP. (3) That is, the ordinate of any point of the cosine curve is equal to the slope of the sine curve at the corresponding point; and the ordinate of the sine curve is equal to the area bounded by the ordinate, the cosine curve, and the axes of coordinates. 83] MAXIMA AND MINIMA 109 Ex. 2. Find the maximum and minimum values of the function fix) = x4 - 4 x8 - 2 x2 + 12 X + 4. Here /'(x) = 4 x^ - 12 x^ - 4 x + 12 and /"(x) = 12 x2 - 24 X - 4. The roots of /'(x) =0 are -1, 1, 3. /"(-I) =32. /"(I) =-16." /"(3)=32. /. /(— 1) = — 5 is a minimum. .-. /(I) = 11 is a maximum, •*• /(3) = — 5 is a minimum. The roots of /"(x) = are 1 ± f ^3,- which are the distances of the points of inflection from the ^/-axis. In the solution of problems in maxima and minima, we must first obtain an algebraic expression, /(x), for the quantity whose maximum or minimum is required. We may then proceed as in the preceding examples. Ex. 3, Find the maximum rectangle that can he inscribed in a given triangle. Let h = the base of the given triangle ABC, h the altitude, and x the altitude of the inscribed rectan- gle. Then from similar triangles, FG:b = ih-x):h. ,:EG = \ih-x). h Then - {hx — x^) is the area of h the rectangle, which is to be made a maximum. Any value of x that will make (^x — x^) a maximum will also make -Qix — X') a maximum. Hence we may put Then Also /(x) =hx- x2. /'(x) = /i - 2 X = when x = \h. /"(x)=-2. •*• f{\^) = 4 ^^ ^s ^ maximum. Therefore the altitude of the maximum inscribed rectangle is one-half the altitude of the triangle. 110 MAXIMA AND MINIMA [83 Ex. 4. Find the area of the largest rectangle ivhich can be inscribed in the ellipse Y • a;2 y2 + ^ = 1. (1) Let K denote the area of the rectangle. Then K=2X'2y='^ Va^x^ - x* (2) a is the function of x which is to be a maximum. Any value of x which will make a^x^ — a;* a maximum, or a mini- mum, will also make K a maximum, or a minimum. Therefore, let f(x) = a^x^ - a;*. Then f<{%) = 2 a^a; - 4 x^ = o when x = 0, or ± ^ a V^, and /"(x) = 2 a^ - 12 x2 = - 4 a2 when x = ^ a ^2. .\x — \a y/2 will make K a maximum. Therefore K=2ab is the area of the maximum rectangle, which is half the rectangle whose sides are the axes of the ellipse. Ex. 5. Find the dimensions of a cone of revolution which shall have the greatest volume with a given surface. Let X = the radius of the base, y = the slant height, V = the volume, and /S' = the total surface. Then and S = 7rx2 + TTxy ; whence y = — — x, TTX (Altitudey = ?/2 _ a;2 = -^ _ 2^. 7r2x2 IT VS^x^ - 2 ttSx* Let Then . y^^7rx2 r^2 2S^ 3 >'7r2x2 T fix) = Sx'^-2 TTX*. /'(x) = 2 6^x - 8 7rx3 = when x = 0, or ± - J-, ' /"(x) = 2 19 - 24 7rx2 = - 4 aS' when x = -J^- • 2 \7r .-. F is a max. when x = -^/-, and y = --./-• That is, if the surface is constant, the volume of the cone is a maximum •when the slant height is three times the radius of the base. and 83] MAXIMA AND MINIMA 111 EXAMPLES Find the maximum and minimum ordinates and the points of inflection (points of maximum or minimum slope) of the curves 1. y = x3 - 3 ic'^ + 4. 2. y = x^- 9x^ + 15 x-S. ^^ _ 3. y = ic8 - 3 x2 + 6 a; + 7. 4. y = x^ - 9 x^ + 2i x + 16. 5. Find the sides of the maximum rectangle which can be inscribed in a circle ; in a semicircle. 6. Find the sides of the maximum rectangle which can be inscribed in a semi-ellipse. 7. Find the altitude of the maximum rectangle which can be inscribed in a segment of a parabola, the base of the segment being perpendicular to the axis of the parabola. 8. What is the least square that can be inscribed in a given square ? 9. Find the altitude of a cylinder inscribed in a cone when the volume of the cylinder is a maximum. la What are the most economical proportions for a cylindrical tin can ? That is, what should be the ratio of the height to the radius of the base that the capacity shall be a maximum for a given amount of tin ? 11. What are the most economical proportions for a cylindrical tin cup ? 12. What are the most economical proportions for an open cylindrical water tank made of iron plates, if the cost of the sides per square foot is two-thirds of the cost of the bottom per square foot ? 13. An open box is to be made from a sheet of pasteboard 12 inches square by cutting equal squares from the four corners and bending up the sides. What are the dimensions of the largest box that can be made ? 14. If a rectangular piece of pasteboard, whose sides are a and 6, have a square cut from each corner, find the side of the square so that the remainder may form a box of maximum capacity. 15. A person being in a boat 3 miles from the nearest point of the shore, wishes to reach in the shortest possible time a place 5 miles from that point along the shore ; supposing he can walk 5 miles an hour, but can row only at the rate of 4 miles an hour, required the place where he must land. 16. The cost per hour of driving a steamer through still water varies as the cube of its speed. At what rate should it be run to make a trip against a four- mile current most economically ? 1^7. Find the altitude of the greatest cylinder that can be cut out of a given sphere. 112 MAXIMA AND MINIMA [83 18. Find the altitude of the maxim uin isosceles triangle that can be inscribed in a given circle. 19. Find the altitude of the greatest cone that can be inscribed in a given sphere. 20. Find the altitude of a cone inscribed in a sphere which shall make the convex surface of the cone a maximum. 21. If the slant height of a cone is constant, what is the ratio of the radius of the base to the altitude when the volume of the cone is a maximum ? 22. Find the dimensions of a cone with a given convex surface and a maximum volume. 23. Find the altitude of the least cone that can be circumscribed about a given sphere. 24. Find the altitude of the maximum cylinder that can be inscribed in a given paraboloid. 25. What is the diameter of a ball which, being let fall into a conical glass of water, shall expel the most water possible from the glass ; the depth of the glass being 6 inches and its diameter at the top 5 inches ? Ans. m in. 26. The sides of a rectangle are a and b. Show that the greatest rectangle that can be drawn so as to have its sides passing through the corners of the given rectangle is a square whose side is ^-X_2. 27. The strength of a beam of rectangular cross-section, if supported at the ends and loaded in the middle, varies as the product of the breadth of the cross- section by the square of its depth. Find the dimensions of the cross-section of the strongest beam that can be cut from a log 18 inches in diameter. 28. A Norman window consists of a rectangle surmounted by a semicircle. If the perimeter of the window is given, show that the quantity of light admitted is a maximum when the radius of the semicircle is equal to the height of the rectangle. 29. What are the most economical proportions for a cylindrical tin can, and a cylindrical tin cup, if the top and bottom are cut out of regular hexagons, and allowance is made for waste ? Ans. tt^ = 4 r V 3, and irh =2 r VS. 30. Show that the curve (x^ + a^)y = a^x has three points of inflection, and that they all lie on the line x = 4.y, 97^ V{^ CHAPTER VII CONIC SECTIONS 84. The general equation of the first degree and also some special cases of the equation of the second degree have been considered in Chapters II and III. We now proceed to the general equation of the second degree, and the standard forms to which it can be transformed. It will presently be shown that the locus of such an equation is al- ways a curve that can be obtained by making a plane section of a right circular cone. For this reason the locus is called a Conic Section.* 85. The Fandamental Property of a Plane Section of a Eight Cir- cular Cone, or a Conic Section. Let VO be the axis of a right circular cone, and APB any section made by a plane not passing through F. Inscribe a sphere in the cone tangent to the plane of the section at F'j then the line of contact HRK of the sphere and cone is a circle with centre C in VO, whose plane is perpendicular to VO and meets the plane of the section APB in the line ES. Pass the plane FJO^ through VO perpendicular to the plane APB, meeting it in the line AB, meeting the plane HKR in HK, and the line ES in Z>; then the plane VMN is also perpendicular to the plane HKR, and therefore perpendicular to ^*S'. * After studying the straight line and the circle, the old Greek mathematicians turned their attention to the conic sections, and by investigating them as sections of a cone soon discovered many of their characteristic properties. The most important of these discoveries were probably made by Archimedes and.Apollonius, as the latter wrote a treatise on conic sections abovit 200 b.c. These curves are worthy of careful study, not only on account of their historic interest, but also on account of their importance in the physical sciences and their frequent occurrence in the experiences of everyday life. For example, the orbit of a heavenly body is a conic section. For this reason they were thoroughly studied by the astronomer, Kepler, about 1600 a.d. The path of a projectile is a parabola. The graphical representations of the law of falling bodies, the pressure-volume law of gases, the law of moments in uniformly loaded beams, are all conic sections. The bounding line of a beam of uniform strength, the oblique section of a stove-pipe, the shadow of a circle, the apparent line dividing the dark and light parts of the moon, etc., are conic sections. The reflectors in head-lights and search-lights are parabolic. 113 114 CONIC SECTIONS E [85 N I ^ 85] CONIC SECTIONS 115 Let P be any point on the section. Draw PF, and the element PV which will be tangent to the sphere at R. Through P draw a line perpendicular to the plane HKRy which will meet CR produced in Q ; and through PQ pass a plane perpen^ dicular to ES meeting it in S. Let /3 = Z PRQ = Z AHD, the complement of the semi-vertical angle of the cone. Let a = Z ADH= Z PSQ. Then, since tangents from an external point to a sphere are equal, PF= PR. From the right triangles PQR and PSQ we get Pq = PR^\np = PS^ina. p $ . PF_sin« f/ . -^) ... ••:^-^i^* i..4 ^''' ^^^ So long as we consider any particular section, the point F and the line ES are fixed, a is constant, and therefore the ratio of PF to PS is constant. Equation (1) expresses the Fundamental Property of a Conic Sec- tion, which is used as the defining property. Moreover, all curves which have this property are plane sections of some cone ; for all possible curves satisfying this condition are gotten by giving this constant ratio all possible values, and also letting the distance, FD, from the fixed point to the fixed line have all possible values. We can do this with a conic section. For any particular value of )8, i.e. for any particular cone, the ratio can vary from zero (when a = 0) to CSC (3 (when « = ^ tt). For any particular value of a the ratio can vary from sin a (when P = \tt) to oo (wheu (i = 0). Thus the ratio can have any value from to oo . Also the distance of F frora ES, depending as it does upon the size of the inscribed sphere, for any particular cone and any particular value of a can vary from zero to oo . Therefore the property expressed by (1) is indeed a defining property of a conic section, that is : A Conic Section, or a Conic, is the locus of a point which moves in a plane so that its distance from a fixed point in the plane is in a con- stant ratio to its distance from a fixed line in the plane.* • * This is generally known as Boscovich's definition of a conic section, bnt, in the article on Analytic Geometry in the Encyclopedia Britannica, nintli edition, Cay ley calls it the definition of ApoUonius. 116 CONIC SECTIONS [8f- The fixed point F is called the Focus; the fixed line ES is called the Directrix; the constant ratio is called the Eccentricity, and k denoted by the letter e ; the line BFD, through the focus perpen- dicular to the directrix, is called the Principal Axis of the conic. 86. Classification of the Conic Sections. Using e to denote the eccentricity, we have, by (1) of § 85, PF sin a z^n When a ^, e > 1, and the plane APB (§ 85) meets NV produced on the other sheet of the conical surface ; the section is then com- posed of two infinite branches, one lying on each sheet of the cone, and is called a Hyperbola. Thus the parabola is the limiting case of both the ellipse and the hyperbola. Let the plane of the section pass through the vertex of the cone. Then if e < 1, the section is a point ellipse or a point circle. If e = 1, the plane is tangent to the cone, and the parabola reduces to two coincident straight lines. If e > 1, the hyperbola becomes two intersecting straight lines, which approach in the limit two parallel lines as the vertex of the cone moves off to an infinite distance. Hence a point, two intersecting straight lines, two parallel straight lines, and two coincident straight lines are all limiting cases of conic sections. Under the head of conic sections we must therefore include : (1) Tlie Ellipse, including the circle and the point; (2) The Parabola; (3) Tlie Hyperbola; (4) The Line-pair. 87] CONIC SECTIONS 117 EXAMPLES 1. Inscribe a sphere* tangent to the plane APB (fig. § 85) on the other side and thus show that the ellipse has another focus and a corresponding directrix; and that the two directrices are parallel and equidistant from the foci. 2. By means of these two inscribed spheres, prove the property of the ellipse given in § 34. 3. Inscribe spheres* in both sheets of the cone and show that the hyperbola also has two foci and two directrices. 4. Prove the property of the hyperbola stated in § 36. 5. Where are the foci and the directrices of the circle, the parabola, and two intersecting straight lines ? iX, General Equation of the Conic Sections 87. To find the equation of a conic section in rectangular coordinates. I. Let the equation of the directrix EC be X cos a -\- y sin a — }} = 0. Let F(k, I) be the corresponding focus. Let P(x, y) be any point on the conic. Draw PS perpendicular to EC, and join P and F. Then from equation (1) of § 86 we have PF=e'PS. (1) (2) * For complete diagrams see Some Mathematical Curves and their Graphical Construction, by F. N. Willson, pp. 45, 46. Also his Descriptive Geometry, pp. 44, 45. - 118 CONIC SECTIONS [87 Now PF'' =(x- ky 4- (2/ - l)% [(2), § 7.] and PS = xGOsa-\-y sina^p. [(4), § 47.] Therefore the required equation is (x — ky -\- (y — ly = e^(x cos a -{-y sin a— py. (3) Expanding (3) and collecting terms we have (1 — e^ cos^ a)a^ — 2{e^ sin a cos a)xy + (1 — e^ sin^ a)y^ + 2(e^p cos a - k)x + 2{fp sin a - V)y -[■ 1^ -\- 1"^ - eY = 0. (4) Since equation (4) contains five arbitrary constants, k, Z, a, p, e, it may be any equation of the second degree. That is, any equa- tion of the second degree represents a conic section. The most general equation of a conic is, therefore, the complete equation of the second degree, and may be written ax' + 2hxy + bf + 2gx + 2/^/ + c = 0. (5) II. Let the directrix be taken as the y-Sixis, the principal axis, FD, (§ 85) as the o^-axis. Then a = l=p = 0, and k = DF. There- fore the equation of the conic (3) takes the simple form. (x — A;)^ -\-y^ = e'x^f or (1 — e^)a^ -\-y' — 2kx If a^ = in (6), then y = ± kV^^. Hence a conic does not intersect its directrix. If 2/ = 0, then there are two real values of x, viz., Therefore a conic section cuts' its principal axis in two points. These points are called the Vertices of the conic. The point mid- way between the vertices is called the Centre of the Conic. The Latus Rectum of a conic is the chord through either focus perpendicular to the principal axis. To find its length, \Qt x = k in (6), then y = ± ekf and 2y = Latus Rectum = 2 ek. The different cases corresponding to the different values of e will now be separately considered. ^'^' I X (6) 88] CONIC SECTIONS 119 Standard Equations of the Conic Sections 88. The Parabola. e = l. When e = 1, equations (7) of § 87 give a^ = |A; = Z>0, 072 = - =00. Hence the parabola has one vertex midway between the focus and directrix, and the other at infinity.* When e = ly equation (6) of § 87 gives for the equation of the parabola referred to its axis and directrix f = 2k(x-i7c). (1) Let a = ik = DO= OF-, then this equation becomes y^ = 4:a(x-a). (2) Now write a; + a in the place of x ; this moves the origin to the vertex 0(af 0) [§ 53, (1)], and the equation becomes y^ = ^ax, (3) which is the standard form of the equation of the parabola. When a; = a in (3), y = ±2a. .-. L'L = 4 a = Latus Rectum. Ex. Construct the parabola, having given the focus and the directrix. ♦ Compare this result with the position of B in the figure of § 86 when a = /S. 120 CONIC SECTIONS [89 89. The Ellipse. ek. Hence the ellipse has two vertices lying on the same side of the directrix, but on opposite sides of the focus. c R B Y P R' X '-f L /> \^ D .A P O Q r /A' D' ^- NJ |i; ■■ B' y'^ Let O be the centre, and let AA = 2 a. Then whence Also 2a = X2 — Xi k ^ 2ek 1-e l+e l-e" a k e a 1-e' k = ae. e l,0 = i(., + ..) = i(^ + ^) a 1-e' e .: FO=DO-DF=~~k = ae. (1) (2) (3) (4) 8»] CONIC SECTIONS 121 Substituting in equation (6) of § 85 the value of k given by (2) gives for the equation of the ellipse referred to DC and DX (x---\- ae\+ f = 6^x2. (5) The origin may be transferred to the centre, ( -, J, by writing aj + - in the place of x [§ 53, (1)] ; this gives (a; + ae)2 + / = e2(^aj + fj, or a^(l - e'-^ + / = a'(l - e^- .-. ^ + t = 1. y^ (6) When a; = 0, we have 2/ = ± a Vl — e^ ; which gives the ?/-intercepts OB and OB?. If these lengths are denoted by ± 6, we have 62=a2(l-e2), (7) and equation (6) takes the standard form Since e < 1, 6 < a from (7) ; therefore Hence the line AA} is called the Major Axis, and BB^ is called the Minor Axis of the ellipse. Take OF' = FO and OD' = DO; draw D'C perpendicular to OX. Then F' is the other focus, and D'C the corresponding directrix (Ex. 1, p. 117). Hence the foci are the points F' (ae, 0) and F{— ae, 0) from (4) ; and the equations of directrices are, from (3), a <» = ±~- (9) e Let P(x, y) be any point on the ellipse ; draw a line through P parallel to AA' meeting the directrices in R and R\ and draw PQ perpendicular to AA'. * For a discussion of this equation see § 35. 122 CONIC SECTIONS [89 Then FP^e-RP, and F'P = e'B'P. [(2), § 87.] .-. FP = e'DQ = e(DO+OQ) = ef^-\-x) = a + ex, (10) and F'P= e • QD' = e(OD' - OQ) = el xj = a — ex. (11) Whence FP + F'P =2 a. (Cf. § 34.) (12) From equations (7) and (4) we get ae = Va^ - b- =F0= OF'. To find the length of the latus rectum we put a; = ± ae in (8) ; this gives 2/' = b\l- e") = - . from (7) .: X'i = ?|^. (14) If tt = 6, equation (8) reduces to a;2 + 2/2 = a^, and equations (13), (4), and (3), respectively, give e = 0, FO=OF'=0, DO=OJ)' = oo. That is, the circle is the limiting form of the ellipse, as the eccen- tricity approaches zero, and the directrices recede to infinity. Ex. Construct an eUipse, having given the foci and the length of the major - T n • distance between foci , ^, ,. , , . « .^ . .i. * In all comes e = 37-^ i— ; ^. — ; both distances become infinite in the distance between vertices parabola, and both become zero in the case of two intersecting lines. (See also (11), §90.) 90] CONIC SECTIONS 123 90. The Hyperbola. e>l. From equations (7) of § 87 we have for the vertices aji = and X2 k 1+e 1-e Since e > 1, Xi=DA < k, and X2 = DA' is negative. Therefore, the hyperbola has two vertices lying on the same side of the focus but on opposite sides of the directrix. Let be the centre, and let A'A = 2 a. Then 2 a = A'D + DA = — x2 + Xi k . k 2ek e-1 a k e + e + 1 e'^ — l e'-l and k = ae — 30 - (1) (2) 124 CONIC SECTIONS [90 The equation of the hyperbola referred to DC and DX is, from (2), and (6) of § 87, /aj-ae-h-Y + Z^eV. (5) Moving the origin to the centre 0( — -, ] gives (a; — aef -\-y^ = e'^[x i > or — H ^ = 1. (6) Since e > 1, the quantity a^(l — e-) is negative; if we put -b^ = a-(l-e-), or 52^^2(g2_-|^s^^ ^^^^ equation (6) reduces to the standard form ^_^-l (8) When X = 0, y = ±bV—l. Since these vahies of y are both imaginary, the hyperbola does not meet the line through its centre perpendicular to its principal axis in real points ; but, if B, B' are points on this line such that B'O = OB = b, the line BB' is called the Conjugate Axis. The line A A' joining the vertices is called the Transverse Axis. On the line OX take OF' = FO, and OD' = DO ; then F' is the other focus and D'C, perpendicular to OX, is the corresponding directrix (Ex. 3, p. 117). Hence the coordinates of the foci are ( ± ae, 0), from (4), and the equations of the directrices are, from (3), oo = ±^' (9) e As in the ellipse, we find the latus rectum LL' = ^^' (10) a Equations (7) and (4) give ae=Va:' + b'=OF. _V^:^i ^OF ^F'F _ ,j^. a OA A' A ^ ^ 91] CONIC SECTIONS 125 Let P(a7, y) be any point on the hyperbola ; draw a line through P parallel to AA^ meeting the directrices in R and R', and draw PQ perpendicular to AA\ Then FP^e- RP, F'P = e • WP. [(2), § 87.] .-. FP^e'DQl=e{Oq-OB)=e{x-^ = ex-a', (12) andi<^'P=e.Z)'Q = e(Oe + i>'0)=e('aj + -) = ea; + a. (13) Whence F'P -Fr = 2a. {Of. § 36.) (14) If a = 6, the equation of the hyperbola becomes x'^-y'^ = a^. (15) This is called the Equilateral or Rectangular Hyperbola. (See §§ 169, 170.) Then from (11), (3), and (4) we have, respectively, e = V2, OD = i a V2, OF = a v'2. Ex. Construct a hyperbola, having given the foci and the distance between the vertices. 91. Limiting cases of conic sections. If A; = 0, equation (6) of § 87 reduces to 2/2 = arXe2-l). This equation represents two straight linesy which are real if e > 1, coincident if e = 1, and imaginary, but with a real point of intersec- tion, if e < 1. From (7) of § 87 we then have a^j = x^ = 0. Hence the foci, the vertices, and the centre of two intersecting lines all coincide on the directrix. The two directrices also coincide. When e=cc (a being finite), the equation of the hyperbola [(8), § 90] reduces to x^ = a', which represents two parallel lines. Equa- tions (3) and (4) of § 90 then show that the foci of two parallel lines (considered as the limiting case of a hyperbola) are at infinity while their directrices coincide and are equidistant from the two lines. Hence we must consider two intersecting lines, real or imaginary {i.e. a real point), two coincident lines, and two parallel lines ns limiting cases of conic sections. (Cf. § 86.) Jv^V 126 CONIC SECTIONS [92 Tangents 92. To find the equation of the tangent to the conic represented by the general equation ax' + 2hxy + bf + 2gx + 2fy-hc = 0. (1) The equation of the tangent to any curve f(x, y) = at the point (x', y') is (§ 62) 2/-2/' = ||!(^-«^'). (2) For equation (1) we have found in § 61 dy ^ ax-\-hy-{-g ^ dx hx 4- by -+-/ Therefore the required equation is (3) or axx^ + h (xy' + x'y) + byy' + gx +fy = ax" + 2hx'y'-\-by" + gx'+fy'. (5) Add gx' -\-fy' + c to both sides of (5) ; then, since (x', y') is on the conic, the right member will vanish and we have the required equation, axx' + h (xy' + x'y) + byy' + gr (a? + x') +fiy + y')+c = 0, (6) Observe that the equation of the tangent at (x', y') is obtained from the equation of the conic by writing xx' for xr^ x'y + xy' for 2 xy, yy' for /, x + x' for 2 x, and y -{-y' for 2 y. Note also that putting x for x' and y for y' in (6) reproduces the equation of the curve. E.g. the equation of the tangent to the circle x'^ + y'^ = r^ at the point (x', y') is xx' + yy' = r^, to the parabola y^ = 4ax at the point (x', y') is yy' = 2a(x-{- x'), to the ellipse ^ + 1^ = 1 at the point (x', y') is ^ + ^' = 1, to the hyperbola ^ _ |^ = 1 at the point (x', ?/') is ^ - ^ = 1. /I'' 93] CONIC SECTIONS 127 Jsj^!) 93. Two tangents can be drawn to a conic from any pointy which will be realf coincident j or imaginary ^ according as the point is outside, on, or within the curve. Let the equation of the conic be [§ 87, (6)] aa^ + f-{-2gx-{-g' = 0, (1) where a = 1 — e^, and g= —k. Let (Jij Tc) be any point j then the equation of any line through this point will be (§ 43) y — k = m(x — h). (2) Eliminating y between (1) and (2) gives (a + m^x" + 2(km - hm^ + g)x -f ^V - 2 hkm -{-k^ + g^ = 0. (3) The roots of (3) are, by § 24, the abscissas of the points of inter- section of (1) and (2). If these roots are equal, the points of in- tersection will coincide and, by § 57, (2) will be tangent to (1). The condition that (3) shall have equal roots * is {km - hm^ -f gf = (a +, m^){hV - 2 hkm + k^ -[-g^ (4) or {ah^ + 2 gh -^g") m" - 2{ahk -f gk)m + {ale" + ag''-g^ = 0. (5) Equation (5) is a quadratic in m whose roots are the slopes of the tangents from (/i, k) to the conic. Since a quadratic equation has two roots, two tangents will pass through any point (Ji, k). The conic is, therefore, a curve of the second class. The roots of (5) are real, equal, or imaginary, according as a/i2 4-A;2 + 2^/i + />,=, or <0. (6) Therefore the tangents are real, coincident, or imaginary accord- ing as the point (h, k) is outside, on, or within the conic. (§ 20, II.) (The directrix is outside, the focus inside the conic.) Since equation (3) is a quadratic in x, any straight line meets a conic in two points, which may be real, coincident, or imaginary. Therefore the conic is also a curve of the second order. If e = 1 and m = 0, then a-\- m^ = 0, and hence one root of (3) is infinite (§ 77). Therefore a straight line parallel to the axis of the parabola meets the curve in one point at a finite distance, and in another at an infinite distance from the directrix. * The two roots of ax^ -f 6x + c = will be equal, if 6^ = 4 ac. The method here used is worthy of special attention because of its wide application. 128 CONIC SECTIONS [94 Pole and Polar 94. The equation of the tangent to the conic ax' + y'-^2gx+g' = (1) at the point (a;', 2/'), if this point is 07i the conic, is (§ 92) axao' + yy' + g(.oo + oc') +g^ = 0. (2) Suppose, however, that F'(x'j y') is 7iot on the conic. Then what is (2)? It still has a meaning, still represents a straight line related in a definite way to the point (x', y') and the conic (1). Moreover this line will cut the conic in two points (§ 93). Let these points be Pi(xi, y^) and P^ix^, 2/2)- Then the equations of the tangents at these points are (§ 92) axx^ + yyi 4- g(x + x^) -\-g'^ = 0, and axx2 -\- yy^ -\- g{x -\- ^2) -\- 9^ = 0- The conditions that (3) and (4) shall pass through {x\ y') are ax% + 2/ '2/1 + 9(:^' + ^i) +f = 0, and ax% + y'y, + g{x' + x,) + / = 0. But (5) and (6) are also the conditions that (2) shall pass through both of the points (xi, y{) and (xo, y^)- Therefore (2) is the line passing through the points of contact of the tangents from the point F'(x', y'). The point (x', y') and the line (2) are called Pole and Polar ivith respect to the conic (1). (3) (4) (5) (6) 95] CONIC SECTIONS 129 The tangents from the point (x', y^ will be real or imaginary according as (x', y') is outside or inside the conic (§ 93) ; but the line (2) is real when (x'j y') is real. So that there is always a real line passing through the imaginary points of contact of the two imaginary tangents drawn from a point within a conic. If (x'f y') is on the conic, the two tangents from it will coincide, and each of the points (iCi, y^) and (x2, y^ will coincide with {x\ y'). Tlierefore the tangent is the particular case of the polar which passes through its oimipole. (See demonstration in § 169.) 95. If the polar of a point P\x\ y') pass through P"(x"j y"), then will the polar of P" pass through P'. (See fig. § 94.) Let the equation of the conic be [§ 93, (1)] ax' + y'-^2gx-^g' = 0. (1) The equations of the polars of P' and P" are axx' + yy' +g(x + x')+g' = (2) and axx" + yy" + g(x + x") + g' = 0. (§94.) (3) The line (2) will pass through the point P" if ax'x"-j-yY+g{x' + x") + g' = 0; (4) but this is also the condition that (3) shall pass through P', which proves the proposition. CoR. I. The locus of the poles of all lines passing through a fixed point is a straight line; viz. the polar of the fixed point. CoR. II. If the polars of two points P and Q meet in i?, then R is the pole of the line PQ. 130 CONIC SECTIONS [95 Two straight lines are said to be conjugate with respect to a conic when each passes through the pole of the other. Two points are said to be conjugate with respect to a conic when each lies on the polar of the other. EXAMPLES Find the equations of the tangent and normal to 1. x^ = 2y, at (-2, 2). 2. y^ = Sx, at (2, -4). 3. x2 + ?/2 = 25, at (4, - 3). 4. x^-y^ = 16, at (- 5, 3). 5. ic2 + 4 ?/2 = 8, at (- 2, 1). 6.2 y'^ ~-x'^ = 4, at (2, - 2). Find the equations of the tangents to the following conies at the origin : 7. x2 + ?/2 + 2x = 0. S. x^ + 2x + 3y = 0. 9. 2xy + bx-3y = 0. 10. Sx^ -2xy -\-4:X-2y = 0. 11. State a rule for finding the tangent to a conic at the origin, ind the polar of the point 12. (3, 2) with respect to y^ = 6 x. 13. ( — 2, — 4) with respect to x^ + ?/2 = 4. 14. (1, 1) with respect to 2x^ -\-Sy^ = l. 15. (0, 0) with respect to 2 x^ - 3 y2 _|_ 12 x - 6 y + 21 = 0. 16. Give a rule for writing the equation of the polar of the origin. Find the tangents to the following conies drawn from the given points (see A IS K ' 17. 1/2 = 4 X, (2,3). 18. y^ = 6x, (-3, -1). 19. x2 + ?/2 = 25, (-1,7). 20. 9x2 + 25 2/2 = 225, (10, -3). ^ 21. Show that the polar of the focus is the directrix. What is the locus of the intersection of tangents at the ends of focal chords ? (Use equation (1), § 93.) 22. Show that the line joining the focus to any point on the directrix is per- pendicular to the polar of the latter point. 23. Show that tangents to a conic at the ends of a chord through the centre are parallel. 24. What is the polar of the centre of a conic ? Where is the pole of a line passing through the centre ? \i' 26. What is the pole of a; cos a 4- J/ since =^ with respect to x^ + y'^ = r^? y'^ = 2x? \i 2 CHAPTER VIII THE PARABOLA 96. Standard /equations of the tangent, polar, and normal to the parabola. In studying the properties of the parabola in this chapter we shall use the standard form of the equation found in § 88, viz. 2/2 = 4 ax. (1) Then the focus is the point (a, 0), the directrix is the line x= — a, and the latus rectum is 4 a. Equation (6), § 92, applied to (1) gives yy' =2a(i€ + ic'), (2) as the equation of the tangent at the point («', y'), if (»', y*) is on the curve; but always the equation of the polar of (x, y'), (§ 94), with respect to the parabola (1). The equation of the normal at the point (x', y') on the curve is [(2), § 62] II _ 1/' = — 2a or 2 a{y - y') 4- y\oc - a?') = 0. (4) The tangent at the vertex (0, 0) is the line a; = ; and the normal at the same point is y = 0, i.e. the axis of the curve. Ex. 1. Show that the equation of the parabola is 2/2 = 4 a(x ± a), according as the origin is at the focus or on the directrix. Ex. 2. Change the equations of the parabolas (y -ky = i a(x - h) and (x -hy = 4 a(y - k) to the standard form, and show that their vertices are at the point (h, k). Ex. 3. What relation does the line (3) have to the parabola when the point (x'iy') is not on the curve ? 181 y-y' = -:^(^-^')^ (3) 132 THE PARABOLA [97 97. Geometnc properties of the parabola. M R ^ P-X^^^ \ \ X T D O If n g Let the tangent at the point P{x\ if) meet the axis in jT, the directrix in i2, and the tangent at the vertex in Q. Let PM and PN be the perpendiculars from P to the directrix and axis, re- spectively. Let the normal at P meet the axis in G, Then we have the following properties ; TO=:ON=x\ [(2), §96.] (1) .-. Subtangent = T]Sr= 2 0N= 2 x'. (2) OQ = iNP=^y'. (3) TF=FP=FG = a-hx'. (4) Z.FPR = ZMPR. (5) Z RFP = Z BMP = i ,r. (See Ex. 22, p. 130.) (G) FM is perpendicular to TP. (7) FM, PT, and OY meet in a point (8) 0G = 2a + x\ [(4), §96.] (9) .-. Subnormal = NG = 2 a, a constant. (10) 98] THE PARABOLA 133 The use of parabolic reflectors depends on the property expressed in (5). Let the student explain. Properties (5) and (7) suggest a method of drawing tangents from an exterior point. Show how this can be done. 98. Equations of the tangent and normal in terms of the slope m. The equation of the tangent [(2), § 96] may be written 2a , 2 ax' 2a , 4: ax' ,^. Let — p = m ; then ^ = — , and (2) may be written l/ = ^^^^j (3) which is the required equation. That is, the line (3) will touch the parabola y^ = 4 ax, whatever the value of m may be. In a similar manner it can be shown from (3), § 96, that the equa- tion of the normal expressed in terms of its slope is y = mx — 2 atn — atn^* (4) EXAMPLES 1. Find the equations of the tangents, and the normals at the ends of the latus rectum. 2. Show that the line y = 3 a; 4- - touches the parabola y'^ = Aax', and also that y = Ax-\-- touches y'^ = S ax. 3. Find the equation of the tangent to y^ = 12 x which makes an angle of 60° ""^ with the X-axis. 4. Find the tangent to the parabola y^ = 6 a; which makes an angle of 45° with the X-axis. Find the coordinates of the vertex, of the focus, the length of the latus rectum, and the equation of the directrix of each of the following parabolas : 6. y2_3a;^.6. 6. x2 + 4x + 2y = 0. 7. (y- 4)2 = 6(x + 2). 8. 4(x-3)2 = 3(y-M). . 9. y^-l- 8x- 6y-M = 0. \Xm 134 THE PARABOLA [99 99. Tlie locus of the middle points of a system of parallel chords of a parabola is a straight line parallel to the axis of the parabola. Let ABhe any one of the chords, let P'(x'j y') be its middle point, and let y be the angle it makes with the axis of the parabola. Then the equation of AB may be written [(4), § 43] x — x' y — y = r, (1) (2) cos y Sin y or a; = ic' -|- r COS y, y = y' -{- r sin y. Let the equation of the parabola be y' = 4.ax. (3) Substituting in (3) the values of x and y given by (2), we have for the points common to the chord and the curve (y' + r sin y)^ = 4 a (x' -\- r cos y), or 7^sm^y-^2(y'siny — 2aGosy)r-{-y'^ — 4:ax'=:0j (4) a quadratic equation in r, whose roots are represented by the dis- tances P'B and P'A. Since P' is the middle point of AB, the sum of these roots is zero. That is, 2/' sin y — 2a cosy = 0. (§ ^8.) 2a Whence where m is the constant slope of the chords. y'=2a cot 7 = — J ' m (5) 100] THE PARABOLA 135 The coordinates of P' therefore satisfy the equation y=^=2acoty. (6) Hence the locus of P', as AB moves keeping m constant, is a straight line O'X' parallel to the axis of the parabola. Definition. The locus of the middle points of a system of parallel chords of a conic is called a Diameter; and the chords it bisects are oblique double ordinates to that diameter considered as an axis of abscissas. We have seen in § 93 that a diameter of a parabola meets the curve in only one point at a finite distance from the directrix. This point is called the Extremity of the diameter. CoR. The line (6) meets the curve in 0' where x = —^ = EO', y = (7) The equation of the tangent at 0' is, therefore [(2), § 96], 2/ = ^^^ + ^- (8) Hence the tangent at the extremity of a diameter is parallel to the chords bisected by that diameter. 100. To find the equation of a parabola ivhen the axes are any diameter and the tangent at its extremity. Using the figure of § 99, and keeping the same notation, we will let 0'P' = x, the new abscissa, and P'B = yf the new ordinate. Then y is always the same as r of equation (4), § 99. And since the coefficient of the first power of r in this equation is zero, we have where and „ 4:ax' — y'^ ^- sin^y ' (1) , 2a m [(5), § 99.] f = RO'-\-0'P' = —, rn? + x. [(7), § 99.] „ 4a .'. y^= . „ X. ^ sin^'y (2) 56 THE PARABOLA [100 Now Fa-- 1 + tan^r a [(4), § 97.] (3) - Oi 7) — tan^y sin^y Therefore, if a': sm^y ', the required equation is 2/2 = 4 a'x. (4) Hence the equation y^ — 4iax always represents a parabola, the a>axis being a diameter, the ^/-axis the tangent at its extremity, a the distance from the focus to the origin, and 4 a the length of the focal chord parallel to the ^/-axis. Formula (6), § 92, by means of which equation (2), § 96, was obtained, and also the derivation of equation (3), § 98, from equation (2), § 96, hold good equally whether the axes are rectangular or not. That is, if the equation of a parabola is 2/^ = 4 ax, the line yy^ = 2a{x-\-x^) (5) will be the tangent at the point {x\ y') if the point is on the curve ; but always the polar of (ic', y') with respect to the parabola. And the line ^ y = mx-\-— (6) will also touch the parabola for all values of m, the meaning of d being that given in § 50. Cor. The polar of any j^oint with respect to a parabola is parallel tr> the chords bisected by the diameter through the point. Conversely, the locus of the poles of parallel chords is the bisecting diameter. For the polar of any point {x\ 0) is, by (5), x = — x\ EXAMPLES ON CHAPTER VIII 1. Find the equation of that chord of the parabola y'^ — Qx which is bisected by the point (4, 3). / 2. Find the equation of the chord of x'^ = —^y whose middle point is (-3,-2). 3. Find the equations of the tangents drawn from the point (—2, 2) to the parabola y'^ = Qx. 100] THE PARABOLA 137 4. Show that the axis of the parabola y^=:Sx divides each of the chords whose equations are . p = — ottb iijl^ two segments whose product is 64. 6. For what point on the parabola y^^'iaxm (1) the subtangent equal to the subnormal, and (2) the normal equal to the difference between the sub- tangent and the subnormal ? '"'^ 6. Show that the lines y = jt (a; -f 2 a) touch both the parabola y'^ — %ax and the circle a^ +,j/2 _ 2 ^2. ' ' ^ 7^ Find the equation of the common tangent to the parabolas y"^ = iiax and »2 = 4 hy. Show also that if a = &, the line touches both at the end of the latus rectum. E 8. Two equal parabolas, A and 5, have the same vertex and their axes In opposite directions. Prove that the locus of the poles with respect to -B of tan- , gents to A is the parabola A. 9. Show that the locus of the poles of tangents to the parabola y^ = 4:ax with respect to the parabola y^= 4:bx is the parabola ay^ = 4 b^x. •\ y 10. Show that for all values of m the line / . . . . a / 1/^ y — m(x + a) + — will touch y^z=:4 a(x + a); tTl y = m(x — a) -h — will touch 2/2 — 4 q^^x — a) ; and (y — k) = m{x — h) -\ — will touch (y — k)'^ = 4 a(x — h"). 11. If (a;', y') and (ic", y") are the points of contact of two tangents to y2 = 4 ax, show that the coordinates of their point of intersection are X = Vx'x", y = i(y' + y"). 4' 12. Show that the directrix is the locus of the vertex of a right angle whose sides slide upon a parabola. (§ 98. ) 13. Two lines are perpendicular to one another; one of them is tangent to ?/2 = 4 a(x + a), and the other is tangent to y^ = 4 6(a; + 6) ; show that these lines intersect on the line x + a + b = 0. 14. Show that the line Ix + my + n = will touch the parabola 2/^ = 4 ax, if In = am^. 15. If the chord PQR passes through a fixed point Q on the axis of the parabola, show that the product of the ordinates, and also the product of the abscissas of the points P and i?, is constant. T 16. Find the coordinates of the point of intersection of y = mx -\ and y = m'x H — -. . Show that the locus of this point is a straight line if mm is m' constant. What is the locus when mm' = — 1 ? 138 THE PARABOLA [100 17. K perpendiculars be let fall on any tangent to a parabola from two points on the axis which are equidistant from the focus, the difference of their squares will be constant. 18. The vertex ^ of a parabola is joined to any point P on the curve, and PQ is drawn at right angles to AP to meet the axis in Q. Prove that the projection of PQ on the axis is always equal to the latus rectum. 19. If P, Q, and B be three points on a parabola whose ordinates are in geometrical progression, the tangents at P and B will meet on the ordinate of q. 20. Show that the locus of the intersection of two tangents to a parabola at V points on the curve whose ordinates are in a constant ratio is a parabola. 21. Prove that the circle described on a focal radius as diameter touches the tangent drawn through the vertex. 22. Prove that the circle described on a focal chord as diameter touches the directrix. 23. Find the locus of the point of intersection of two tangents to a parabola which make a given angle a with one another. If a = 45°, show that the locus is ?/2 — 4 ax = (x-\- a)^. lia = 60°, show that the locus is y- - 3 x^ - 10 aa: - 3 a^ = 0. [Suggestion. The line y = mx + — will go through (x', y') if m^' — my' + a = 0. The roots of this equation are the slopes of the two tangents which meet in (x', y'). Let mi, m^ be these roots, then see § 68.] 24. The two tangents from a point P to the parabola y"^ = 4: ax make angles tan-%i and tan-im2 with the ai-axis. Find the locus of P, (1) when wij + mi is constant, (2) when wii^ + m^^ is constant, and (3) when m\m2 is constant. 25. If K is the area of a triangle inscribed in the parabola y^ — 4 dx, and K' is the area of the triangle formed by the tangents at the vertices of the inscribed triangle, prove that 8 a^=r 16 aK' = (yi ~ 2/2) (^2 ~ 2/3) (2/3 ~ Vi), where 2/1, ?/2, ys are the ordinates of the vertices of the inscribed triangle. (See Ex. 11.) Find the locus of the middle points 26. Of all ordinates of a parabola. 27. Of all focal radii. 28. Of all chords through the fixed point (h, k). As special cases, let (h, k) be (1) the focus, (2) the vertex, (3) the point (4 a, 0), and (4) the point (— a, Q). 29. Show that the parabola is concave towards its axis. CHAPTER IX ^_ THE CIRCLE XOl. Equations of the circle, and the corresponding equations of the tangentj polar, and normal. We have seen in § 32 that the equation of the circle whose radius is r takes the simple form a^ + / = v'2, (1) when the origin is at the centre ; while if the centre is at the point (a, b) the equation may be written (x-ay-^(y-by='r'. (2) Moreover, we have found in § 87 that the locus of any equation of the second degree is a conic. Now the conic represented by the general equation (5), § 87, will be a circle if a = 6 and h = 0. For this equation may then be written x^-\-y' + 2gx + 2fy-^c = 0. (3) Equation (3) may be put in the form of (2), which gives {x + gy+(y+fy = 9'-{-r-c. (4) Hence the locus of (3) is a circle whose centre is the point (— g, — /), and the radius is equal to ^ g^ -\-f^ — c. The circle will therefore be real, a point, or imaginary according a'S(/2+/^-c>, =, or <0. By applying the rule of § 92 to equations (1), (2), and (3), re- pectively, we obtain xoc' 4- yy' = r'^9 ® {X - a)(x' -a) + {y- h){y< -b)= r^, (6) and XX' + yy' + g(x + x') + f(y + y') + c = 0. (7) These are the equations of the tangent to the circles (1), (2), (3), respectively, at the point (x', y') if this point is on the curve; but, 139 140 THE CIRCLE [101 by § 94, they are always the equations of the polar of the point (x\ y') with respect to the circles represented by (1), (2), (3). Since the normal (§ 57) at any point (x', y') of the circle a:F-\-y^ = 7^ is perpendicular to (5), its equation is [(2), § 62'] or • scy' - oc'y = 0. (8) That is, the normal at any point of a circle passes through the centre. The equations of the normals to the circles (2) and (3) at the point (x'j 2/') are, respectively [(2), § 62], y-y' = ^\^(x-x'), (9) x' — a and y-y' = yL±f(x-x^): (10) x' + g or xy' — x'y — b(x — x')-\-a(y — y')=0, (11) and xy' — x'y -\-f(x — x') — g(y — y') = 0. (12) The general equation of the circle (3), or (2), contains three parameters, or constants. Therefore a circle can be made to satisfy three conditions, and no more. If we wish to find the equation of a circle which satisfies three given conditions, we assume the equation to be of the form (3), or (2), and then determine the values of the constants g, f, c, or a, b, r, from the given conditions. Ex. Find the equation of the circle passing through the three points (0, 1), (2,0), and (0, -3). Let the equation of the required circle be x^ + y^ + 2gx + 2fy + c = 0. (1) Since the given points are on the circle, their coordinates must satisfy equation (1). .-. l + 2/+c = 0, 4 + 4^ + c = 0, 9-6/+c = 0. Whence we find gr = — ^, /= 1, and c = — 3. Substituting these values in (1) the required equation becomes x^ + y^ — I x +2^-3=0. The centre is the point (^, - 1), and the radius is ^VOd. 102] THE CIRCLE 141 102. A geometrical construction for the polar of a point with respect to a circle. Let the equation of the circle be x'^,/ = 'i^, (1) Let P(x', y') be any point, BC its polar, and let OP and BC intersect in Q. Then the equation of BC is [(5), § 101] xx' + yy' = r^, and the equation of the line OP is (§ 44) xy' — x'y = 0. Hence BC is perpendicular to OP (§ 45), and therefore (2) (3) 0Q = Vx^' + y'' Also [(5), §47.] [(4), §7.] (4) (5) (6) OP=Vx'^-{-y''. .-. OP' OQ = r'. We therefore have the following construction for the polar of a point P. Draw OP and let it cut the circle in R; then con- struct a third proportional, OQ, to OP and r, i.e. take Q on the line OP, such that OP: OP = OR:OQ, and draw a line through Q perpendicular to OP. Ex. 1. Construct the pole of a given line. 142 THE CIRCLE [103 103. To find the equation of the tangent to the circle a? + f = i^ (1) in terms of its slope m. The line y = mx + h (2) will touch the circle (1) if the perpendicular distance from it to the origin is equal to the radius r of the circle ; that is, (§ 47) if ^ :, or 6 = rVrT^. (3) Vl + m^ Therefore the straight line y = mx + ^Vl + in^ (4) will touch the circle (1) for all values of m. Since either sign may be given to the radical Vl + m^ in (3), it follows that there are two tangents to the circle for every value of m ; i.e. there are two tangents parallel to any given straight line. Ex. 1. Derive equation (3) by treating (1) and (2) simultaneously and taking the condition for equal roots. EXAMPLES Find the equation of the circle passing through the three points 1. (1, 0), (6, 0), (0, 4). 2. (0, 0), (1, 1), (4, 0). 3. (2, -3), (3, -4), (-2, -1). 4. (1,2), (3, -4), (5,6). Find the equations of the tangents to the circle 5. x2 + 2/2 = 4 parallel to2x + 3?/ + l=0. 1^6. a;2 + ^2 _ 6 a; parallel to3x-2y + 2 = 0. Find the polar of the point 7. (1, 2) with respect to a;2 + y2 _ 5. 8. (3, - 2) with respect to 3(x2 + y2) _ 14. 9. (-4, 1) with respect to ic2 + y2 _ 2 aj 4- 6 y + 7 = 0. Find the polar of the line 10. 2 oj + y = 1 and x — ^y = \ with respect to ic2 + ^2 _ 2. 11. X — 2 ?/ = 3 and 2 x + 2/ = 4 with respect to x^ + if = 6. 12. x + 2/+l=0 with respect tox2 + ?/2 + 4x-6i/ + ll=0. \l.jr^ 104] THE CIRCLE 143 104. To find the length of a tangent drawn from a given point P{x\ y') to a given circle. Let the equation of the circle be (x-ay-h(y-by-r^ = 0. ^ (1) Let C be the centre and PT one tangent from P. Then, since CPT is a right triangle, PT^=CP^-CT\ But OT2=r2, and CP'= (x' - ay + (y' - by. [§ 7, (2).] .-. PT^ = («' - a)2 + (2/' - 6)2—^2. (3) That is, the square of the tangent is found by substituting the coordinates x', y' of the given point in the left member of equation (1). Since the general equation of the circle, a^H-/ + 2i/x + 2/2/ + c = 0, (4) can be put in the form of (1) by merely adding and subtracting g^ and /^ in the first member, it follows that if the coordinates of any point are substituted in the first member of (4) the result will be equal to the square of the length of the tangent drawn from the point to the circle ; or the product of the segments of any chord (or secant) • drawn through the point. (See proof of § 154.) >- .-^ Ex. 1. What is the meaning of (3) when the second member is negative Ex. 2. What is represented by c in equation (4)? Ex. 3. Where is the origin if c is positive ? if c is zero ? if c is negative ? ,^ 144 THE CIRCLE [105 105. If a circle passes through the common points of two given circles, tangents drawn from a7iy point on it to the two given circles are in a constant ratio. Let S = x^-^y'-h2gx + 2fy-\-c = (1) and S'^x'-\-y^-\-2g'x-\-2f'y-\-c' = 0, (2) be the equations of the two given circles. Then the locus of aS' = kS^, i.e. (See Ex. 5, p. 62.) a?-\-f-^2gx + 2fy-\-c = X(x' + f + 2g'x-\-2fy-\-c'), (8) for all values of X, will pass through the common points A, B, of (1) and (2). Moreover, (3) is a circle (§ 101), and therefore, for different values of X, represents all circles through the intersection of (1) and (2). Let P(x', y') be any point on (3) ; let PT and PT^ be the tangents to (1) and (2) respectively. Then the coordinates x', y' must satisfy (3), and we therefore have x" -\-y" + 2 gx' + 2/?/' + c = \(x'' + y"-^2 g'x' + 2f'y' + c'). (4) Therefore PT' = X'PT'% (§104.) (5) which proves the proposition, since X is constant for any particular circle. 106] THE CIRCLE 145 When A. = 1, it is easy to show that the radius and the coordinates of the centre (§ 101) of the circle represented by equation (3) all become infinite. In this case the equation reduces to 2(flr-fir')« + 2(/-/')2/ + c-c' = 0, _^ (6) which is of the first degree, and therefore represents the straight line AB through the common points of the two given circles. Let QR and QR' be tangents to /iS = and S' = 0, respectively, from any point Q on AB] then, since ABQ is the circle through the common points of (1) and (2) corresponding to X = l, it follows from (5) that QR = QR'. (7) That is, tangents drawn to the two given circles from any point on the line (6) are equal. It is to be noticed that the straight line given by (6) is in all cases real, provided g, f, c, g\ f\ c' are real, although the circles /S' = and >S' = may not intersect in real points ; in fact one or both of the circles may be wholly imaginary. We have here, therefore, the case o f a real sti aijght line p assing thr ough the imaginary points of inter- is ection of tw o real or imaginary circles. (Cy. § 94.) Definition. The straight line through the points of intersection (real or imaginary) of two circles is called the Radical Axis of the tAvo circles. From equation (7) it follows that the radical axis may also be defined as the locus of the points from which tangents drawn to the two circles are equal to one another. Cor. If the coefficients of o^ in S and S' are unity, the equation of the radical axis of the two circles S = and S' = is S — S' = 0. Ex. 1. Show that the radical axis of two circles is perpendicular to the line joining their centres. Ex. 2. If tangents are drawn to two circles from any point on a line parallel to their radical axis, show that the difference of the squares of these tangents is constant. Ex. 3. Show that the radical axis of two circles divides the line joining their centres into two segments, such that the difference of their squares is equal to the difference of the squares of the radii. 146 THE CIRCLE [106 106. The radical axes of three circles, taken in pairs, meet in a point. Let Si = 0, /iS'2 = 0, ^iSg = be the equations of three circles, in each of which the coefficient of ic^ is unity. Then the equations of their three radical axes are (§ 105, Cor.) Si — /S'2 = 0, S2 — ^3 = 0, Sq — Si = 0. The sum of any two of these equations is equivalent to the third. Hence they form a consistent system, and therefore their loci meet in a point. Or, prove by § 49, letting A = 1. TMs point is called the Radical Centre of the three circles. EXAMPLES ON CHAPTER IX Find the length of the tangents (or the product of the segments of the chords^ drawn from the points V 1. (3, 2), (5, - 4) to the circle x"^ + y^ = 4. 2. (- 3, 2), (4, - 4) to the circle x^ -h y^ = 25. 3. (3, - 2), (1, 3) to the circle x^ -\- y"^ - 2x - 4y = 0. i 4. (2, 1), (0, 0) to the circle 2 (x2 + ?/) _ 12 x - 4 y + 15 = 0. 6. (0, 0), (- 2, - 5) to the circle x'^ -j- y^ - 6x -\- 4y + 4 z=0. 6. (0, 0), (6, - 3) to the circle x'^ + y"^ -\- 6 x - 8y - U =0. Find the radical axis of the circles I 7. x2 + ?/2 + 6 x - 4 ?/ - 3 = and a:2 + ?/2 - 4 X + 8 ?/ - 5 = 0. 8. a;2 + ?/2 - 8 X - 10 y + 25 = and x2 +?/2+8x-2y + 8 = 0. 9. x2 + ?/2 + ax + 6y - c = and rtx2 + ay"^ + a^x + b^y = 0. 10. Find the radical axis and the, length of the common chord of the circles x^ + y^ + ax-\-by + c = o'and x^ + y^ -^ bx + ay + c = 0. 11. Show that the three circles x2 + y2_2x-4y = 0, x2 + 2/2_6x + 4^ + 4 = 0, x2 + ?/2_8x + 8i/ + 6 = have a common radical axis. Find the equation of a fourth circle such that the four shall have a common radical axis. Find the radical centre of the three circles 12. af2 + 2/2 _ 4 a; ^ 8 y _ 5 ^ 0, x2 + y2 _ g x - 10 ?/ + 25 = 0, ' x2 + y2 + 8x + ll?/- 10 = 0. 13. x2-}-y2_|.6x-8?/ + 9 = 0, x2+?/2 + 8x + 2y + 9 = 0, 2(x2+2/2)_5(3a; + y) + 18 = 0. hMA 106] THE CIRCLE 147 / 14. What is the equation of the normal in terms of its slope ? ^ 16. How many normals can be drawn from a point to a circle ? 16. Find the equation of a circle passing through (0, 4) and (6, 0), and hav- ing V 13 for radius. 9 17. Find the equation of a circle whose centre is (3, 4) and which tmiches the line 4x-3y + 20 = 0. 18. Find the equation of the circle passing through the point ( — 3, 6) and touching both axes. ^ 19. Find the equation of the circle touching the line y — c and both axes. Write down the equation of the tangent to the circle J80. x2 + y2 _ 2 a; + 3 y - 4 = at the point (2, 1). '^21. a;2 ^_ y2 ^_ 4 aj _ 6 y _ 13 = at the point (- 3, - 2). 22. Show that the lines y = m{x — r) ±r V 1 + wi^ touch the circle x2 + ?/2 = 2 rx, whatever the value of m may be. Find the equation of the tangent to the circle 23. 9 (x2 + y2) _ 9 (6 X - 8 y) + 125 = parallel to 3 x + 4 j^ = 0. 24. Show that the line x — 2 y = touches the circle a;2 _|_ ^2 _ 4 a; 4- 8 2/ = 0. -4. 25. The line y = 3 a; — 9 touches the circle ic2_|_2/2 + 2x + 4y-6 = 0. Find the coordinates of the point of contact. 26. Find the equation of the tangent to x2 -f 2/2 _ ,.2 ^j) which is perpendic- ular to y = mx 4- 6, (2) which passes through the point (c, 0), (3) which makes with the axes a triangle whose area is ifi. Find the polar of the point . 27. (2, - \) with respect to x2 + 2/2 + 3 x - 5 ?/ + 3 = 0. 28. (- a, 6) with respect to x2 + ?/2 - 2 ax + 2 6y + a2 - 52 = 0. Find the pole of the line •( ^ 29. 2 X -I- 14 2/ = 15 with respect to 2 (x2 + ?/2) _ 3 ^j 4. 5 j, _ 2 = 0. 30. 3 (ax - 6y) -a^^-lfi with respect to x2 + 2/2 _ 2 ax -f- 2 6?/ = a2 -l- h\ ^ 31. Show that the circles x2 + ?/2_4a;^2i/ = 15 and x^ ■\-y'^ — ^ touch one another at the point (—2, 1). 32. Show that the radical axis of two circles bisects their four common tan- gents. 33. The distances of two points from the centre of a circle are proportional to the distances of each from the polar of * other. 148 THE CIRCLE [106 / "^ 34. What is the analytic condition that the origin shall be the radical centre of three given circles ? v^ 35. Find the equation of the circle through the origin and the points of inter- section of the circles x'2-\-y'i-^x-1y + 6 = and x^ + y"^ + ix + dy - 12 = 0. -; What is the ratio of the tangents drawn from any point on it to the two given circles ? / >^ 36. Find the equation of the circle which touches the line 4 ?/ = 3 x and passes through the common points of x2 4-2/2 = 9 and ic^ + ?/2;+ x + 2y=U. 37. What is the ratio of the tangents drawn from any point on the third circle in Ex. 11 to the other two circles ? ^_ 38. Find the equations of the straight lines which touch both of the circles x2 + 2/2 = 4 and (x - 4)2 -f ^2 _ i, ^^g^ Sx±V7 y = S and x ± vT5 y = 8. 39. Find the equations of the common tangents to the circles x'^ + y^-\-6y-\- 6 = and x"^ -h y^ - 12 y -\- 20 = 0. 40. If the length of the tangent from the point (x', y') to the circle x2 + ?/2 = 9 is twice the length of the tangent from the same point to x2 + 2/2 _j_ 3 ^j _ g ^ _ q^ show that ^,2 + 2/'2 + 4 X' - 8 y' + 3 = 0. 41. If the tangent from P to the circle x2 -i- ?/2 -f 3 y = is four times as long as the tangent from P to the circle x^ + y^ = 9, show that the locus of P is 5(x2-f?/2) = ?/ + 48. 42. The length of a tangent drawn from a point P to the circle x'^ + y'^+4:X-6y + 4 = is three times the length of the tangent from P to the circle x2 + 2/2-6x + 2?/ + 6 = 0. Find the locus of P. 43. Find the locus of a point whose distance from the origin is equal to the length of the tangent drawn from it to the circle x2 + ^2_8x-4?/ + 4 = 0. 44. Find the locus of a point P whose distance from a fixed point is in a constant ratio to the tangent drawn from P to a given circle. Under what condition is the locus a straight line ? 46. Show that the polar of any point on the circle x2 + 2,2 _ 2 ax - 3 a2 = 0, with respect to the circle x^ + ^2 ^ 2 ax - 3 a^ = 0, will touch the parabola 2/2 + 4 ax = 0. 106] THE CIRCLE 149 46. Show that the polars of the point (1, 0) with respect to the two circles x^ -^ y^ + 4:X — 14: = and x^ -\- y^ = 4 are the same line ; show that the same is true of the point (4, 0). 47. Find two points such that the polars of each with respect to the two circles x'^ + y^-2x-S = and x^ -{- y^ -i- 2 x - 17 = coincide. 48. A certain point has the same polar with respect to two circles ; prove that any common tangent subtends a right angle at that point. Show also that there are two such points for any two circles. 49. Find the locus of the intersection of two tangents to jc2 4. ^2 _ ,.2 which are at right angles to one another. 50. Find the locus of the intersection of two tangents to x^ -\- y^ = r^ which intersect at an angle a. 61. Show that if the coordinates of the extremities of a diameter of a circle are (xi, yi) and (x2, y^), respectively, the equation of the circle will be {x - xi){x - X2) -\- (y ~ y\) (y - 2/2) = 0. [Suggestion. Lines joining any point (a;, y) on the circle to (xi, yi) and (X'z, y-z) are at right angles to one another.] Find the equation of the circle which touches 62. the lines x = 0, x = a, and 3y = 4x + 3a. OneAns. 4 {x^ + y^)- 4: a {x-h 6 y)+ 26 a^ = 0. 53. both axes and the line - + | = 1. a 64. Prove analytically that the locus of the middle points of a system of parallel chords of a circle is the diameter perpendicular to the chords. (See § 99.) ' 65. Show that as a varies the locus of the intersection of the lines X cos a + y sin cc = a and a; sin a — y cos a = b is a circle. 66. A circle touches the y-axis and cuts off a constant length (2 a) from the X-axis ; show that the locus of its centre is x'^ — y^ = a^. 57. Two lines are drawn through the points (a, 0) and ( — ,a, 0) and make an angle a with one another. Show that the locus of their point of intersection is x^ -\-y^ ±2 ay cot a = a^. 58. If the polar of the point (x', y') with respect to the circle x^ + y^ = ^2 touches the circle x^-i- y^ = 2 ax, show that y'^ + 2 ax' = a^. 59. Show that if the axes are inclined at an angle w, the equation of the circle is (§ 8) (a; _ a)2 + (y - by + 2 (x - a) (y- 6) cos w = »^, where (a, 6) is the centre and r the radius. CHAPTER X THE ELLIPSE AND HYPERBOLA 107. Standard equations of the tangent^ polar, and normal to the ellipse and hyperbola. It has been shown in § 89 and § 90 * that, if the axes of the curve are taken as coordinate axes, the equations of the ceyitral conies may be written in the standard form Then the coordinates of the foci are (± ae, 0); the equations of a 2b^ the directrices are « = ± - ; the length of the latus rectum is — ; and e = a For equation (1) formula (6), § 92, gives Equation (2) is the equation of the polar (§ 94) of the point («', y') with respect to the central conic (1), which polar is a tangent at the point {x'y y') when (x', y') is on the conic. The equation of the normal at any point (x', y') on the conic (1) is "-"' = ffc("-^'>' "' ^ = "'• f (')' « ^2-] (3) Ex. 1. Find the equations of the central conies when the origin is at either focus ; at either vertex ; at the point (^, k) , the coordinate axes being parallel to the axes of the conic. Ex. 2. What relation does the line (3) have to the conic when (x', y') is not on the curve ? * These sections should now be carefully reviewed. t We shall use this form of the equation, although the simpler form ax^ + by^ = 1 is sometimes more convenient. When the double sign db or =p is prefixed to b^, the upper sign holds for the ellipse and the lower for the hyperbola. All results are true for both curves unless the contrary is expressly stated. Furthermore, results for the ellipse include those for the circle as the special case when a = 6. 160 108] THE ELLIPSE AND HYPERBOLA 151 108. To find the equation of the tangent to the conic in terrns of its slope m. " Assume the equation of the tangent to be y = mx + c, (2) where m is known, and c is to he determined so that (1) and (2) shall intersect in two coincident points (§57). t Eliminating y between (1) and (2) gives ^ + (^^ + cf _ -I / a' 6^ or x" {a^m" ± 6' ) + 2 ahmx + a^ {^ ^ ^a^^2 Whence c" = a'm^ ± h\ (4) That is, the points of intersection of the straight line and the conic will coincide if c = ± V a'm' ± h\ (6) Hence the line whose equation is y = mx ± V a2m2 ± 62, (6) will touch the conic (1) for all values of m. The double sign before the radical in (6) shows that there are two tangents for every value of m ; i.e. there are two tangents to a central conic parallel to any given straight line; and these two parallel - tangents are equidistant from the centre of the conic. Ex. 1. Derive equation (6) by the method used in § 98. Ex. 2. In a similar manner show that the equation of the normal to (1) expressed in terms of its slope is w(a2 T 62) Va2±62w2 Ex, 3. How many normals can be drawn from a given point to a central conic ? 152 THE ELLIPSE AND HYPERBOLA [109 109. Geotnetric properties of the ellipse and hyperbola. Y ^ Let the tangent at P(x', y') meet the axes in T and T"; let the normal at P meet the axes in N and N' ; let BP be the ordinate of P and F, F' the foci of the conic. Draw FG, F'G', and O/iT perpendicular to the tangent PT. ^^ThenOr=^,^ OT'^-^.^;.^ ^[(2), § 107.] y a^— X 2/' .1^ 0N= eV, ON' = ^^y'- 'l^^ [(3), § 107.] Subnormal = i2iV= (e^ — l)x' = ?/ die' N^ T OK'NP=FG . JF^'G^' = ± 61 PN^PM'= FP.'F'P:=± (a' - eV). (§§ 89, 90.) F^G and FG^ bisect ^JV. The locus of G^ and G^' is a^ + 2/' = «^ [Use (6), § 108.] (1) (2) (3) (4) (P) (6) (7) (B) (^) F^N __ F' 6 +777r 2 ae + e^x' ^ a + e x' ]^F " OF- 0N~ ae- e^x' ~ a - ex' F'N^ F'P y *** NF ±FW (§§89,90.) (10) 109] THE ELLIPSE AND HYPERBOLA 153 Therefore the tangent and the normal bisect the angles between the focal radii FP and F'P. Hence, if an ellipse and a hyperbola have the same foci, the tangent and the normal to one of the curves at any one of their four common points are, respectively, the normal and the tangent to the other. That is, the two conies intersect orthogonally. Conies having the same foci are called Confocal Conies. Ex. 1. Explain what would happen if a light were placed' at one focus of an ellipse ; a hyperbola. Ex. 2. What is the limit of ON, ON', and BN asx' = a? asx' = 0? Ex. 3. Show that equations (1), (3), and OK' NP=b^ are also true when P is any point, TT' the polar of P, and PN is perpendicular to TT'. Ex. 4. Show that the equation 1 represents a system of coit- al ^\ 62 4. X focal conies, where X is the arbitrary parameter. Investiijate the nature of these conies for values of X ranging from — oo to + oo . Show that two confo- cals, an ellipse and a hyperbola, pass through every point in the plane, and that these meet at right angles. \ 154 THE ELLIPSE AND HYPERBOLA [109 EXAMPLES Find the eccentricity, foci, and latus rectum of each of the following conies : 1. a;2 + 2 2/2=:4. 2. ix^-9y^ = S6. 3. 4 x2 + y2 = 8. y 4. 3 a;2 - 2/2 = 9. f 6. 3(x- 1)2 + 4(^ + 2)2=1. y/ 6. 3(2/ - 1)2 -4(«+ 1)2 = 1. Find the equation of an ellipse referred to its axes 7. if the latus rectum is 6 and the eccentricity ^. 8. if the latus rectum is 4 and the minor axis is equal to the distance between the foci. 9. Find the equation of the hyperbola whose foci are the points (+4, 0) and whose eccentricity is ■y/2. 10. Find the eccentricity and the equation of the ellipse, if the latus rectum is equal to half the minor axis. 11. Find the equation of the hyperbola with eccentricity 2 which passes through (-4, 6). / 12. Find the equation of the ellipse passing through the points ( — 2, 2) and (3, — 1); also the equation of the hyperbola through (1, — 3) and (2, 4). Through how many points can a central conic be made to pass if its axes are given ? Why ? 13. Find the eccentricity and the equation of a central conic if the foci lie midway between the centre and the vertices ; if the vertices lie midway between the centre and the foci. 14. Show that the tangents at the ends of either axis of a central conic are parallel to the other axis ; and also that tangents at the ends of any chord through the centre are parallel. 16. Find the equations of the tangents and normals at the ends of the latera recta. Where do they meet the a;-axis ? One Ans. y + ex = a. / 16. Show that the line ?/ = 2 a; — y'| touches the conic 3 x2 - 6 2/2 = 1. 17. Find the equations of the tangents to the ellipse x"^ + 4ty'^ = \Q which make angles of 45° and 60° with the x-axis. 18. Show that the directrix is the polar of the focus. 19. If the slope of a moving line remains constant, the locus of its pole with respect to a central conic is a straight line through the centre of the conic. 110] THE ELLIPSE AND HYPERBOLA 155 110. Conjugate Hyperbolas. The two hyperbolas whose equations are ^_r_i a' W~ ' and or x" -1, 62 -".=1, (1) (2) are so related that the trans- verse axis of the one is the conjugate axis of the other. The two hyperbolas are then said to be conjugate to one another. The eccentricity of the Conjugate Hyperbola* is ei = the coordinates of its foci are (0, ± be^) ; the equations of its direc- trices are 2/ = ± — ; and its latus rectum is — ^• When a = b, equations (1) and (2) become, respectively, V6' + a' /2 = a\\ and y^ — Qi? = c?.) Hence if a hyperbola is equilateral or rectangular [§ 90, (15)], its conjugate is also rectangular. Two conjugate hyperbolas are not, in general, similar (§ 116), i.e. of the same shape, but two conjugate rectangular hyperbolas are equal." * The hyperbola (2) is usually called the Conjugate Hyperbola, while (1) is called the Original, or Primary Hyperbola. It is to be noticed that the equation of the conjugate hyperbola is found by changing the sign of one member ot the equation ol the primary hyperbola. Likewise the equation of the conjugate ellipse is found to be Henoe the conjugate of an ellipse is imaginary. 156 TPIE ELLIPSE AND HYPERBOLA [111 111. To find the locus of the point of intersection of two perpendicu- lar tangents to the conic or ¥ The equation of any tangent to (1) may be written (§ 108) y = mx + ^ahn^ ± h^. (2) If this line (2) passes through (ccj, 2/i)> we shall have which when rationalized becomes (x,'-a')m'-2x,y,7n+(y,'Tb')=0. (3) This equation is a quadratic in m whose two roots are the slopes of the two tangents which pass through the point (x^, y^), whose locus is required. Let mi and mg be the two roots of (3) ; then (§ 68) mimg — — • Xi — a^ The two tangents will be at right angles if mimg = — 1 (§ 45) ; i.e. if or , aj/ + 2/i' = a'±&2. (4) The required locus is, therefore, the circle x^ + y^ = a^±b^, (5) which is called the Director Circle of the conic. Cor. I. Ifaj y2 = ci sin <^. Since 2/1 = -2/2 = & sin , the coordinates of P are a Xi = a cos «t), Vi-h sin <|>. (6) II. The circle described on the transverse axis of a hyperbola as diameter may be called the Auxiliary Circle of the hyperbola. Let P(x, y) be any point on the hyperbola and NP its ordinate. Draw NQ tangent to the auxiliary circle at Q, so that P and Q are on the same side of the transverse axis when P is on the right branch, and on opposite sides when P is on the left branch of the curve. Then, as P describes the complete hyperbola in the direc- tion indicated by the arrows, Q will move consecutively around the circle in the direction indicated. Thus, for every position of P on the hyperbola, there is one and only one corresponding position of Q on the circle. Hence P and Q may be called Corresponding Points, and the angle XOQ = _ y — 6 sin <^ ^^x a cos ' b sin — b sin ' Since cos — cos <^' = — 2 sin |( + <^') sin ^{ — ') and sin <^ — sin <^' = 2 cos J(<^ + <^') sin ^(<^ — <^'), equation (1) reduces to ^-"^^^ ^ |-^^^^ (2) -2sinK')* .-. ^ cos i(<|> + 4.0 + 1 sin !(+ y b tan <^) and (a sec <^', b tan tf>') on the hyperbola can be shown to be ^ cos ^(4. - 4>') - 1 sin |(<|> + 4>') = cos 1(4. + 4»'). W 160 THE ELLIPSE AND HYPERBOLA [114 To find the equation of the tangent at the point (}>, we put <^' = <^ in equations (3) and (4), and we obtain for the ellipse ^cosc|» + |sm<|> = l, (5) and for the hyperbola -sec<|>-gtan<|» = l. (6) From equation (3) we see that if the sum of the eccentric angles of two points on an ellipse is constant and equal to 2 a, the equation of the line joining them is - cos « + T sin a = cos J( — <^'). (7) Hence the chord is always parallel to the tangent i cos a + 1 sin « = 1. (8) Conversely, in a system of parallel chords of an ellipse, the sum of the eccentric angles of the extremities of any chord is constant. Similarly from equation (4) we see that if the sum of the eccen- tric angles of two points on a hyperbola is constant and equal to 2 a, the equation of the chord through these points is - cos i( — <^')— I sin a = cos a, (9) and therefore the chord, and the tangent at the point a, viz., *t sin a — cos a, (10) ah ' ^ always meet the 2/-axis in the same fixed point. 114. To find the equation of the normal at any point in terms of the eccentHc omgle of the point. Let (a cos = :^-^^(x-acos4>), (1) Similarly we find the equation of the normal to the hyperbola at the point (a sec , b tan then — = cos sec a, ^ = sin sec a. Eliminate for the locus.] What is the corresponding theorem for the hyperbola ? A (1) 162 THE ELLIPSE AND HYPERBOLA [115 115. Def. An Asymptote* to a curve is the limiting position of the tangent line as the point of contact moves off to an infinite distance, while the line itself remains at a finite distance from the origin. Tojind the asymptotes of the hyperbola. a^ b'" As in § 108, the abscissas of the points where the line y = mx 4- c (2) meets the hyperbola are given by the equation x" (aV - 5') + 2 a'cmx + a' (c" + b^) = 0. (3) If the line (2) becomes an asymptote, both roots of equation (3) must become infinite. Hence the coefficients of a^ and x must both approach zero (§ 77). That is, a^cm = 0, and a^m^ — b^ = 0. .-. lim c = 0, and lim m = ± — (4) a ^ ^ Substituting these limiting values in (2), we have for the required equations of the asymptotes or expressed in one equation \) C^ Therefore the hyperbola has two asymptotes, both passing through the centre and equally inclined to the transverse axis. The equations of the asymptotes to a hyperbola can also be found by considering the limiting form of the equation of the tangent as the point of contact moves off to an infinite distance. The equation of the tangent to (1) at (x'j y') is xx' vv' . ,_ ^-f = l- ■ (7) * Greek, dtriJ/xn-Twros, not falling together. 115] THE ELLIPSE AND HYPERBOLA 163 Since the point (x\ y') is on the conic (1), we have 0/ Hence quotation (7) may be written H _ If now the point of contact («', y') moves off to an infinite distance so that x' becomes infinite, the limiting position of the line (8) is given by the equation x v 5±f = 0, (9) which is the same as equation (5) above. CoR. I. Two conjugate hyperbolas have the same asymptotes, which are the diagonals of the rectangle formed by the tangents at their vertices. CoR. II. A straight line parallel to an asymptote will meet the conic in one point at infinity. For, if c is not zero, only one root of (3) is infinite. Cor. III. The line y = mx will cut the hyperbola in real or imagi- nary points according as m<,or^-- It will meet either the hyperbola a or its conjugate in real points for all values of m. Cor. IV. The asymptotes of an ellipse are imaginary. For, if we change the sign of 6^, the values of m for infinite roots in (3) become imaginary. It is to be noticed that the equations of two conjugate hyperbolas and the equation of their common asymptotes, viz., -2-^=±l and ^-^ = 0, a^ Ir a^ Ir differ only in their constant terms. Moreover, this must always be true ; for any transformation of coordinates will affect the first mem- bers of these equations in precisely the same way. Hence the new equations will differ only in their constant terms (not usually by unity) ; and the value of the constant in the equation of the asymp- totes will be equal to half the sum of the constants in the equations of the two hyperbolas. 164 THE ELLIPSE AND HYPERBOLA [116 116. Similar and Coaxial Conies. Since a^Ksbnd 6^^ are the semi-axes of the ellipse S+P=^' (1) its eccentricity is given by the equation e = (§ 107.) a^K a That is, the eccentricity of (1) is the same as the eccentricity of the ellipse represented by the standard equation 5 + S = l- (2) Two conies having the same eccentricity are said to be similar ; for one is then merely a magnification of the other. Conies having their axes on the same lines are said to be Coaxial. Hence if K is an arbitrary parameter, (1) will represent a system of similar and coaxial ellipses. For any particular value of K the equations i-t=±K (3) represent a pair of conjugate hyperbolas (§ 110). If, however, jfiTis arbitrary, equations (3) will give (as in the case of the ellipse) a system of similar and coaxial hyperbolas, together with their corresponding conjugate hyperbolas, which are also similar. It follows from § 115 that these two infinite systems of hyperbolas all have the same asymptotes. Moreover, the asymptotes are the limit which both systems approach as K becomes zero. Thus two intersecting lines are not only one of a system of similar and coaxial hyperbolas, but may also be regarded as a pair of self-conjugate hyper- bolas. It is also to be noticed that although both axes of two intersecting lines are zero, the limit of their ratio as they approach zero is the tangent of half the angle between the lines. Cor. The axes of similar conies are proportional. 117] THE ELLIPSE AND HYPERBOLA 165 117. To find the locus of the middle points of a system of parallel chords of a central conic. I. Let AB be any one of a system of parallel chords of the ellipse ^ + t = K. (1) Let P(x', y') be the middle point of ABj and y its inclination to the aj-axis. Then the equation of AB may be written [§ 43, (4)] x—x' y — y' cos y Sin y ot x = x' -\-r cos y, y = y' -; r sin y, (2) where r is the distance from (a;', y') to any point (a;, y) on the line. If the point (x, y) is on the ellipse, these values (2) may be sub- stituted in equation (1) ; this gives (x' + r cos y^*^ , (7' -f r sin y)'^ j^ ^^ -p ~ 7^ "-^ = -fl., or /cos ^2 ^ ^2 y T- ^ ^2 -^ 52 y ^^2-^52 ^ "• W The values of r lound by solving this quadratic equation are the lengths of tka lines PA and PB, which can be drawn from P 166 THE ELLIPSE AND HYPERBOLA [117 along AB to the ellipse. Since P is the middle point of the chord, these two values of r must be equal in magnitude and opposite m sign ; i.e. the sum of the roots of (3) must be zero. Hence (§ 68) a;'cos Y y' siny^r. .^. a' ^ b' ' ^ ^ The required locus is, therefore, the straight line y= cot 7 • a?. (5) Henee every diameter (§ 99) of an ellipse passes through the centre. CoR. I. All chords intercepted on the same line, or on a series of parallel lines, by a system of similar and coaxial ellipses are bisected by the same diameter. Since equation (5) is independent of K, the locus of P is the same whatever value may be given to K in (1). (§ 116.) CoR. II. If a straight line meets each of two similar and coaxial ellipses in two real points, the tic o portions of the line intercepted between them are equal ; i.e. AA = BB'. CoR. III. Chords of an ellipse which are tangent to a similar and coaxial ellipse are bisected at the point of contact. CoR. IV. The tangent at either extremity of any diameter is parallel to the chords bisected by that diameter. II. In like manner, if "y is the inclination to the a>-axis of a system of parallel chords of the hyperbolas a^ b^ ' ^ ^ we find the locus of the middle points of the chords to be the straight line y=-^cot7-a^, (7) for all values of K, including the case K— 0. Hence all diameters of a hyperbola pass through the centre. The preceding corollaries apply also to similar and coaxial hyper- bolas. 117] THE ELLIPSE AND HYPERBOLA 167 Cor. V. Cliords intercepted on the same line, or on a system ofpar- ellel lines, by two conjugate hyperbolas, and their asymptotes, are bisected by the same diameter. Cor. YI. If a straight line meets each of two conjugate hyperbolas in real points, the two poHions of the line intercepted between the curves are equal. TJie portions intercepted between either hyj)erbola and the asymptotes are also equal; i.e. A" A = BB" and A' A = BB'. Hence the part of a tangent to a hyperbola included between the two branches of its conjugate, and also the part included between its asymptotes, are bisected at the point of contact. Ex. 1. Find the locus of the middle points of chords of the ellipse 4 x2 + 9 y2 = 36 parallel to 3 a; - 2 ?/ = 1. Ex. 2. Find the equation of the chord of the hjrperbola 26 x^ - 16 j/2 = 400 which is bisected at the point (2, - 6). Ex. 3. Find the equation of the chord of the ellipse 4 a;^ + 8 t/^ = 32 which is bisected at the point ( — 2, 1). \ A.l^L/ 168 THE ELLIPSE AND HYPERBOLA [118 Conjugate Diameters 118. We have seen in § 117 that all diameters of a central conic pass through the centre. Conversely, every chord which passes through the centre is a diameter, i.e. bisects some system of parallel chords. For, by giving y a suitable value, equations (5) and (7) of § 117 may be made to represent any chord through the centre. If y' is the inclination to the aj-axis of the diameter which bisects all chords whose inclination is y, we have, from (5) and (7) of § 117, h- tan y'—^— cot y, a^ or tan y tan y' = q= (1) a^ Let y = mx and y = m'xhe any two diameters. Then, if the first bisects all chords parallel to the second, we have from (1) , o mm'=T^' ' (2) Since this is the only condition that must hold in order that the second may bisect all chords parallel to the first, it follows that, if one diameter of a conic bisects all chords parallel to a second, the second diameter will also bisect all chords parallel to the first. Def. Two diameters, so related that each bisects every chord parallel to the other, are called Conjugate Diameters.* For example, the axes are a pair of conjugate diameters. From equation (2) we see that the slopes of two conjugate diameters of an ellipse have opposite signs, whereas in the hyper- bola the signs are the same. (See figures under § 117.) If m < -, then m' > -, numerically, a a Hence conjugate diameters of an ellipse are separated by the axes, and also by the lines ay= ±bx; while conjugate diameters of a hyperbola are separated by the asymptotes, but not by the axes. * It is evident that none but central conies can have conjugate diameters, since in the parabola all diameters have the same direction (§ 99) . 118] THE ELLIPSE AND HYPERBOLA 169 If m = - , then m' = in the ellipse. The two diameters are then equally inclined to the major axis, and, from the symmetry of the curve, the two diameters are equal in length. The equations of the equal conjugate diameters of an ellipse are, therefore, i, v=±l^. (3) If m = ± -, then in the hyperbola m' = ± -, respectively. a a Therefore equi-con jugate diameters of a hyperbola coincide with an asymptote, so that an asymptote may be regarded as a self-conju- gate diameter. The equi-con jugate diameters of a conic, therefore, in all cases -V coincide in direction with the diagonals of the rectangle formed by >a: tangents at the ends of its axes. ^ CoR. I. If tivo diameters are conjugate with respect to one of two con- ^" jugate hyperbolas, they will be conjugate with respect to the other also. ^%m. and (7), § 117.] v^ >^5^ CoR. II. One of two conjugate diameters of a hyperbola meets the y * j f curve in real points, and the other meets the conjugate hyperbola in real JTT^ points. (Cor. Ill, § 115.) r ^^^ For this reason we will call the extremities of any diameter of a ^ *^ ^ hyperbola the points in which it cuts either the primary or the con- K^J jugate hyperbola, as the case may be ; and the length of the diameter ^ O will be the distance between these points. ^^Jv Cor. III. Tangents at the ends of any diameter are paraMel to the ^ n conjugate diameter. /N^ ^% _ ^'^^^ •1 ^ Ex. 1. Write down the equations of the diameters conjugate to f^ \iJi*^ ^ x-y = 0,x-\-y = 0,by = ax,ay = bx. ^^ ^ Ex. 2. In the ellipse 2x^ + ^y^ = S, find two conjugate diameters, one of which bisects the chord as + 2 y = 2. g^^^^ Ex. 3. Find the equation of the diameter of the hyperbola 16x^ — 9y^ = 144 TC conjugate to x -f- 2 y = 0. •^>i^ y^ ^^* "*• ^^^^ ^^^ conjugate diameters of the ellipse 4 a? -f- 25 y* = 100, one of /which passes through the point (3,-1). '^^.A Ex. 6. Find the equation of the chord of the hyperbola x^ — y^ = l6, whose ^«^ddle point is (- 2, 'N^ftW-'"^ 170 THE ELLIPSE AND HYPERBOLA [119 119. Oiven the extremity of any diameter , to find the extremities of the conjugate diameter. I. Let Pi(fl7i, 2/i)j -Pa (''^2) 2/2) be the extremities of two conjugate diameters of an ellipse. Then the equations of OP^ and OP2 are or = —x and y = ^x. Xi X.2 XiX.2 .2 ~^ a^ = 0. [(1),§118.] (1) a^ ■ h^ Let <^i, <^2 be the eccentric angles of P^ Pg, respectively. Then ajj = a cos <^i, 2/i = ^ sin <^i, 0^2 = a cos <^2> 2/2 = ^ sin <^2- (§ 112, 1.) Substituting these values in (1), we have cos <^i cos Hence the corre- sponding diameters OQi, OQ2 of the auxiliary circle are perpendicu- lar to one another. 119] Since THE ELLIPSE AND HYPERBOLA 171 <^2 = <^i±90°, sin <;^2 = ± cos <^i, cos 2 — T sin <^i. Therefore the extremities of two conjugate diameters of an ellipse may be written JPi (a cos <|)i, 6 sin <|»i) and 1*2 ( ^ « siii<|>i, ± 6 cos <|>i) , '] (3) or ^i(a?i, Vi) and Pgf T ^2/1? ± -^ij- (4) II. If Pi, F2 are the extremities of two conjugate diameters of a hyperbola, equation (1) becomes Then from § 112, II, and § 118, Cor. II, we also have Xi = a sec ^1, 2/1 = 6 tan <^i, X2 = a tan <}>2, 2/2 = 6 sec ^2- Substituting these values in (4) gives sec <^i tan 2 — tan <^i sec <^2 = 0, sin 2 sin <^i or cos ^1 cos <^2 cos <^i cos 2 ,; ^2 = <^ or <^2 = TT — i = 0. (6) (7) 172 THE ELLIPSE AND HYPERBOLA [120 That is, the eccentric angles of the ends of two conjugate diame- ters of a hyperbola are either equal or supplementary. Therefore the corresponding diameters OQi, OQ2 of the auxiliary circles are equally inclined to the transverse axes of the two conjugate hyper- bolas. Since tan <^2 = ± tan <^i and sec <^2 = ± sec «^i, the extremities of any two conjugate diameters of a hyperbola may be expressed in the form Pi (a sec <|>i, b tan 4>i) and J*2 ( ± « tan c|>i, ± b sec «|>i), Pi(^ 2/1) and J^2(±^yi, ±-^1)' or (8) 120. The sum of the squares of two conjugate semi-diameters of an ellipse is constant. Let the extremities of any two conjugate diameters be [§ 119, (3)] Pi (a cos -}-b^ sin^ <^, [(4), § 7] h"' = a^sm^-\-b^cos^. ... a'^ + 6^2 = a2 + 62. 121. The area of the parallelogram formed by tangents at the ends of conjugate diameters of an ellipse is constant. Let Pj (a cos <^, b sin <^) and P2 ( =F « sin , ±b cos <^) be the extremities of any two conjugate diameters, and let ABCD be the parallelogram formed by tangents at the ends of these diameters. Draw P^N perpendicular to OP2 ; then Area ABCB = 4.0P2' P^N= 4 6' . P,N. Since OP2 is parallel to the tangent at Pj [§ 118, Cor. Ill], the equation of OP2 may be written [(5), § 113] - cos A 4- 1 sin d> = 0. X a mm \ 121] P,N= THE ELLIPSE AND HYPERBOLA \ cos^ <^ + sin^ <^ ah * 173 ab Vcos^ -5 52 .-. Area ABCD = ^ab. Y Cor. If angle P1OP2 = w, then -x;^ ^ a' a'V EXAMPLES 1. The difference of the squares of two conjugate semi-diameters of a hyper- bola is constant. 2. The area of the parallelogram formed by tangents to two conjugate hyper- bolas at the ends of two conjugate diameters is equal to 4 ab. 3. If w denotes the angle between two conjugate diameters of a hyperbola, then sin w = ab a'h' \f 4. Show that the acute angle between two conjugate diameters of an ellipse ^is least when the diameters are equal. 5. Show that the eccentric angles of the extremities of the equi-conjugate diameters of an ellipse are 45° and 136°. 6. Conjugate diameters of a rectangular hyperbola are equal, and equally inclined to the asymptotes. 7. Tangents to two conjugate hyperbolas at the extremities of two conjugate diameters meet on the asymptotes. (See Fig. § 117, H.) 174 THE ELLIPSE AND HYPERBOLA [122 8. The area of the triangle formed by two conjugate semi-diameters and the chord joining their ends is constant. 9. Prove that for all values of m the line passes through the extremities of two conjugate diameters of an ellipse. "What is the corresponding equation for the hyperbola ? 10. The product of the focal radii of a point P is equal to the square of the semi-diameter parallel to the tangent at P. 122. To find the eqvxition of a hyperbola when referred to its asymp- totes as axes of coordinates. The equation of the asymptotes, referred to themselves as axes of {coordinates, is xy = 0. Therefore the equations of any two conjugate hyperbolas referred to them is of the form (§ 115) xy = ±K. (9) Hence the equation xy = K, where K is any constant, always represents a hyperbola referred to its asymptotes as axes of coordi- nates; so that, if the axes of coordinates are at right angles, the hyperbola xy = K is rectangular. 123. To find the polar equation of a central conic, the pole being ai the centre. The formulae for changing from rectangular to polar coordinates are (§6) /i • z, ^ ^ a; = pcos^, 2/ = psm^. These values substituted in or n2= ±^'^' - ±^'^' r a? sin2 e±W cos'' B a"- (a? T b"") cos^ 6 which is the required equation. ^ l-e2cos2e' 123] THE ELLIPSE AND HYPERBOLA 175 EXAMPLES ON CHAPTER X 1. Show that the sum of the squares of the reciprocals of two perpendicular diameters of an ellipse is constant. (See § 123.) 2. If an equilateral triangle is inscribed in an ellipse, the sum of the squares of the reciprocals of the diameters parallel to the sides is constant. 3. Find the inclination to the major axis of the diameter of an ellipse the .sqii ire of whose length is (1) the arithmetical mean, (2) the geometrical mean, and (3) the harmonical mean between the squares on the major and minor axes. (§123.) ^ws. «o (3), 45°. 4. The locus of the poles of a series of parallel chords is the diameter which bisects the chords. Hence the line joining the intersection of two tangents to the centre bisects the chord of contact. i<^ 6. Find the equations of two conjugate diameters of the hyperbola b'^x^ — a-y- = a-b'^, one of which bisects the chord through (0, b) and (ae, 0). 6. In the hyperbola 4 aj2 _ 5 ^-2 _ 20 find the equations of two conjugate diameters, one of which bisects the chord 2 a; + 3 y = C. 7. If straight lines drawn through any point of an ellipse to the ends of any diameter POP' meet the conjugate diameter P\OP\' in Q and .B, show that Oq'OR = OPx^. 8. Show that the locus of the intersection of the perpendiculars from the foci upon a pair of conjugate diameters of an ellipse is a similar concentric ellipse. 9. Two conjugate diameters of an ellipse are drawn, and their four extremi- ties are joined to any point on a given circle whose centre is at the centre of the ellipse. Show that the sum of the squares of these four lines is constant. 10. P, is a point on a branch of a hyperbola, P2 is a point on a branch of its conjugate, OPi and OP-i being conjugate semi-diameters. If F\ and F^ are the interior foci of these two branches^ respectively, show that F2P2 ~ FiPi = a'^b. 11. Find the equation of the chord passing through the extremities of two conjugate diameters. 12. The lengths of the chords joining the extremities of two conjugate diameters of an ellipse are Va2 + 6* ± a2g2 sin 2 If, as is generally possible, x' and y' be so chosen that ax'-\-hy'-{-g = 0, (3) and hx' + by'-\-f=0, (4) the coefficients of x and y in (2) will both vanish, and the equation referred to (a;', y') as origin will then be aa^ + 2 hxy + by"^ + c' = 0, (6) where c' = ax'^ + 2 hx'y' + 6y '2 ^ 2 gx' + 2 /?/' + c. (6) The locus represented by (5) is symmetrical with respect to the origin [§ 28, (9)] ; i.e. the origin is now at the centre. Hence the coordinates of the centre of the conic represented by (1) are the values of x' and y' which satisfy equations (3) and (4), ab -h^ ab- h^ Hence, if h'-^ab^ 0, the coordinates of the centre are both finite^ and this transformation is possible. 179 180 GENERAL EQUATION OF THE SECOND DEGREE [125 Multiply equations (3) and (4) by a;' and y\ respectively, and sub- tract the sum from the right member of (6) ; we thus get c' = gic' +fy' +c. (8) where A = dbc + 2 fgh - af^ - hg'^ - ch^, (10) If A = 0, then c' = 0, and equation (5) may be written ax + hy= Ty^fi^ — ab. (11) Hence the locus is two straight lines, which will be real, coinci- dent, or imaginary according as ^^ — a& >, =, or < 0. If A = 0, and also ah — h? = 0, then c' is not necessarily zero. The first three terms of equation (5) are then a perfect square. The equation may therefore be written Vaic -f Vfty ± V^^ = 0, (12) and represents two parallel lines, which coincide when c' = 0. The function of the coefiicients denoted above by the symbol A is called the Discriminant of the General Equation. Hence an equation of the second degree will represent two straight lines if its discriminant vanishes. 125. Whenh^-ah^O. In order to reduce the equation [(5), § 124] ax^ + 21ixy + hy^ + c' = 0, (1) to any one of the standard forms (§§ 89, 90) we must remove the term 2 hxy. For this purpose we turn the axes through a certain angle 6, keeping the origin fixed. To turn the axes through an angle 6 we substitute for x and y, respectively [§ 54, (1)], « cos d — 2/ sin 6 and XB>mB + y cos B. (2) Substituting these values in (1), expanding and collecting terms, we have 125] GENERAL EQUATION OF THE SECOND DEGREE 181 (a cos^ ^ H- 2 ^ sin ^ cos ^ + & sin^ ^)ar' + 2[(6 - a) sin ^ cos ^ + ^(cos^ 6 - sin^ e)'\xy + (a sin^ ^ - 2-^ sin d cos ^ + 6 cos^ e)y^ + c' = 0. (3) The coefficient of xy in equation (3) will vanish if B be so chosen * ^* 2(6-a)sindcosd + 2A(cos2^-sin2d) = 0. (4) i.e. if (a-6)sin2^ = 2^cos2^. (5) .-. tan2e=-^. (6) a-b ^ ^ Whence sin 2 ^ = ± ^^ — , (7) V(a-6)2 4-4^2 ^ and cos 2 ^ = ± ^~^ (8) Any value of obtained from (6) will reduce (3) to the form a'aj2 + 6'2/2 + c' = 0, or-^ + -^ = 1, (9) a' &' where a' = a cos^ ^ + 2 ^ sin ^ cos ^ + & sin^ 6, (10) and 6' = a sin* d - 2 fe sin ^ cos ^ + & cos* 0. (11) Equation (9) is therefore the required result. The values of a' and 6' may be expressed in terms of a, 6, and h as follows : From (10) and (11), by addition and subtraction, we obtain a' + 6' = a4-&, (12) and a'-6' = (a-6)cos2^+2Asin2^. (13) Substituting (7) and (8) in (13) gives 2h a'-6' = ±V(a-6)* + 4/i2 = ^^=^. (14) sin2d Whence, from (12) and (14), a' = ||a + 6±V(a-6)2 + 4^|, (15) and. 6' = ||a + 6q:V(a-6)2 + 4/i2}. (16) 182 GENERAL EQUATION OF THE SECOND DEGREE [125 The ambiguity in the values of a' and b' given by (15) and (16) may be removed by (14). From the many values of $ which satisfy (6) we will agree always to choose that one which lies between 0** and 180°. Then will always be an a^ute angle, and sin 2^ will always he positive. Therefore it follows from (14) that a'—b' will always have the same sign as h. It is also worthy of notice that the values of a' and b' given by (15) and (16) are the two roots of the equation x'-(a-^b)x- Qv" - ab) = 0. (17) Hence a' and b' will have the same sign or opposite signs, i.e. the conic will be an ellipse or a hyperbola according as A^ — a&<, or>0. If a 4- 6 = 0, then a' = — b' and the conic is a rectangular hyperbola. Ex. transform the equation S x^ -\- 4:xy + 5 y^ + S X - 16 y - IQ = to the standard form, and construct the conic. \ ^^ \ It > O / X The equations for finding the centre are4a; + y + 2 = and 2 x + 5 y = 8. .-. «' = -!, y' = 2. Then c' — gx^ +fy' -\- c = - 36. Therefore the equation referred to parallel axes O'JP, O'T through the 126] GENERAL EQUATION OF THE SECOND DEGREE 183 centre is 8x^ + ixy -{■ 5y'^ = Z6. Also a' = ^\a + b± V(a -6)2 + d/i^ j = J (13 ± 5) = 9 or 4, 6' = I {a + 6 T V(a-6)2 + 4^2J = J(13 T 6) = 4 or 9. Since h is positive, we take a' = 9 and 6' = 4. Hence the equation of the curve referred to its own axes O'X", 0' T' as axes of coordinates is « ,,„ - + ^ = 1. 4^9* Also tan2^=»^^=i a-b 3 Therefore the line O'X" must be drawn so that Z X'O'X" = ^ tan-i |, 126. Whenh^-ab = 0. In this case the coordinates of the centre [(7), § 124] are both infinite, and therefore the first degree terms cannot be removed by changing to a new system of axes parallel to the old. Since the second degree terms now form a perfect square, the general equation may be written (l3y-{-axy + 2gx + 2fy + c = 0, (1) where a = -y/ay p=^b, a has the same sign as h, and fi is always P°«^^i^®- .-. h = a/3. (2) First Method. From equation (6), § 125, we have tan2fl- ^^' __2«^_^tan^ .„. tan2^---^^-^-^^_^— ^-^. (3) .-. taiixis (any other line) is that part of the axis intercepted between two planes through the ends of the line perpendicular to the axis. The projections of a line on a series of parallel axes are evidently all equal. The angle between two lines which do not intersect is equal to the angle between two intersecting lines parallel respectively to the two given lines. 131] COORDINATES 197 Hence, as in plane geometry, the projection of a limited line on any axis is equal to the line multiplied by the cosine of the angle between the line and the axis. Also, the projection of a broken lijie (in space) on any axis is equal to the projection^ on the same axiSj of the straight line joining the ends of the broken line. For example, let p be the distance from the origin to the point (Xj y, z). Then, projecting on any line we get Proj. of p = Proj. of a? + Proj. oty + Proj. of z, (1) This equation is evidently true if p is the diagonal, and a;, y, z are the three dimensions of any rectangular parallelopiped. We shall frequently have occasion to use this special case of the last theorem. Polar Coordinates. Direction Cosines 131. Let P(x, y, z) be any point in space referred to rectangular axes. The position of P will evidently be determined if we know its dis- tance p from the origin and the angles «, )8, y, which OP makes with the axes. The four quantities 0>> «> Pf y) a.re the Polar Coordinates of P. The distance p is called the Radius Vector of the point P, and a, /8, y are called the Direction Angles of the line OP. Since a;, 2/, and z are the projec- tions of p on the three axes, we have 05 = pcos a, y = p cos P, » = p COSY, (1) Cos a, cos Pj and cos y are called the Direction Cosines of the line OP. Hereafter we shall represent them by the letters Z, m, and n, respectively. Then equations (1) become aj = ip, y = mp, z = np. (2) It is to be carefully noticed that Z, m, n are the direction cosines of a directed line; that if the signs of l, m, n are all changed, the 198 COORDINATES [132 direction of the line is reversed. It is evident from equation (2) that the signs of I, m, and n for any line through the origin will be the same respectively as the signs of the rectangular coordinates x, y, and z of any point P on the line, provided OP be taken as the posi- tive direction of the line. Hence we may always choose the polar coordinates of a point so that p shall be positive, and each of the angles a, /8, y shall be less than 180°. The direction cosines of any line are evidently the same as the direction cosines of a parallel line through the origin, since parallel lines make the same angles with the axes. Squaring and adding equations (2) we get p\V^ + w?-\-n^^;>?-^f + z'', (3) and, since p^ = a^ + / + z\ [(2), § 129] we have l^ + m^ + n^^l. (4) That is, the sum of the squares of the direction cosines of any line is equal to unity. Hence the four polar coordinates of a point are equivalent to only three independent conditions. If we divide each of the three numbers a, 6, c by the square root of the sum of their squares, we get Since these results are numbers which satisfy equation (4), they are the direction cosines of some line, whatever the values of a, 6, c may be. That is, any three numbers are proportional to the direction cosines of some line. Note. — Custom is not uniform in regard to the use of the name Polar Coordinates. Many authors apply the name to the system described in § 132. Spherical Coordinates 132. Let OX, OF, OZ, be a set of rectangular axes, and P any point. Then OP, or p, the angle 6 which OP makes with OZ, and the angle <^ which the plane ZOP makes with the fixed plane XOZ are the Spherical Coordinates of the point P, and are written (p, 6, <^). 133] COORDINATES 199 Since OC = p sin Oj the relations between rectangular and spherical coordinates are 0? = p sin 6 cos , 2/ = p sin 6 sin , iS = pCO80. (1) Whence the relations between polar and spherical coordinates are found by equation (1) § 131 to be cos (x = sin 6 cos <|>, cos § = sin sin , 7 = e. (2) If P is a point on the surface of the earth and Z the pole, then $ is the co-latitude and the longi- tude of P. If P is a point on the celestial sphere and Z the pole, is the co-declination and <^ the right ascension of P; if ^ is the zenith, then is the zenith distance and <^ is the azimuth of P. 133. Cylindrical Coordinates. — If the position of the foot of the coordinate z in the plane xy is defined by the polar coordinates (p, 6) instead of (a?, y), then (p, 0, z) are called Cylindrical Coordinates. EXAMPLES 1. Find the direction cosines of a line equally inclined to the three axes. 2. A line makes an angle of 60° with each of two axes. What angle does it make with the other axis ? 3. If one direction angle of a line is 135°, another 120°, what is the third ? 4. What are the direction cosines of a line perpendicular to the a>axis ? the y-axis ? the 2;-axis ? 5. What are the direction cosines of a line parallel to the x-axis ? the ?/-axis ? the 5j-axis ? 6. Find the direction cosines of the line joining the origin to the point (3, - 2, - 1). Of the line joining the points (- 2, 4, 2) and (1, 2, - 4). 7. Find the direction cosines of the line joining the two points (xi, yi, zi) and (X2, yz, 22)- 8. Show that the square of the distance between the two points whose polar coordinates are (pi, «!, Pi, 71) and (p2, a2» P2, .'2) is pi^ + />2^ — 2 pi/)2(cos «! cos a2 + COS ft COS P2 + cos 7i cos 72). CHAPTER XIII LOCI 134. We have seen in § 127 that x = ais the equation of a plane parallel to the 2/2;-plane ; that x = a, y = b are the equations of a line parallel to the 2;-axis ; and that a; = a, y = b, z = c represent a point. So that here we have a plane represented by one equation, a straight line by two equations, and a point by three. We shall now show that, in general, one equation represents a sur- face of some kind ; two equations represent a line of some kind ; and three equations represent one or more points. Let the equation of the locus be F(Xj y, z) = 0. We have seen that the equations of the line through the point (a, b, 0) parallel to the z-axis are x = aj and y = b. Hence, if we put x = aj and y = b in the equation of the locus, we get the equation F(a, 6, z) = 0, which must be satisfied by the coordinates of all points common to this line and the locus. Let the roots of this equation be Zi, 22, etc. Then the locus is met by this line in the points (a, b, z^), (a, b, z^, etc. Since, in general, the number of roots of the equation F{aj bjZ) = is finite, the straight line will meet the locus in a finite number of points. Hence the locus, which is the assemblage of all such points found by assigning different values to a and 5, is a surface and not a solid figure. If the coordinates of a point (x, y, z) satisfy two equations F{x, y^ z) = and <^(ic, y, z) = 0, simultaneously, the point must be on both of the surfaces which these equations represent. Therefore the locus is the curve determined by the intersection of the two sur- faces. When three equations are used simultaneously, they are sufficient to determine absolutely the values of the unknown quan- tities oj, 2/, z. Hence three equations represent one or more points. 135. Equations involving only one or two variables. If an equation contains only one variable, x say, let it be put in the form (x) = 0. We know that this equation is equivalent to (x — a) 200 135] LOCI 201 (a? — 6) (x — c)"' = 0, where a,b,c, ••• are roots of <^(«). Hence such an equation represents one or more planes parallel to the coordinate plane x = 0. Let only one of the variables be absent, so that the equation is of the form F(x, y) = 0. Let P{x, y, 0) be any point in the icy-plane whose coordinates satisfy the equation F{x, y) = 0. Draw a line through P parallel to the 2J-axis. Then all points on this line have the same x and y as P. That is, they are all on the surface. Hence the locus of the equation F(Xj y) = is the cylindrical surface, or cylinder, traced out by a line which is always parallel to the z-axis, and which moves along the curve in the a^-plane defined by the equation F(x, y) = 0. In like manner the equations /(y, z) = and ), which become zero if k = —C± V-4' -\- B^ -\- C^ — D^ and imaginary if k>-C+^/A^ + EP-^C'-D, or if kK-C-^/A'-^B'-^-C-D. 204 LOCI [139 The a»-coiitours are circles whose equations may be written (x-\-Ay-\-(z-\-Cy = A'+C'-(J<^ + 2Bk + D). (3) Likewise the 2/2!-contours are circles whose equations are (y-{-By-{-(z+Cy = B'+C'-{lc'-{-2AJc-\-B). (4) Moreover, the centres of these three systems of concentric circles are the projections of the point (—A, — B, — C), and the radius of each system is -y/A^ -\-B^-{-C^ — D when k is equal respectively to — O, —B, and — A. Hence these contours indicate that the sur- face is a sphere with centre at the point (—A, —By — C) and radius equal to V^^ 4- ^ + C^ — D. This can be shown to be true by writing the given equation (1) in the form (x-^Ay-\-{y->rBy + {z + Cy = A' + B'-{-C'-D, (5) and comparing with equation (1) § 129. Hence the equation of the sphere whose centre is the point (a, 6, c) and radius r is {nc - a)2 + (y - &)2 + (s - C)2 = r2. (6) If the centre is at the origin, the equation is i»2 + 2/2 + s2 = r2. (7) 139. Trace the surface whose equation is [Frost's S. G. p. 5.] {x + yy = az. When x = 0,y^ = az; therefore the trace on the 2/2;-plane is a parabola OQ, whose axis is OZ and vertex O. Similarly the trace OP on the x^-plane is the equal parabola x^ = az, having the same vertex and axis. If z = Jc, (x + yy = ak. That is, any xy- contour is two parallel straight lines, equally inclined to the x and y-axes. Hence the surface is a cylinder generated by a straight line PQ moving along the two equal parabolas y"^ = az and x^ = az, and always parallel to the straight line a; + y = in the cc?/-plane. The other two systems of contours are parabolas which are all equal to the traces OP and OQ. 140] LOCI 205 EXAMPLES Trace the surfaces represented by the following equations : 1. 2x-^Sy-4z = 12. ^^ ^^t-^^i 2. X2 + 2/2 + ;j2 = 16. ■ a2 52 c2 ' 3. x^ + y^ + z^-ix + 6y-2z = ll. 15. ^_l^_?! = i 4. .2 + ,2 = «2. «^ ft^ c2 • 5.y^^z^ = 2az. '' (x^yy^z^^a^ B.y^ = 4az. 1^- ^^ = 2... 7. ;22_y2 = «2. 18. (aj + y)2 = 2(a2-02). 8. x'^+y^ = z\ 19- (a; + y - a)2 + 2;2 = a2. 9. X2 + 2/2 = , =, or (0, -a, 0), ^(0,0, a), and F(0, 0, — a) form a regular octahedron. Find the locus of a point P in space such that (1) ^P2 + 5P2 + ^P2=CP2 + 2)P2 + i?'P2; Ans.z = 0. (2) u4P2+C'P2 + JSrP2 = PP2 + 2)p2 + PP2; Ans.x + y-\-z = 0. (3) ^P2+C'P2=PP2+2)p2+^p2+i^p2. Ans, x'^+y^-\-z'^+2a(x+y)+a'^=0, (4) ^p2 + PP2=OP2 + i)P2+^p2 + pp2; Ans. z^ + y^ + z^ + tt"^ = 0. (5) ^P2 + PP2 =01^ + DI^= EP^ + PP2 ; Ans. All space. 14. If ABCD is a regular tetrahedron, show that the locus of a point P, such that 2 P^2 _ pj52 _}. pc'2 + P2)2^ ig a sphere passing through the points, P, 0, D, and having a radius equal to twice the face altitude of the tetrahedron. 15. Show that the equation 8 + ^8' — represents a surface passing through all the common points of the two surfaces ^S' = and 8^ — 0. Show also that 88' — represents both of the surfaces /S' = and 8' — 0. 16. Find the equation of the surface of the blade of a screw-auger. CHAPTER XIV THE PLANE AND THE STRAIGHT LINE 142. To Jind the equation of a plane. Let OH be perpendicular to the given plane ABC, intersecting it in jfiT; and let I, m, n be the direction cosines of OH. Let OK=p be the distance measured from the ongin to the plane, and let P(xj y, z) be any point in the plane. Draw PR perpendicular to the plane XOFand BQ perpendicular to OX. Then OK, the projection of OP on OH, is equal to the sum of the projections of OQ, QB, and MP on OH [(1), § 130] Therefore Ix + my ■\-nz = Pf (1) which is the equation of the plane in the Normal or Distance Foi-m. Since changing the signs of all its direction cosines reverses the direction of a line, the equation of a plane may always be written so that p shall be measured along the positive direction of OH-, i.e. so that p shall be positive. The positive side of the plane is 209 210 THE PLANE AND THE STRAIGHT LINE [143 found by going from the plane in the positive direction of p. Hence when p is positive the origin is on the negative side of the plane. Equation (1) may also be written in the form ^ + i^ + ^ = l. (2) P P P I m n T) T) If) If now we let a = ^, 6 = — , and c = -, we have V m n' which is the equation of the plane in terms of its intercepts on the Les. The general equation of the first degree may be written By ^ Cz _ -D (4) =' (5) V^2 + _B2+C2 V^2 + jB2_,.Cf2 V^2 + ^2+(72 -^A^ + B^+C^ in which the coefficients of a;, y, and z are the direction cosines of some line [(5), § 131]. Comparing this with equation (1) we see that (5) is the equation of a plane in the distance form. 143. The distance from a given plane to a given point. The demonstration is precisely the same as that for the corre- sponding proposition in Plane Geometry. If d represents the distance and (x^, y^ z^ is the given point, the required formula is d = lxi + mm + nzi -p, or g^ ^xi ^ Byi + Czi + D ^ .^ according as the equation of the plane is lx-\-my-\- nz =py or Ax + By + Cz-\-D = 0. As in Plane Geometry a point (x'j y\ z') is on the positive or negative side of the plane Ax -{- By -{- Cz -\- D = 0, according as Ax' + By' -\-Cz' -{-D is positive or negative. X, y, 2^, 1 Xi, Vh «1, 1 X2, ^2, 5?2, 1 X3, 2/3, ^3, 1 144] THE PLANE AND THE STRAIGHT LINE 211 EXAMPLES 1. Show that the equation of a plane through the three points (xi, yi, 01), (3^2, 2/2, 22), and (X3, 2/3, 5?3) is = 0. 2. Find the equation of the plane through the three points (1, 2, 2), (2, —4, —3), and (— 6, 2, 5). Find j9, the intercepts, and traces of the plane. Ans. 2x — 3y + 42; = 4. 3. Find the equation of the plane through the point (3, 2, — 4) parallel to the plane 2x — 3y — 5^ = 0. Ans. 2x — Sj/ — 6^ = 20. 4. If >S^ = and S' = are the equations of two planes, show that S+\S' = will be the general equation of a plane through their intersection. 5. Find the equation of a plane through the origin and through the inter- section of the two planes 3x + 4y — 2^ + 4 = and 4x — 5x — ^ = 6. Ans. nx-\-2y-Sz = 0. 6. Show that the four planes x — y — 2z = lj 2x — 2/4-2 + 1 = 0, x-\-2y — z = 6, and 4x4-y + 60 = O meet in a point. Find the general condition that four planes shall meet in a point. 7. Show that the four points (0, 1, 3), (1, 1, 1), (-2, -3, -5), and (4, 2, —2) are in the same plane. (Use the determinant in Ex. 1.) 8. Show that the two points (1, —4, —2) and (—1, 2, 3) are on opposite sides of the plane 7x — 3«/ + 42; = 5, and equidistant from it. 9. Show that the equations of the planes which bisect the angles between the two planes Az + By + Cz -]- D = and A'x + B'y + C'z + Z>' = 0, are Ax -h By -[■ Cz ■}- D ^ ^ A'x -^- B'y + C'z + D' V^2 + ^ + 02 V^'2 4. B'^ + C'^ 144. Equations of a straight line^ We have seen in § 134 that it requires two equations used simul- taneously to represent a line in space. Since two planes intersect in a straight line we may take the two general equations of the first degree Ax-{-By + Cz-^D = Oy and A'x + B'y-{-C'z-\-D' = 0, (1) as the most general equations of a straight line. If we treat these equations simultaneously and eliminate 2, y, x, 212 THE PLANE AND THE STRAIGHT LINE [145 respectively, we obtain three other consistent equations which may be reduced to the form b~^' h' ' c~"' c' ' a'~" (^) Since each of these equations (2) is satisfied by the coordinates of every point on the line, they will each determine a plane through the line. These planes are seen to be the projecting planes of the line, while their equations also represent the projections of the line on the coordinate planes. The equations of any two of the project- ing planes may be chosen as the equations of the line. If the line is parallel to one of the coordinate planes, two of the projecting planes coincide and the equations of the line will be of the form bx -\- ay = ab, z = c; if the line is parallel to one of the axes, one of the projecting planes is indeterminate, and the other two are of the form x = a, y=b. From the equations (2) of the projecting planes we see that the coordinates of the points where the line meets the coordinate planes x = 0, 2/ = 0, z = Oj are respectively (0, 6, 0% (a, 0, c), (a', b', 0). The equations of a straight line contain four independent constants. 145. The symmetrical equations of a straight line. Let P'(x', y', z')he a fixed point on the line, and P(Xj y, z) any other point on the line at a distance r from P' ; let I, m, n be the direction cosines of the line P'P. Through P ' and P draw planes parallel to the coordinate planes, making a parallelopiped whose edges P'Q, QR, and MP are respectively equal to the projec- tions of P'P, or r, on the axes. Since these edges are respectively equal to x — x', y — y', and z — z\ we have aj — «' = Zr, y — y^ — mr, z — z^ — nr, (§ 130) (1) or a?-ac' I m, n which are the required equations of the line. 147] THE PLANE AND THE STRAIGHT LINE 213 146. To find the equations of a straight line through two given points (»!, yi, z^ and (a^g, Vz) ^2)- Since the line passes through the point (ajj, y^ Zi) its equations will be of the form [(2), § 145] I m n Then, since the point (ajg, 2/2? ^2) is also oli the line, we have X2- xi _ y2- yi ^ Z2- Zi ^2) I m n Dividing (1) by (2) gives the required equations, x-xi y-vi z-zx (3) a?2-a?i y^-vt z^-zi Hence, the direction cosines of the line are proportional to the differences of the coordinates of the two given points. 147. The equations of any two straight lines in rectangular co- ordinates can be written in a very simple form by a proper choice of axes. Take the middle point of the shortest distance between the two lines for the origin, and the z-axis along this line. Take the yz and xz planes so that they bisect the angles between the two planes determined by the z-axis and the two given lines. Then the equations of the two lines can be written y = mx, z = Ct and y = -mx, s = -c, (1) or in the symmetric form EXAMPLES 1. Find the symmetric equations, and the direction cosines, of the line of intersection of the planes 6x — y-\-z + 6 = and x — y — z-\-l = 0. Eliminating z and y in turn between these equations, we get Zx = y-S and 2x-{-z-\-2 = 0. Whence ^ = 1L^ = ^.±1. 13-2 Hence the direction cosines of the line are proportional to 1, 3, and — 2 ; and 1 o 2 their actual values are , , Vli Vli V^lT 214 THE PLANE AND THE STRAIGHT LINE [148 Find the projections, the symmetric equations, the points where they pierce the coordinate planes, and the direction cosines of the lines whose equations are 2. x-{-y-z + l=0 and ix + y + z = 5. 3. x + y-z + l = and ^x-\-y-2z-\-2 = 0. 4. 2x-y-\-z-S = and x-{-2y + z = 5. 5. 8x-2y + ^z = 12 and 6 x - 4 y - S z -{■ 2i = 0. 6. 5x-Sy + 2z-{-6 = and Sx-5y-2z = 7. 7. Write the symmetric equations of a line perpendicular to a coordinate axis; a coordinate plane. 8. Write the symmetric equations of the line through the point (2, — 3, 1) equally inclined to the axes. 9. Find the equations and direction cosines of the line through the two points (- 1, 3, 2) and (2, - 3, 0). 10. Find the equations of the line through the origin perpendicular to the plane lx + my + nz= p. 11. Find the coordinates of the point where the line meets the plane 2a; — y — 355 + 15 = 0. x-2 1 y - 2 ^ g + 3 -2 -3 148. To find the angle between two straight lines whose direction cosines are given. Draw OP and OP through the origin parallel respectively to the two given lines. Let l, m, n and V, m'j n' be the direction cosines of OP and OP' respectively, and let 6 represent the angle POP'. Let p be the distance from the origin to the point P{xy y, z). Then projecting p, x, y, and z on OP' we get [(1), § 130] p cos d = I'x 4- m'y + n'z. (1) But x^lp, y = mp, and z = np. . [(2), § 131.] (2) .-. cos9 = ir -\-tnni'-\-nn'. (3) It = 90°, cos d = 0. Hence the condition for perpendicularity is W + mm' + nn' = 0. (4) 149] THE PLANE AND THE STRAIGHT LINE 215 It should be noticed that equation (3) gives the angle between two lines directed from the origin. If the signs of Z, m, n are all changed, the direction of OP will be reversed, the sign of IV + mm' -\- nn' will be changed, and will be the supplement of its former value. But if the signs of V m' n' are also changed, the direction of both lines will be reversed, the sign of cos will not be changed, and 6 will be un- altered. 149. To find the angle between two planes. The angles between two planes are evidently equal to the angles between the lines through the origin perpendicular to the planes. Let the equations of the planes in the distance form be Ix + my -\- nz = py (1) and l'x-{-m'y-{-n'z=p', (2) Then cos 6 = W + mn' + nn'. [(3), § 148.] (3) If the planes are at right angles, cos ^ = ; i.e. IV + mm' + nn' = 0. (4) If l = V, m = m', and n=:n', then cos = 1, and the planes are parallel. ^ If the equations of the planes are Ax + By-^Cz + D = Oy (5) and • A'x + B'y-\-az-\-D' = 0, (6) eos9= AA' + B B ' + Ca _ , [(5), § 142] (7) V^2 + ^2 + (72 . VJL'2 + JB'2 + Cf'2 and the condition for perpendicularity is AA' + BB' + CC' = 0. ' (8) If ^ = kA', B = kB' and C= kC, the planes are parallel. Let the equations (1) and (2) be written so thatp and jp' have the same sign. Then, when cos 6 is positive the angle between p and p' is acute, and the angle between the planes in which the origin lies is obtuse. If cos 6 is negative^ the origin lies in the a^mte angle between the planes. 216 THE PLANE AND THE STRAIGHT LINE [149 EXAMPLES 1. Show that the lines 4x = — ?/ = 3;s and Sx = — 4ty = — z are perpen- dicular to each other. 2. Find the angle between the lines Ans. cos-^ . 26 3. Find the angle between any two of the four lines through the origin equally inclined to the axes. 4. Find the angle between any two of the lines which bisect the angles between the axes. 5. Find the angle between one of the lines in Ex. 3 and one of the lines in Ex. 4. 6. Find the angle between the planes x + y-\- z = 1 and x — y — 2z = 2. Is the origin in the acute or the obtuse angle ? the point (1, 3, — 1)? V2 Ans. cos-i — — , Acute, Obtuse. o 7. Find the equation of the plane through the line x-\- y — z = 2j 2x — Sy-\-4:Z-\-6 = and perpendicular to the plane x — 2y + z = 0. Ans. Sx-\-Sy -2z = 7. 8. Find the equation of the line througji the point (1, 4, 3) perpendicular to the plane 3x — 2^ + 42; = 0. 9. Find the equation of the plane through the point (2, —1, 3) and perpen- dicular to the line 2x + Sy — z=2, x-2y + z = S. Ans. x-Sy-7z + lQ = 0. 10. Find the dihedral angles of a regular octahedron. 11. Show that the line - = — = - will be parallel to the plane I m n I'x 4- tn'y + n'z =p, if W + mm' + nn' = 0. 12. Show that the equations of the straight lines which bisect the angles between the lines ? = l = i and | = J^ = 1. I m n V m' n' are ^-^V^^ and * - ^ - '^ l + V w -r m' w + n' l — V m — m' n — n' 151] THE PLANE AND THE STRAIGHT LINE 217 Transformation op Coordinates 150. To change the origin of coordinates loithout changing the direc- tion of the axes. This transformation is evidently similar to the corresponding one in Plane Geometry. Hence if we wish to find the equation of a locus referred to new axes parallel respectively to the old, and passing through the point (ajo, 2/o? ^o)) we have only to write in the place of X, 2/, 2;, respectively, 151. To change the direction of the axes without changing the origin. Let Zi, mi, «i ; Zg? ^2j ^2; a,nd Zg, mg, %, be the direction cosines of the new axes OX', Y^, OZ' respectively, referred to the old axes. Let P be any point (a;, y, z) referred to the old axes, and let its coordinates referred to the new axes be OQ = a;', QR = y\ RP = z'. Then projecting the lines OP, x', y', z' on the old axes OX, OT, OZ, respectively, we get [(1), § 130] y = mtoc' + m^y' + mgs', I (1) and z = n\x' + n^y' + ns^'. J These are the required formulae. The student should compare them with the corresponding for- mulae in Plane Geometry. It is evident that the degree of an equation will not be altered by either of these transformations. The direction cosines of the old axes referred to the new are respectively l^, I2, /g ; m^ m^, m^ ; and Wj, 712, Wg. Hence we have the six relations h^ + mi^ + V = 1, ] (2) 218 THE PLANE AND THE STRAIGHT LINE [151 n^ + rii + ni = 1. J (3) Since both sets of axes are rectangular, we also have the six equations 1^2 -H m-i^mi + 921^2 = 0, Uz + wigms + W2W3 = 0, • (4) ^1 + Wgrni + Tiarii = ; - liTTix -h Zgma + ?3m3 = 0, miWi + mgWa 4- mgWg = 0, (5) Wi^i + 712^2 + WgZg = 0. EXAMPLES ON CHAPTER XIV 1. Transform the equation (x + yy^ = az by turning the axes of x and y around the 0-axis through an angle of 45^. Ans. 2x^ = az, 2. If P is a fixed point on a straight line through the origin equally inclined to the axes, any plane through P will intercept lengths on the axes the sum of whose reciprocals is constant. 3. The equation of the plane through the line - = ^ = -, and which is per- l m n pendicular to the plane containing the lines — = ^ = ? and - = ^ = — , is m n I n I m (m — n')x + (w — X)y + (Z — m)z — 0. 4. Show that the three straight lines x_'f_z x_y_z ^-.y_-.^ a^y^abc^lmn Will lie in one plane if a{bn — cm) + /3(d — an) + y{am —^l) = 0. 6. If a, 6, c and a', b', d are the intercepts of a plane on two sets of rectan- gular axes having the same origin, then i + l + l = i + -l + l. a2 62 c2 a'2 &'2 ^ c'2 6. The locus of a point whose distances from two given planes are in a constant ratio is a plane. 151] THE PLANE AND THE STRAIGHT LINE 219 7. Show that the locus of a point which moves so that the sum of its distances from two fixed planes is constant is a plane parallel to one of the planes which bisect the angles between the two fixed planes. "What is the locus if the difference of these distances is constant ? 8. Find the locus of a point which moves so that the sum of its distances from any number of planes is constant. 9. Transform the equation z"^ = ax -{■ by by turning the axes of x and y around the «-axis until the new y-axis coincides with the line ax + by = 0, z = 0. Ans. 02 - xVa^ + 52. 10. What is the equation of the surface x^ + y^ -{- 2 z^ - 2 z(x + y) = a^ when referred to new axes such that the new ic-axis is equally inclined to OX, O F, and OZ, and the new y-axis is the line x + y=0, = 0? Ans. y^ + Sz^ = a^. 11. Show that the six planes, each passing through one edge of a tetrahedron and bisecting the opposite edge, meet in a point. 12. Through the middle point of every edge of a tetrahedron a plane is drawn perpendicular to the opposite edge. Show that the six planes so drawn will meet in a point such that the centroid of the tetrahedron is midway between it and the centre of the circumscribed sphere. 13. Through two fixed straight lines in space two planes are drawn at right angles to one another. Find the locus of their line of intersection. (See § 147.) 14. A line of constant length has its extremities on two given straight lines. Find the equation of the surface generated by it, and show that any point on the line describes an ellipse. 16. A straight line meets two given straight lines and makes the same angle with both of them. Find the equation of the surface which it generates. 16. Three straight lines mutually at right angles meet in a point P, and two of them intersect the axes of x and y respectively, while the third passes through the fixed point (0, 0, c) . Show that the equation of the locus of P is x^ + i/^ + z^ = 2cz. 17. Show that when the new axes are chosen, as in Ex. 10, the equation of the surface xy + yz + zx=:0 becomes 2x^ — y^ — z^ = 0. CHAPTER XV CONICOIDS 152. A surface whose equation is of the second degree is called a Conicoid. In this chapter we shall investigate some of the properties of the conicoids by taking the equations of these surfaces in their Standard Forms. We shall begin with the Sphere, which may be defined as the locus of a point whose distance from a fixed point is constant. From this definition it follows at once from equation (2), § 129, that, if the centre is at the origin and the radius is r, the equation of the sphere is x^-]-y^ + z^ = r^; (1) and if the centre is at the point (a, b, c), the equation is (X - a)2 + (2/- 6)2 + (;s _ c)2 = ^2, (2) Moreover, the general equation x^ + y^ + z^ + 2Ax-h2By + 2Cz + D = 0, (3) may be written in the form (x-^Ay + iy + Bf-hiz-^Cy^A' + B'+C'-D, (4) which shows that the equation represents a sphere whose centre is the point (—A, —Bj —C) and whose radius is -y/ A^ -{- B^ -\- C^ — D. That is, every equation of the second degree in which the coefficients of a^, 2/^ and z^ are equal, and in which the terms containing xy, yz, and zx do not appear, represents a sphere. 153. To find the equation of the tangent plane at any point (a;', y\ z') of the sphere. Let the equation of the sphere be x^ -\- y"^ -\- z^ = i^. (1) The equations of the radius drawn to {x\ y\ z') are ^=2^ = ^. (2) x' y' z' ^^ Since the tangent plane passes through the point (x\ y\ z') and is perpendicular to this radius, its equation is 220 154] CONICOIDS 221 x'(x-x')-]-y'(i/-y')+z'{z-z') = 0, (3) Since x'^ -\-y'^ + z^ = 7^, this equation reduces to icx' + yy' + zz' = r^, (4) In like manner, the equation of the plane tangent to the sphere ix^-{-y^ + z^-^2Ax + 2By-^2Cz-\-D = (5) at the point (a;', y', z') can be shown to be xx'-\-yy' + zz' + A(x + x')+B(y + yf) + C(z + z')+n = 0. (6) 154. Interpretation of the expression (x' - ay + (y' - by + (z' -cy-cP (1) when the point P(x', y\ z') is not on the sphere (x^ay+(y-by-{-(z-cy-d' = 0. (2) Let I, m, n be the direction cosines of any line through P. Then the equations of this line may be written (§ 145) I m n ' ^ ' Let this line intersect the sphere in the points Q and B. Then at the points Q and E [from (2) and (3)] (lr + h'-ay'^(mr-\-y'-by-{-(nr-\-z'-cy-d^ = 0. (4) If Ti and ra are the roots of this equation, we have nrg = (X' - a)2 + (y' - &)2 + (2,/ _ c)2 -d^ = rQ. JPB, (5) That is, the expression (1), or (5), is always equal to the product of the distances from P to the sphere measured along any straight line passing through P. If 7'i Vz is negative, P is inside the sphere. Then (5) is the product of the segments of any chord passing through P; it is also numei-i- ■cally equal to the square of the radius of the small circle on the sphere, whose centre is at P. If ri rj is positive, P is outside the sphere. In this case the expres- sion (5) is equal to the product of the whole secant by the external segment ; and therefore it is also equal to the square of any tangent PT drawn from P to the sphere, (Cf § 104.) Cor. All tangents drawn from an external point to a sphere are equal. 222 CONICOIDS [155 155. If a sphere passes through the line of intersection of two given spheres, tangents drawn from any point on it to the two given spheres are in a constant ratio. Let S = x' + y' + z' + 2Ax + 2By + 2Cz + D = 0, (1) and S' = x'-}-y'-{-z' + 2A'x-^2B'y-{-2C'z + D' = 0, (2) be the equations of two spheres, in each of which the coefficient of x^ is unity. Then the equation of any sphere through their line of intersection is <^ __^^i-.q (3) If PTf and PT' are tangents drawn from any point on (3) to (1) and (2) respectively, it follows from § 154 that JPT2 = X.1>T'2, (4) which proves the proposition, since X is constant for any particular sphere. If X = 1, equation (3) reduces to 2iA-A')x + 2(iB-B')y + 2(C-C')z + D-I>' = 0, (5) which is of the first degree, and therefore represents a plane. The plane through the line of intersection of two spheres is called their Radical Plane. The radical plane of two spheres may also be defined as the locus of all points from which tangents drawn to the two spheres are equal. EXAMPLES 1. What does the constant term D represent in the general equation of the sphere ? Where is the origin if D is positive ? if Z) is zero ? if Z> is negative ? Where is P in § 154 if nrz = - fZ^ ? 2. How many independent conditions can a sphere be made to satisfy ? 8. Find the equation of a sphere through four given points. Find the centres, radii, position of the origin, length of tangents from the origin, and the intercepts of the following spheres. 4. a;2 + ?/2 + 2;2_2ic_42/-60 + 5 = O. 5. x^ + y^ + z^ + 10x-2iy = 0. 6. x:^-\-y^ + z'^ + 6x-8y + 2z-10 = 0. 7. a;2 + y2 + 02 _ 4 X + 6 2/ + 10 5! = 0. 155] CONICOIDS 223 Find the equation of a sphere 8. With centre on one of the coordinate axes and passing through the origin. 9. Touching two of the coordinate planes. 10. Touching the three coordinate planes. — - 11. Touching two of the coordinate axes. 12. Touching the three coordinate axes. 13. Touching the three axes and passing through the point (2, 4, 0). How many such spheres are there ? 14. Show that if the coordinates of the extremities of a diameter of a sphere are (xi, yi, zi) and (x2, Vi, Zi) its equation may be written ix - xi) (x - Xi) + (y- yi) (y - 2/2) + (« - zi) {z - Zi) = 0. 16. Show that the polar equation of the sphere x2 + 2/2 + 02^_2^4ic + 2^y + 2Cfe + 2> = O is p2 + 2 p{Al + 5m + Cw) + D = 0. What property of the sphere follows from the fact that the product of the roots of this last equation is constant ? 16. Show that the radical plane of two spheres is perpendicular to their line of centres, and bisects all their common tangents. 17. Show that the radical planes of three spheres meet in a line which is perpendicular to the plane through the centres of the spheres. This line is called the Radical Axis of the three spheres. 18. Show that the radical planes of four spheres meet in a point. This point is called the Radical Centre of the four spheres. 19. What is the geometric property of the radical axis of three spheres ? of the radical centre of four spheres ? What is the analytic condition that the origin shall be the radical centre of four spheres ? 20. A and B are two fixed points, and P a variable point such that PA = n ' PB. Show that the locus of P is a sphere. Show also that all such spheres, for different values of n, have a common radical plane. 21. Show that the spheres whose equations are x*+2/2 + «2 + 2^ + 2JBy + 2(7« + 2> = and a;2 + ya _^ 2;2 ^ 2 ^'a; + 2 B'y + 2 C'2? + 2>' = will cut one another at right angles if 2 ^14' + 2 JB£' + 2 CC" - 2> - D' =t a 224 CONICOIDS [156 The Cone 156. To find the equation of a cone generated by a straight line passing through the origin, of which the guiding curve is a conic. Let the equations of the guiding conic be «?^y' 1, z = c. (1) a' ' b' Let Q(xi, i/i, c) be any point on the guiding conic ; then ^4-^ 1, (2) a- b^ and the equations of the generating line OQ are X _y z Xi~yi c Whence (3) ^1 = -, yi = -, and -, . = 1. (4) r cr ^ ^ Substituting these values in equa- tion (2) gives ^ ^ f ^ z^ a'r^'^b'r' c'r' 2 ^ 62 c2 0, (5) (6) which is the required equation. By putting x, y, and z respectively equal to zero in (6), we find the equations of the traces of the cone to be b' (^ ^' ^ + 2^ = 0. (7) Each of the first two of these sections is a pair of straight lines through the origin, and the third is a point ellipse. By putting cc, y, and z respectively equal to fc, we find the equations of the three sets of contours to be ?!_^ — ^ ^ — — x^ y k' ¥ c^ (8) 156] CONICOIDS 225 each of which for different values of k represents a system of similar and coaxial conies (§ 116). The first two are hyperbolas with trans- verse axes along the 2J-axis, and whose asymptotes are the traces on the corresponding coordinate planes. The last are ellipses which increase indefinitely in size as the cutting plane recedes from the origin. As a check it should be noticed that the section made by the plane 2 = c is the guiding conic. If we take as the guiding conic the hyperbola the equation of the surface will be which is a cone extending along the 2/-axis, since the sections per- pendicular to this axis are ellipses with centres on this axis. Similarly, if the guiding conic is the hyperbola -2-^2 = 1, 2; = c, (11) the resulting equation will be 052 1/2 «2 which represents a cone extending along the aj-axis. If we take as the guiding conic the parabola 2/2 = 4aajj z = c, (13) the equation of the cone will be cy'^ = 4 axz, (14) The traces of this surface show that the cone is tangent to the coordinate planes a;=0, 2=0, along the 2;-axis, and a^axis respectively ; I.e. these axes are elements of the cone. The a»-contours are the rectangular hyperbolas 4 00:2; = cT^. The other two sets of contours are the parabolas ^, ^ ^ ^^^ ^^^ ^ ^ 4 ^^^ ^^^^ Observe that these parabolas are sections made by planes parallel to an element of the cone. CONICOIDS [157 If we transform equation (14) by turning the axes of x and z clockwise through an angle of 45°, the new equation will be ^ + |l_?! = 0, (16) c 2a c ^ ^ which is of the same form as equation (6), and therefore represents a cone extending along the new 2;-axis. It follows from equations (6), (10), (12), and (16) that the conical surface generated is essentially the same, whatever the form of the guiding conic. The equations of the cone found above are all homogeneous. Moreover, if they are referred to any new set of rectangular axes having the same origin [(1), § 151], the new equations will also be homogeneous. Furthermore, any homogeneous equation represents a cone whose vertex is at the origin. For if the coordinates of the point (a;, y, z) satisfy a homogeneous equation, so also will the coordinates of the point (kx, ky, kz), whatever the value of k may be. Hence a line through the origin and any point on the surface lies wholly on the surface. 157. Definitions. — The form of equations (6), (10), (12), and (16) of § 156 shows that the surfaces which they represent are sym- metrical with respect to each of the coordinate planes, and also with respect to the origin. That is, each of these planes bisects all chords of the surface which are perpendicular to the plane. A plane which bisects all chords of a conicoid which are perpendicular to it, is called a Principal Plane. The sections made by the principal planes are called the Principal Sections of the conicoid. The lines of inter- section of the principal planes are called the Axes of the conicoid; they are also the axes of the principal sections. The point of inter- section of the principal planes is called the Centre of the conicoid. It follows from these definitions that the cones in § 156 have three principal planes and three axes. These are the coordinate planes and the coordinate axes, with the single exception of the locus of equation (14). Moreover, we have also found in § 156 that if the guiding conic is an ellipse, a hyperbola, or a parabola, the cone is such that sections perpendicular to one of its axes are ellipses. Such a cone is called an Elliptic Cone to distinguish it from the cone of revolution, or circular cone. 158] CONICOIDS 227 The Ellipsoid 158. Let + -, = 1, y = 0; and ^ + ^' = 1,2 = 0, or Ir (1) (2) be two fixed ellipses, XZ, XY^ having a common major axis; and let ABC be a variable ellipse which moves so that its plane is always parallel to the 2/2;-plane, and which changes in size so that the ends of its axes, A and B, always lie in the two fixed ellipses. The surface generated by this variable ellipse is called an Ellipsoid. Let Pix, y, z) be any point in the ellipse AB^ whose semi-axes are CA, CB ; and let PD be drawn perpendicular to CA. Then, since DP = y and CD = Zy yi ^2 CB^'^CA' ' (3) Since A and B are also on the two fixed ellipses (1) and (2), respectively, and their coordinates are (x, 0, CA) and (x, CB, 0), we have ^ + M' = l, and ^ + «^ a* y" 1. (4) 228 CONICOIDS [15d Substituting in (3) the values of Gj^ and CI^ given by equa tions(4),weget ^ ^ ^^ '^^-^'--=1, (5) which is the standard equation of the ellipsoid. The surface is symmetrical with respect to each of the coordinate planes, and also with respect to the origin. Hence these are the principal planes of the surface, the coordinate axes are its axes, and the origin is the centre. The principal sections are the ellipses .+% = ^> ^. + ^ = 1^ fe + :i = i- (^) The equations of the three sets of contours are t^t^l-^ tj^t^l-t t^t^l-^. m y" c" a"' a^ & h^ a^ W- (? ^^ Each set is a system of similar ellipses which vanish, respectively, when Tc is equal to ± a, ± 6, ± c. In general, it is here assumed that a';>h'> c. If c = &, the equation (5) becomes The 2/2;-contours are now concentric circles, and the surfaxje is an ellipsoid of revolution generated by revolving the ellipse 6V H- ay = a^W about its major axis. This surface is called a Prolate Spheroid. If 6 = a, the equation of the surface (5) takes the form 05^ J/2 «2 of which the a;2/-contours are concentric circles. The surface is an ellipsoid of revolution generated by revolving the ellipse (1) about its minor axis, i.e. the z-axis. Such a surface is called an Oblate Spheroid. If c = 6 = a, the ellipsoid becomes the sphere a^ + 2/^ + 2^ = a\ (10) 159] CONICOIDS 229 159. Let and The Hyperboloid of One Sheet a? ,2 ^2 T2-^ = l^ 2/ = 0; (1) (2) .^ '^ z /• ^^ V Hv y / ^ ^ \ v"-""\ / \ c /r D ~'-^ V"^— I / '' / /J*" \-""l ---- J / / o \ /'"' Zt""-A / ^~— i / HA / -7 — - — i \ /"" / z^-'-X 'v^ // / be two fixed hyperbolas, EFy HK, having a common conjugate axis OZ', and let ABC be a variable ellipse which moves so that its plane is always parallel to the an/-plane, and which changes its size so that the ends of its axes, A and JB, always lie in the two fixed hyperbolas. The surface generated by this variable ellipse is called a Hyperboloid of One Sheet. 230 CONICOIDS [169 Let PiXy ?/, z) be any point in the ellipse AB^ and let PD be drawn perpendicular to AC) then, since CD = x, DP = yj and CA, CB are the semi-axes of the ellipse, ^ I y _ i /Q\ CA^'^ CB"" ' ^^ Since ^ and B are on the fixed hyperbolas (1) and (2), which is the standard equation of the hyperboloid of one sheet. The surface is symmetrical with respect to each of the coordinate planes, and also with respect to the origin. Hence the coordinate axes are the axes of the surface, and the origin is its centre. The principal sections made by the planes x = and y = are the two fixed hyperbolas (2) and (1), and the section made by the plane 2 = is the ellipse ^ ^,2 ^ + 1 = 1- (6) a^ Ir The intercepts on the axes of x and y are ± a a,nd ± 6, but the surface does not intersect the 2-axis. The equation of the a^-contours made by 2; = A; is -2 + r2 = i + l'- (7) a^ W (f These sections are similar coaxial ellipses for all values of A;, which increase in size without limit as the cutting plane recedes in either direction from the origin. The equation of the contours made by the plane a; = A; is ^-^=1-^- (8) These sections are hyperbolas for all values of A;. If — a < A; < a these hyperbolas have their transverse axes along the 2/-axis, but if A; > a, or A: < — a, their transverse axes lie along the z-axis. When A; = ± a, the contour is a pair of straight lines, which are the asymp- totes of the entire system of hyperbolas. 160] CONICOIDS 231 Similarly, the contours made by y = A; are the hyperbolas which have their transverse axes along the a>axis, or 2-axis, according as A;<, or >6, numerically, and whose common asymptotes are the contours made by the planes y = ±b. When b = a, the equation (5) of the surface becomes ^+IL*_5j = i, (10) a2 a2 c2 ' ^ ^ which is a hyperboloid of revolution generated by revolving the hyperbola (1) about its conjugate axis. 160. Asymptotic cone of the hyperboloid of one sheet. Let the equation of the hyperboloid be be the equation of a cone along the z-axis [(6), § 156]. The equations of the contours of these two surfaces made by the plane z = k are, respectively, ^ + 2^-1 + ^ (3) ^a y2 J^ and a^ + h = r W A comparison of equations (3) and (4) shows that, for the same finite value of k, the section of the cone is smaller than the section of the hyperboloid. Hence the cone may be said to lie inside of the hyperboloid. Equation (3) may also be written in the form which shows that the sections of the two surfaces become equal, i.e. they approach the same limit, when the cutting plane recedes in either direction to an infinite distance from the origin. That is, the 282 CONICOIDS [161 cone is tangent to the hyperboloid at infinity, and is, therefore, called the Asymptotic Cone of the hyperboloid of one sheet. 161. Let Thh Hyperboloid of Two Sheets and = 1, 2/ = 0; 1, . = 0, (1) be two fixed hyperbolas, EF, EHy having a common transverse axis; and let ABC be a variable ellipse which moves so that its plane is always parallel to the ysi-plane, and which changes its size so that the ends of its axes, A and B, always lie in the two fixed hyperbolas. The surface generated by this variable ellipse is called a Hyperboloid of Two Sheets. Let P{Xy y, z) be any point on the ellipse AB, and let PD be drawn perpendicular to CA\ then, since CD = Zy DP = y, and CA, CB are the semi-axes of the ellipse, CB^'^CA' ^' Since A and B are also on the fixed hyperbolas (1) and (2), -i ;;^ = l,and- ^ = 1. a' a' b' (3) W 161] CONICOIDS 233 which is the standard equation of the hyperboloid of two sheets. The surface is symmetrical with respect to each of the coordinate planes and the origin. Hence the axes of coordinates are the axes, and the origin is the centre of the surface. The intercepts on the a^axis are ± a, but the surface does not intersect either of the other axes. The equation of the contours made by the plane a; = A; is These sections are imaginary for all values of k between 4- a and — a. Hence there are no real points on the surface between the planes x — a and x— — a. If A: is numerically greater than a, these sections are real ellipses which increase indefinitely in size as k increases without limit, but reduce to points when k = ±a. Hence the planes x = ± a are tangent to the surface. Thus the surface is shown to consist of two distinct parts, and for this reason the hyperboloid is said to have two sheets. The xy and a;2;-contours are hyperbolas with transverse axes along the iP-axis, and whose asymptotes are the traces of the asymptotic cone on the xy and a;2j-planes. From § 160 it is evident that the equation of the asymptotic cone is a2 62 c2 "• ^^^ If c = 6, equation (5) becomes a;2 y^ 2:2 ^~62-^-l' W which is the equation of a two-sheeted hyperboloid of revolution generated by revolving the hyperbola (2) about its transverse axes. Two conicoids are similar if their principal sections are similar conies. Hence, if K is an arbitrary parameter, the equations, represent systems of similar conicoids (§ 116). 234 CONICOIDS [162 The Elliptic Paraboloid 162. Let ABC be a variable ellipse whose plane is always parallel to the a^-plane, and whose vertices A, B move along the two fixed parabolas OA and OB, whose equations are x' = 2az, 2/ = 0; (1) and 2/' = 2&^, a; = 0. (2) •The surface generated by this moving ellipse is called the Elliptic Paraboloid. Let P(xj y, z) be any point in the ellipse AB, and let PD be perpen- dicular to AC ; then since CD = x, SiJid DP = y CA^^CB" ' (3) Since A and B are also on the para- bolas (1) and (2), respectively, and OC=z CA^ = 2az, and CB' = 2bz. (4) X y a o ' (5) which is the standard equation of the elliptic paraboloid. The surface is symmetrical with respect to the xz and yz planes, and the 2;-axis. Hence the 2;-axis is called the axis of the paraboloid. The surface passes through the origin, cutting the z-axis once, the x and y axes each twice, but does not cut the axes at any other point. If we put z = km (5), we get ? + f = 2^- a (6) Hence a section parallel to the aJ2/-plane is imaginary if k is negative. If k is positive, the section is an ellipse which increases in size as the plane recedes from the origin, and diminishes to a point when fc = 0. Therefore the surface is tangent to the a^-plane, and lies wholly above this plane. 163] CONICOIDS 235 The equations of the xz and 2/2;-contours are a^ = 2az-^,ajidf = 2bz-—' b a (7) From equations (1) and (2) we see that, for all values of A;, these sections are respectively equal to the two fixed parabolas OA and OB. If 6 = a, equation (5) may be written ac2 + 2/2 = 2a«, (8) which represents a paraboloid of revolution about the 2;-axis. The Hyperbolic Paraboloid 163. Let a^=:2az, y = 0, (1) be the equations of a fixed parabola OA, and let AE be another given parabola with a constant latus rectum 2 6. Let the parab- ola AE move, keeping its vertex A in the fixed parabola OA, its plane parallel to the 2/2;-plane, and its axis AR in the os^-plane, the concavities of the two parabolas being turned in opposite directions. The surface generated by this moving parabola AE is called a Hyperbolic Paraboloid. Let P{x, y, z) be any point on the parabola AE. Draw PD per- pendicular to AR ; DC and AB perpendicular to OZ. Then BA^^x'^^^a - OB, and DP^ = y''=^2h • DA. Whence ^-^= OB-DA = OC=z. 2a 26 a b ' which is the standard equation of the hyperbolic paraboloid. The surface is symmetrical with respect to the planes x = and y = 0, and the z-axis. Hence the 2-axis is called the axis of the sur- face. The surface cuts the z-axis in one point, the x and y axes each in two coincident points at the origin. (2) (3) (4) 236 CONICOIDS [163 If we put z = km equation (4) we get a b ' (5) which represents a hyperbola with transverse axis on the aj-axis or y-axis according as k is positive or negative. When A; = 0, the section is two straight lines, HK and LM (large figure), which are the asymptotes of all these contours. The equations of the xz and yz-contovLTs are a^ = 2a2 + ^, and2/' = -26« + ^, b a (6) which for all possible values of k represent two systems of parabolas. The first are all equal to the fixed parabola OA with axes turned upward, the second are all equal to the movable parabola AE with axes turned downward. 164] CONICOIDS 237 164. The paraboloids are the limiting forms of the central conicoids as the centre recedes to infinity. Let the equations of the central conicoids be _ If the origin is moved to the point (— a, 0, 0), the new equation may be written ^ ^^ ^^2 52 c* Let — = l, and ~ = l' j then i, V are respectively the semi-latera recta of the principal sections made by the planes 2 = 0, and i/ = 0. Equation (2) may then be written aJ* V^ 25* f» /ON Now, if a becomes infinite, while I and V remain finite, equation (3) becomes in the limit, for the ellipsoid, hyperboloid of two sheets, and one sheet, respectively, f+F=2»' f+f=-2., t-t=2.. (4) The first two are elliptic paraboloids, the last is a hyperbolic parabo- loid, all with axes coinciding with the aj-axis. EXAMPLES 1. Show that a hyperboloid degenerates into a cone when its axes become indefinitely small, preserving a finite ratio to eacb other. 2. Show that the traces of the asymptotic cone are the asymptotes of the contours of the hyperboloids. 3. Compare the section of the hyperboloid of one sheet [(5), § 159] made by the plane x = k with the section of its asymptotic cone made by the plane X = y/k^ - a*. What does this show ? 4. Show how an elliptic paraboloid may be generated by a moving parabola. 6. Show how a hyperbolic paraboloid may be generated by a moving hyper- bola. 6. Show that all planes parallel to the axis of a paraboloid cut the surface in parabolas. 7. Show that the projections, on a plane perpendicular to the axis of a para- boloid, of all plane sections not parallel to the axis, are similar conies. 238 CONICOIDS [165 8. Show that all parallel parabolic sections of a paraboloid are equal. 9. Let ri, r2, rg be any three semi-diameters of an ellipsoid which are mutually at right angles. Show that i. + ^ + ± = l + l + l. n^ n^ rs^ a2 &2^c2 10. Show that the equation of the cone whose vertex is at the origin and which passes through all the points of intersection of the ellipsoid [(6), § 158] and the plane Ix + my + nz = 1 is /v2 f<2 «2 11. Show that the two conjugate hyperboloids have a common asymptotic cone, and show how they are situated with respect to this cone. 12. What are the limiting forms of the asymptotic cones as the hyperboloids pass into paraboloids in § 164 ? Tangent Planes 165. To find the equation of the tangent plane at any point {x\ y\ z') on a conicoid. Let the equation of the conicoid be Let the equations of any line through the point (x', y\ 2') be I m n \ / \ / or x = x^ + lry y = y'-{-mr, z = z'-\-nr. (3) The distances from the point (ic', y'^ z') to the points where this line meets the conicoid are the values of r given by the equation (x' + lry (y'^mrf (z' + nry _ a' + P "^ ? -^' ^^^ JP , m2 n2\ „ fix' . my' . nz'\ . x'^ , y'' , z" ^ ^ .^, Since the point (x', y', z') is on the conicoid, x^ y^.z^ ^2 + 52 + ^2 „12 ,/2 «f2 165] CONICOIDS 239 Therefore one value of r is zero, whatever the direction of the line (2) may be. But if we choose the direction of the line so that we also have 7 , , , ^2+ 52 + ^ -"» — -iJ; the other value of r will also vanish ; that is, the line will then meet the surface in two coincident points, and is therefore a tangent line at the point (a;', y\ z'). The equation of the locus of all the tangent lines which can he drawn through the point {x\ y\ 2') is found by eliminating Z, m, n between equations (2) and (7). We thus obtain ^(»-«')+^(2'-/) + 5(^ -«') = <>, (8) which, by virtue of equation (6), reduces to XX' yy' zz' _ .^. Hence the tangent lines all lie in a plane. This plane is called the Tangent Plane at the point (a;', y\ z'). By a proper choice of signs in (9) we can write the equation of the tangent plane to either of the hyperboloids. It should be noticed that the factors before the parentheses in equation (8) can be obtained by taking half the partial derivatives (§ 61) of equation (1) with respect to x, y, z, respectively, and then substituting in these derivatives a;' for a;, y' for y, and 2' for %. It can be shown that this rule holds for any surface. Assuming this rule to hold for the paraboloids J±f-2. = 0, (10) we have for the tangent plane at the point (a;', y\ z') J(a5-aj')±|'(2/-y)-(^-«')=0, (11) or ^^yf=^^ + z'). (12) This should also be proved independently. Ex. Show by means of equation (5) that every plane section of a conicoid is a conic. 240 CONICOIDS [166 166. The Normal to a surface at any point P is the straight line through P perpendicular to the tangent plane at P. Hence the equations of the normal to the ellipsoid at the point («', y\ z') are [(9), § 165] - {irJ 1M — «#'« — «' (i) a2 y -y' z-z' ~ y' ~ z' ^ and to the elliptic paraboloid [(11), § 165] y -y' z-z' - y -1' (2) a b From these, by a proper choice of signs in the denominators, we easily obtain the normals to the other conicoids. 167. To find the condition that the plane Ix -\- my -\- nz = p (1) shall touch the ellipsoid. The equation of the tangent plane at any point {x\ y\ z') of an ellipsoid is [(9), § 165] Equations (1) and (2) will represent the same plane if p p p a^ ly" c^ ' ^ ^ Equating the coefficients of the identity (3) gives l_x^ 'rfi_y[ ^_^' p a^ p H^ p~^ (4) Whence a^l±^^ri_±^^x- y^ z^^^^ p" a^^ V"^ & ^^ Therefore the plane Ix -\- my -\- nz = p will touch the ellipsoid if a2«2 + 62^2 + ^2^2 =p2, (6) In like manner it can be shown that the same plane (1) will touch the paraboloid ^ • -4-^ = 2« (7) ah if al^ + hm^ + 2 pn = 0. (8) 160] CONICOIDS 241 168. To find the locus of the point of intersection of three tangent planes to an ellipsoid which are mutually at right angles. Let the equations of the three tangent planes be [(6), § 167] ?!» + wiiy + ni« = VaV 4- ft^w^i^ + c^V> — (1) l^c 4- rn^ -\-n^= ^o^li + h'^rti} + (?n}, (2) and l^ -h may + 7132; = VaV + 6^3^ + <^n}. (3) Squaring and adding these equations we get, by virtue of the relations between the direction cosines of mutually perpendicular lines (§ 151), aj2 + y2 + 2;2 = a2 + 62 + c2. (4) Therefore the required locus is a sphere. This sphere is called the Director Sphere of the ellipsoid. Poles and Polar Planes 169. The equation of the plane tangent to the conicoid t^tj^t^X (1) at the point (a;', y\ 2'), if this point is on the surfa^e^ is (§ 165) a^ h^ cf Suppose, however, that the point {x\ y\ 2*) is not on the surface. What, then, is the meaning of this equation (2) ? It still represents a real plane, which is related in some definite way to the point (x\ y'f z^ and to the conicoid, since its parameters involve both the coordinates of the point and the parameters of the conicoid. In order to determine what this relation is, we will let x = x' + lr, y = y' + mr, z = z' + nr [(3), § 165] (3) be the equations of any straight line through the point (a;', y', 2'). Substituting these values of a;, y, z in equation (2), we find the dis- tance from the point («', y', 2') to the point where this line meets the plane (2) to be the value of r given by the equation te| , my^ , nz' r~ (r^^ 7/'* «'2 • \*) - + ^-4- a* ^ 6» ^ c 242 CONICOIDS [169 Let Vi and r2 be the distances from the point (x*, y\ z') to the points where this line (3) meets the conicoid (1). Then from equation (5), § 165, we get Ix^ my[ , n£ 2 11 2ri^2 fa\ .'. - = — + — , or r = =-=-• (6) r ri r^ ^i + rg That is, the plane (2) and the point (x\ y\ z*) divide harmonically every chord of the conicoid (1) drawn through the point (x', y\ z'). This plane is called the Polar Plane of the point («', y' z'), and the point (x\ y\ z') is called the Pole of the plane with respect to the conicoid. (Cf. § 94.) If ri = rj, the line (3) is tangent to the surface. But when Vi = 9*2, we find from equation (6) that r = ri = rg. Therefore the polar plane passes through the points of contact of all tangent lines drawn from its pole to the surface. The assemblage of such tangent lines forms a cone, which is called the Tangent Cone from the point to the surface. Moreover, if ri = and ?'2 =^ 0, then r = also, in whatever direc- tion the line is drawn ; i.e. if the point («', y', z') is on the conicoid, it is also on its own polar plane. If ri = r2 = 0, then r is indetermi- nate ; i.e. when the line is tangent to the conicoid it lies wholly in the plane. Therefore the pole of a tangent plane is the point of contact. When the point (x', y\ z') coincides with the centre of the conicoid, ri = — r2, and therefore ?• = oo. Hence the polar plane of the centre is at infinity. Furthermore, the second of equations (6) shows that r is always realf although ri and ra may be imaginary. This is evidently necessary, since the line will always meet the plane in one real point. In a similar manner it can be shown that equation (12), § 165, is the polar plane of the point (x', y\ z') with respect to the parabo- loids. 170] CONICOIDS 243 170. If the polar plane of a point P, with respect to a conicoid, passes through a point Q, then will the polar plane of Q pass through P. The proof of this proposition is precisely the same as that of the corresponding proposition in Plane Geometry (§ 95). -_ _ Let R and S be any two points on the line of intersection of two planes A and B, whose poles with respect to the same conicoid are P and Q. Then, since E is on both of the planes A and B, the polar plane of R will pass through both P and Q, and therefore through the line PQ. For the same reason the polar plane of S will pass through the line PQ. Similarly, the polar plane of any point Pj on the line PQ will pass through the line RS. The two lines PQ and RS which are such that the polar plane, with respect to a conicoid of any point on the one, passes through the other, are called Polar, or Conjugate Lines. EXAMPLES ON CHAPTER XV 1. Show that every tangent plane to a cone, and the polaj plane of any point (except the vertex) with respect to a cone, passes through the vertex. 2. Show that all normals to a sphere pass through its centre. 3. Show that the line OP joining the centre O of a sphere to a point P is perpendicular to the polar plane of P. If the line OP meets the polar plane in Q, show that OP'OQ = r^. 4. Show that the distances of two points from the centre of a sphere are proportional to the distances of each from the polar plane of the other. 6. Show that the locus of the point of intersection of three mutually perpendicular tangent planes to a paraboloid is a plane. 6. Find the equation of the director sphere of the surface generated by revolving a rectangular hyperbola around its conjugate axis. 7. Show that tangent planes at the ends of a diameter of a conicoid are parallel. 8. Prove that the locus of the poles of a series of parallel planes is a straight line through the centre of the conicoid. 9. Find the equation of a sphere which cuts four given spheres orthogonally. [See Ex. 21, p. 223.] 10. Show that a sphere which cuts each of the two spheres S=0 and S' = at right angles, will also cut the sphere S + \S' = a.t right angles. 244 CONICOIDS [170 11. Find the equation of the sphere which touches the plane y = 0, and cuts the plane 2: = in the circle (x — ay + (y — by = r^. Show that the area of the section of the sphere made by the plane x = is ir{b^ — a^). Why is this result independent of r ? 12. A straight line is drawn through a fixed point 0, meeting a fixed plane in Q, and in this line a point P is taken such that OP- OQ ia constant. Show that the locus of P is a sphere passing through 0, whose centre is on the line through O perpendicular to the plane. 13. A straight line moves so that three fixed points, A, B, C, on the line lie one in each coordinate plane. Show that any other point P on the line generates an ellipsoid whose semi-axes are equal to PA, PB, and PC. 14. Show that the equation of the cone whose vertex is at the centre of the ellipsoid, and which goes through all points common to the ellipsoid and the sphere x^ -{■ y^ -\- z"^ = r% is 16. If a > 6 > c and r = 6 in Ex. 14, show that the cone breaks up into two planes, whose intersections with the ellipsoid are circles. 16. If P and Q are any two points on an ellipsoid, the plane through the centre and the line of intersection of the tangent planes at P and Q will bisect the chord PQ. 17. P and Q are any two points on an ellipsoid, and planes through the centre parallel to the tangent planes at P and Q cut the chord PQ in P' and Q'. Show that PP' = QQ'. 18. The normal at any point P of an ellipsoid meets a principal plane in G. Show that the locus of the middle point of PG is an ellipsoid. 19. The normal at any point P of an ellipsoid meets the principal planes in d, ^2» Cfs. Show that PG^ PG2, PGz are in a constant ratio. 20. The normals to an ellipsoid at the points P, P' meet a principal plane in G, G'. Show that the plane which bisects PP' at right angles bisects GG'. 21. Show that a section of a hyperboloid made by a plane parallel to an element of the asymptotic cone is a parabola. 22. Show that the general equation of a cone referred to three of its generators as axes of coordinates is fyz + gzx + hxy = 0. APPENDIX I. The Direction of a Curve at the Origin. It is often useful to know how to find the direction of a curve at the origin before taking up the formal study of slope. In many instances this can easily be done. For example, let the equation of the curve be „ ... 2/ = ar'. (1) Let P{x, y) be any point on the curve close to the origin. Draw the line OPj and let B represent the angle XOP. Then tan^==^=^. OD X Since the point P is on the curve, we have, from equation (1), tan^: (2) (3) The direction of the curve at the origin is the limiting direction of the line OP as we make the point P move along the curve and approach as near as we please to the origin. From equation (3) we get for this limiting direction of OP J™ tan « = ,«-„(.) 0. W That is, the direction of the curve at the origin is the same as the direction of the x-axis. If the equation of the given curve is then 2/ = a^-ar, or - = a^-l, 245 (5) (6) 246 APPENDIX Hence the direction of the curve at the origin is that of the line y = -x. (See the curve FQ in § 27.) The direction of the curve at the origin can be found in this way whenever the equation of the curve can be put in the form i = *(»'). (7) provided we can find the limiting value of <;^ (x) as a; = 0. Moreover, the direction of a curve at the points where it crosses the axes can be found in a similar manner. For example, the locus of equation (5) cuts the fl>axis at the point (1, 0). Let this point be R, and let P(x, y) be a point on the curve close to B such that x>l. Let $ be the angle XEP. Then tan (9 = -J- = ^ 4- a;. (8) X— 1 .■.^f^t^ne = ^%(^ + x)=2. (9) Hence the curve has the direction of the line y = 2(x— 1). EXAMPLES Find the direction of the following curves at the origin : I. y = x^. 2. y = X". 3. y'^ = x^. 4. y'^ = ax. 5. x^-y\a-x) = 0. 6. a;(x2 + ?/2)- a(x2 - y2) = 0. Find the direction of the following curves at the points where they cut the axes : 7. ?/ = x3-3a;2 + 2x. (See Ex. 2, § 81.) 8. 2/ = a;^ - x^. 9. y = x8-x2-6x. 10. ?/ = x3-2x2-llx + 12. II. Example illustrating § 81. Let f'(x) = 2kx. (1) Then f(x)=kx' + c, ' (2) where c is an arbitrary constant which will disappear when we take the derivative. APPENDIX Then y = 2'kx=f{x) is the equation of the straight line L'M\ and y = kx'^-c = f{x) is the equation of the parabola LGM^ where OG = c. lY My 247 (3) Let OQ = a and OB = h. Then QA' = 2 fca, RB' = 2 fc6, the area of the triangle OQA' — ka^^ and the area of triangle OBB' = kb\ .'. area of QRB'A' = kV - ka\ (5) Also, RB=f(b)=^kh''^c, and qA=f{a)=^ko?^-c [from (4)]. (6) .-. BB-QA = f(b) - /(a) = fc62 _ fcal (7) .-. area of QRB'A' =f(b) -/(a) = RB - QA. (8) That is, the number of square units in the area of the trapezoid is equal to the number of linear units in (RB— QA). Similarly, the area of EFD'C = EC— FD, a negative number. If we put c = 0, the parabola will pass through the origin, and the ordinate QA will be zero when the area of the triangle OQA' is zero. Then the number of units in the ordinate QA will be equal to the number of units in the area of the triangle OQA'. 248 APPENDIX III. Trigonometrical formulce. 1. sin ^ CSC ^ = 1. 8. sin (— ^) = — sin 6. 2. cos ^ sec ^ = 1. 9. Cos (— 6) = cos 0. 3. tan ^ cot ^ = 1. 10. sin (90° ± $)= cos 6. 4. tan^ = ^. 11. cos(90°±^)=Tsin^. 5. sin2 $ 4- cos^ 6 = 1. 12. sin (180° ± ^) = =F sin 0. 6. sec2 6 - tan^ ^ = 1. 13. cos (180° ±6) =- cos $. 7. csc^ ^ - cot^ ^ = 1. 14. sin (270°±^) = -cos^. 16. cos (270° ±^)= ± sin 9. 16. sin (6 ± 6') = sin 6 cos 0' ± cos 6 sin $'. 17. cos (9 ± 9') = cos 9 cos 9' T sin 9 sin ^'. 18. tan(^±g^)= tan^±tan^'^ ^^ ^^^2^= ^^^^^ iTtan^tan^' 1-tan?^ 20. cot (9 ± 9') ^^ot^^ot^-Tl^ 21^ ,ot 2 ^ = 22^iz-_l. ^ ^ cot^'±cot^ 2cot^ 22. sin2^ = 2sin^cos^. 23. cos2^ = cos2^-sin2^ = 2cos2d-l = l-2sin2^. 24. sini-^ = VKl-cos^). 25. cos J ^ = VJ(lTcos^. 26. sin ^ + sin ^' = 2 sin ^(9 + 9*) cos |(^ - 9'). 27. sin ^ - sin ^' = 2 cos ^(9 + ^') sin i(9 - 9^ 28. cos ^ 4- cos 0' = 2 cos ^(9 + ^') cos ^(9 - 9'). 29. cos^-cos^' = -2sini(d + ^')sin|(^-^'). In any plane triangle 30 ^^^ ^ _ sin ^ _ sin C gi a + ^ _ tan -|(.^ 4- ^) a b G ' ' a — b tani(^ — J5)' 32. a^ = b^-\-c^ — 2 be cos A. 33. Area = -^ 6c sin ^. ANSWEKS CHAPTER I Page 4.-7. (x, -y), (-«, y), (-«, -y). 8. (a\/2, 0), (0, aV2), (- aV'2, 0), (0, - ay/2). 9. (0, 0), (2a, 0), (a, aVS) for one position of the triangle. 10. On a line two units to the right of the x-axis. On a line three units below the y-axis. 11. Yes. Yes. No. 13. y = x, or y = — x. 14. On a Circle with centre at the origin and radius equal to a \ x^ -^ y'^ =■ 'k. 15. x= — 5, or X = 1. 16. y = X + 3. Pages. —6. An isosceles triangle. 7. A parallelogram. 8. (VlO, tan-i-^), (- \/2, 45°), ( VIO, tan-i - 3), (3v^, 45°). 9. (0, 0), (2 a, 0), (2 a, 60°) ; (a, 0), (a\/3, 30°), (a, 60°). 10. (0, 0), (2 a, 0), (2aV2, 45°), (2 a, 90°) ; (a, 0), (aVB, tan-4), (aV5, tan-i2), (a, 90°). 11. (0, 0), (2a, 0), (2aV3, 30°), (4 a, 60°), (2aV3, 90°), (2 a, 120°); (a, 0), (aV7, tan-i^V (aVl3, tan-i^V (a\/l3, tan-i2\/3), (avT, tan-i-§^V (a, 120°). 12. p must vary from to co, while d varies from to 2 ir, or ^ must vary from — 00 to 4- CO, while d varies from to tt. 19. p — a sec ^, where a is the distance from the pole to the line ; p = hc&cd. 20. On a circle passing through the pole, with centre on the initial line and diameter a ; on a circle with diameter a, above the initial line and tangent to it at the pole. Page 9.— 1. 13. 3. 2\/7. 4. 3V5, 3V6, 3V2. Page 10. — 3. a^b-2y/2' Page 11. - 1. (- 2, 1) and (~ 1, 2), (4, 7) and (-7,-4). 2. 2 : 3 ; -(3:8). Page 16. — 1. 13. 2. ^VS. 3. 12^ + 6a/3. 4. 2V3. 6. 8\/3. 6. ia2V3. 7. 2ac. 8. a\ 9. 31. 10. 47. 11. (2V3, 2), (\/2, - V2), (!i-fV2). 12. (5,tan-i-f), (13,tan-i-J^), (\/l0,tan-i3). 13. (-^,0), (I, 1). 14. (- J^, J^), (- \, \), 16. (17, 1), (- 13, 6). 16. Sides 13, 13, 7\/2. Medians i V366, \y/m>, ^y/2, isosceles. 17. Sides 4\/2, 3V2, 6V2, area 12, a rt. A. 18. Sides 2V5, 2V6, 3V5, 3\/5, area 24, a parallelogram. 19. p = a. 20. e=a. 21. p = 2asin^tan^. 22. p = asecd ±b. 23. y=mx. 24. x2-2/2 = flf2. 26. (x2 + ya)8 = 4a2xV. 26. (2xH22/Hax)2= a2(x«+y2). 249 250 ANSWERS CHAPTER II Page 19. — 1. x^ + 2/2 >^ <^ or = 9. 2. (x + 3)2 + (y - 1)2 >, <, or =16. 3. (X - ay + {y- 6)2 >, <, or = r^. Page 20. — 1. ?/ - X + 2 >, <, or = 0. 2. y + ic - 3 >, <, or = 0. Page 21. —1. 3x + 5?/-4>, <, or =0. 2. 2(rt - c)x + 2(& - d)y = a"^ + b^ - c^ - cP. Page 22. — 1. The origin. All the plane except the origin. No locus in the plane. 2. The jc-axis. No locus. All the plane except the x-axis. 3. The line X = a. All the plane to the right of the line x = a. All the plane to the left of the line x = a. 4. The line y = b. All the plane above this line. All the plane below this line. 5. The circular ring bounded by the circles x^+y^=4t and x2 + 2/2 = 9. 6. The ring bounded by the concentric circles (x — 2)2 + (y- 3)2 = 9 and (x - 2)2-f (y - 3)2 = 16. 7. All the plane between the two lines x= a and x = b. 8. A circle. All the plane outside of this circle. All the plane inside of this circle. 9. That part of the plane bounded by two circles passing through the pole, with centres on the initial line and diameters a and b. 10. Similar to No. 8. 11. Similar to Nos. 8 and 10. 12. x=2, x=|, x=3, x=|. Page 33, § 25. — 1. n. 2. The values of p corresponding to ^ = are the intercepts of the locus on the initial line. The values of 6 corresponding to p = give the direction of the lines tangent to the curve at the pole. Page 33, § 26. — The points of intersection are : 1. (f|, V)- 2. (-^^, V)- 3. (-2,-3). 4. (0,6), (3, 4). 6. (8,1). 6. Imagin ary. 7. (0,0), (-6,-2). 8. (^av^,av^). 9. la(2 -\-V5), 2a^2 ±Vll 10. (4a,4a). 13. 6|. Page 37. — The loci of these equations are symmetrical with respect to : 1. 2/-axis. 2. j^-axis. 3. x-axis. 4. x-axis. 6. Origin. 6. Origin. 7. x-axis. 8. 2/-axis. 9. Both axes, and the origin. 10. Origin. 11, 12, 13, 14, 15, 16. Both axes, and the origin. 17. Origin. 18. Nothing. 19. ?/-axis. 20. Nothing. 21. The origin, and the lines y = x and y = — x. 22. y-axis. 23. Origin. 24. Origin. 25. The origin, and the lines y = x and ?/ = — x. 26, 27. The line y = x. 28. The origin, both axes, and both lines y = x, y=z — x. 29. Both axes, and origin. 30. Origin. 31. Same as 28. 32. The line y = x. 33. Same as 28. 34. x-axis. 35. x-axis. 36. y-Sixis. 37. Origin. 38. Both axes, and origin. 39. The origin, and the lines y = x, y = — x. Page 41, § 32. — 1. x2 + y^ =_± 2ry. 2. a, b, r. 3. (T 2, 0), 2. 4. (0, T 3), 3. 5. (- 1, 2), V5. 6. (|, - f), ^Vsi. 7. (- 3, 2), 2. 8. (I, -1),3. 9. (-3, -4), 6. 10. (f,4),4. 11. x'^+y^±2rx±2ry+r^=0. Page 41, § 33. — 1. (2), (3), (4) are of the first degree because the pole is on the circle. Hence any line through the pole can cut the circle in only one other point. In (1) the pole is outside if a > r, inside if a < r. 2. See for- mulae in § 32. ANSWERS 251 Page 43. — 1. p^ = — r-^ ; Since the denominator is the sum 6^ cos2 e + a'^ sin2 d of two squares, it can never be zero. Hence p can never be infinite. 2. Out- side, inside. 4. (1) 10,6, (±4,0). (2) 4>/2,4, (±2,0). (3) 10,8,(0, ±3). Page 44, 1 36. -3. .- ,.,„,. /T,. 3,„. , - Infinite. 4. (1) (± 5, 0), 3x + 4?/ = 0, 3a;-4?/ = 0._ (2) (0, ±\/4l), 25 2/2-16 a;2=0. (3) (±2>/6, 0), 4x2-2/2 = 0. (4) (±V2, 0), x2-?/2 = o. (5) (0, ±\/2), x^ - y^ = 0. (6) (±V6,0), 4x2 -2/2 = 0. Page 44, § 37. -(2) (1, 0), x = - 1. (3) (- 2, 0), x = 2. (4) (f, 0), 2x = -3. (5) (0,2), 2/ =-2. (6) (0, - f), 2 2/ = 5. (7) (0, - 3), 2/ = 3. Page 45. — 1. 2/ = 4x. 3. 2(a - c)x + 2(6 - d)y = a2 + 52 _ ^2 _ ^. 5. x2 + 2/^ + 6x-62/ + 9 = 0. 6. x2 + 2/2 ± 8x ± 8?/ + 16 = 0. 7. (x-2)2 + (2/ + 6)2 = (4 ± 2)2. 11. (1), (2), (4) a straight line ; (3) a hyperbola. 12. (1) a circle ; (2) a hyperbola ; (3) two hyperbolas ; (4) two straight lines. 13. x2 + 2/2 = c2 - a2. 14. x2 + 2/2-4x = 0. 15. 2ax = c2. 16. A circle with centre at the centre of the square. CHAPTER in Page 51. — 1. When the line goes through the origin. When the line is n 1 X XI- . « 1 10 X V - X VSv parallel to the 2/-axis. 2. y = -—x -—, —^^ + -^=1, - + -^= - 5 V2 3. 2/-fx + 6, 3g + | = l, -|a: + t2/ = ¥. 4. y = x-6, ^ + "^^ = 1 V2 \/2 . 4 ,5 V41 V41 \/41 6. 2/ = Ax-l|, ^^ + _^ = l,,5^x-H2/ = l. 7. 2/ = ix + |, 1^ + 1=1 ^-A.y = 9_. 8. y=ix-i, ? + -iL=i, .2_x-_3_'^ = ^_, V6 V5 2V6 * ^ 2^-t ' Vi3 VI3^ Vl3 »• 2/ = -|a;, 4^ + 4^ = 0. 10. ? = 1, x = a. 11. 2/ = 4, ? = 1, y = 4 \/l3 Vl3 a 4 12. ^^ :e+ ^^ 2/+ ^^ =0;i^x + ^2/+^^ = ^ + Jg+0 ^ A + B+0^^ A + B +C ' A A^ ^ A^ J2A{A + B+C)^ I 2B{A+B + C) j 2C(^ + ^+C) _ \ BG ^y CA ^^y AB 13. _iOa;-82/ + 40 = 0; -^x-^2/ + l = 0. 14. |x-32/ + 9 = 0; ± fa; 1=22/ ±6 = 0. 16. -6x + 22/±3 = 0, or 20x - 82/ - 12 = 0; 10x-4 2/-6 = 0, or - 15x + 62/ + 9 = 0. Page 52. — 1. a = 30°, p = 2. 2. a = 60°, ;) = 1. 8. « = - 45°, p = 3. 4. a= -120°, j)= -4. 6. « = 120°, p = - 1. 6. « = - 60°, i? = - 5. 252 ANSWERS 7. pcoa{e — a)=p. 8. pcoB0 =Pt psine—p. 9. ^ = A;, where A; is any con- stant angle ; ^ = 0. 10. ^ = ; /2, ±1. 9. 0, 00 , ± |V3. 10. - .3, - .1, - .6, .5, - 1, 1, .... Page 78. — 1. 3a; + 4y = 25. 2. 12a; - 5y + 169 = 0. 3. 2^ = a; + 2, 2y = a; + 8. 4. 2a; + 2/ = 6. 5. y = 8a;-16, a; + y = 2. 6. a; + 3y = 4. 7. 2a5-3y = 0, 2x + 3y+18 = 0. 8. 3y = 4a;-8, 4x + 32/ = 26, 4a; + 3y + 24 = 0, 3y = 4a;4-42. 9. x-^2y = S, 2y = x + 5, 2y = x-5, x + 2y = lS. 10. a; = 3, 3a; + 4y = 15, 4y-3a; + 17. 17. a;a;' - yy' = 1. 20. a;a;'~-i + yy'^-^ = 1. 21. a = 1 ; y = 1. 22. a; + 2 y = 3. 23. y = 2 a; - 8. 24. 2y = x + 9, « + 22/ + l=0. 25. 3!/=4a;-l, 4a; + 3y = 13. 26. a; — 2/ = 0, a; + 2/ = 0, a; = ± a. 27. The equation of the tangent at any point (a, /3) on the curve is ^^^ + ^^ = 2, at the point (a, 6), - + ^ = 2. a« &»* a b 30. tan-if. 32. 90°. 33. 90°. 34. tan-i3^3j. 35. 2 y = a; + 4, 2x + y = 12, 46°. 39. -^^. 41. -r^^^' 42. -J^^ll-. 63. -sin 2 a;. (1 - a;2)2 (a - bxy (1 + a;)"+» 55. tan2 x. 56. 8 sec* x. 57. x cos x. 59. 2 sec2 x tan x(sec2 x + tan^ x). 61. 4x8 + 2x(a + 6). 62. 15x* + 3x(2a + 6x). 63. ~ ^ "'^ . a - 3 X 2 X -4- 3 x8 ^^ "^ ^^ 65- , 67. / ' 68. 6x(2x« + 3)(l-3x2)2(2x-12x8-9). 2Va-x Vl+x2 V -T- yv j\ j ^ 71. ^,f~^^ ' 73. _i^i±J_. 74. (m sin2 X + n cos2 X) (i + x2)t a+«^^)* ( 8in**-^x ■ cos"*-^x \ cogm+ix sin»+ix ] 254 ANSWERS CHAPTER VI Page 86. — 1. 0(ic) = 2x ^ -5x^-6, B = ^. 2. 2, 4. 3. 4, 14. 4. ± V2. 5. 2±V3. 6. '^^Yo~^^ ' '^' 1,-3,4. 8. ±1,3,5. 9. Two real roots, one between and 1, the other between 4 and 5. 10. 2, — 4, 10. 11. 3, and one root between and 1, the other between — 1 and — 2. 12. 4, and one root between and 1, the other between — 2 and — 3. 13. — 3, 5, between and 1, and between 3 and 4. 14. x^ + x^ - 17 x + 15 = 0. 15. x* - 3 cc^ - 28 x"^ + 36 a; +144 = 0. 16. 30^3 + 77 x2 - 92x + 21 = 0. 17. x* - 17 a;2 + 16 = 0. 18. x* - 2 x3 - 19x2 + 20 X = 0. 19. x* - 5 x2 + 6 = 0. 20. x* + 2 x^ + 2 x2 + 4 X = 0. 21. x3 - 13 x2 + 50 X - 60 = 0. 22. x* - 6 x^ - 8x2 - 66x - 65 = 0. 23. x5-3x*-23x3 + 61x2 + 94x-120 = 0. 24. x^ + 2 x* - 16x3 + 18x2 + 15x = 0. 25. 6x6 -11x5 -10 x* + 3x3 + 2x2 = 0. 26. x* + 2 x^ + 9x2 + 2x + 66 = 0. 27. x5 + 5x4-20x2 + 71x + 231 = 0. Page 87. — 1.-2. 2. - V3. 3. - 4. 4. 4. 5. 0, 0, 0, 3 + V^^. 6. 0, - f . 7. 0, 0, - |. Page 90.— 1. 1.879, -.347, -1.532. 2. 1.356,1.692,-3.048. 3. 1.939. 4. 3.264. 5. 1.769,2.672, -4.441. 6. .593,2.047. 7. 2.382,4.618. 8. 3.128, 1.201,-1.33. 9. .494,2.861,-3.112. 10. 2.583,7.169,-3.399,-6.353. Page 92. 2. 3, -5, 2 - y/3. 3. f , 3 - V^. 7. x* + 2 x2 + 25 = 0. 8. ±V2±V3I,3±V363. 4 Page 95.-1. x2 + 4x-5 = 0. 2. x3-6x2- 7 x + 60 = 0. 3. x3 + 8x2 - 28 X - 80 = 0. 4. X* - 12 x2 - 12 X + 3 = 0. 5. x^ - 4 x2 - 60 x = 0. 6. x* - 4 x3 + x2+ 6 X = 0. 7. x* - 11 x3 + 36 x2 - 30 X = 0. 8. x2 - 25 = 0. 9. a;3 _ 4 a; _ 2 = 0. 10. x* - 35 x2 - 90 x + 304 = 0. 11. (1) c = - 2, (2) c = 1, or - 5. 12. x3 + 6 x2 - 32 = 0, or x3 - 6 x2 = 0. 13. 9 x2 + 8 x - 1 = 0. 14. 12x3 + 13x2-3x -2=0. 15. 24 x4 + 20x3 -30x2-5x + 6 = 0. 19. 1,2, 4. 20.-1,11. 21. -1, -1±2V^ . 22. i,l±2^. 23. 1,^I±^. o 5 4 25. ±1, 1±^-Z^. 26. -2, -I, 2 + V3. 27. -1,^,2, ^i±V^n5. 28. ±1,4.2, ^i±|^E«. 14 5 Page 97.-1. (2, -16), (-2, 16). 2. (1, 16), (4, -11). 3. (0, 32), (4, 0). 4. (-1, -67), (3, 189). 5. (0, 10), (2, -54), (-2, 74). 6. (1,73), (-3, -567), (4, -224). Page 100. — 1. - 6, - 6, 1. 2. 3, 3, - 4. 3. 3, 3, 3, - 2. 4. 2, 2, 3, 4. 6.3,3,3,-4. 6. 2, 2, 2, - 3, - 3. 7.-1,-1,-1,2,2. 8.3,3,-2,-2. 9. 1, 1, 1, -3 + V-15 , ^Q 2, 2, 2, -3. 11. 1, 1, 1, 1. -4. 12. -4,-4, ANSWERS 255 -4,2^ 13. _5, -6, -5, -5,1. 14.2,2,2,-1,-1,1. 16. 1 ± v^ 1± V2, - 2. ^16. - 2 ± V3, - 2 ± VS, - 2, 3. 17. 1 ± V^, 1 ± V^, - 4. 18. _l±V-2, -l±V-2, 4. 19. 4g8 + 27r2 = 0. Page 104. — 1. 4, - 4, 0. 2. - 36, 18. 3. 108. 6. 34^2^. 6. ^^ a'^. 7. f|V2. 8. I. 10. 1. 11. |. 12. 9a2. 15. i- Page 111. —1. Max. 4 ; min. ; (1, 2). 2, Max. 4 ; min. -28 ; (3, -12). 8. Neither a max. nor a min.; (1, 11). 4. Max. 36; min. 32; (3, 34). 6. rV2; rV2 and lrV2. 6. aV2 and ^6V2. 7. f the altitude of the segment. 8. The square with corners at the middle points of the sides of the given square. 9. ^ the altitud e of the cone . 10. h=r. 11. h=2r. 12. /i = fr. 13. 2x8x8. 14. l(a -h h— \^a'^ — ab -\- b'^), where a, b are the sides of the given rectangle. 15. He must walk one mile. 16. 6 miles an hour. 17. | rVS. 18. | r. 19. fr. 20.fr. 21. V2. 22. r=J-^, h = J^. 23. 4 r. 24. i the altitude of the paraboloid. 27. 10.392 in. and 14.697 in. CHAPTER VII Page 117. — 5. The two foci of a circle coincide at the centre, and the direc- trices are at an infinite distance from the centre ; one focus and one directrix of a parabola are in the infinite region of the plane ; in the case of two intersecting lines the two foci coincide with the point of intersection of the lines, and the two directrices also coincide and pass through this same point. Page 130. —1. 2x + y + 2 = 0, 2y = a; + 6. 2. x-^y-{-2 = 0, y = x-6. 8. ix-3y = 25, Sx-{-iy = 0. 4. 6a; + 3?/ + 16 = 0, Sx-6y-\-30 = 0. 5. «-2?/ + 4 = 0, 2x + y + S = 0. 6. x + 2y + 2 = 0, 2x-2^ = 6. 7. x = 0. 8. 2 ic + 3 ?/ = 0. d. 5x-Sy = 0. 10. 2x-y = 0. 11. Put the first degree terms of the equation of the conic equal to zero ; the result will be the equation of the tangent at the origin. 12. 2y = Sx + 6. 13. x + 2y-\-2 = 0. 14. 2 a; + 3 ?/ = 1. 15. 2 x - y + 7 = 0. 16. The equation of the polar of the origin with respect to a conic is found by putting half of the first degree terms plus the constant term of the equation of the conic equal to zero. 17. y = x + l, 2y = x + 4. 18. 6y = 5x + 9, x + 2y-f-5 = 0. 19. 3y = 4x + 25, 3x + 42/ = 25. 20. 2^ + 3 = 0, 4 X + 5 2/ = 25. 24. The line at infinity. At infinity. 25^ /r^cos« r!lllL^V(-Psec«, -tana). \ P P J CHAPTER VIII Page 131. — 3. It passes through the point (x , y') and is perpendicular to the polar of this point. 256 ANSWERS Page 133. •— 1. Tangents, y =x-\-a, y = — x — a; normals, y = - a; + 3 a, y = x-Sa. 3. 2/ = \/3(a; + l). ^.2y = 2x + 3. 5. F(- 2, 0), i^C- f, 0), L.R. = 3, directrix a; = - 2|. 6. F(- 2, 2), F^- 2, f), L.R. = 2, directrix y = 21. 7. F(- 2, 4), .F(- i, 4), L.R. = 6, directrix x = 3 J. 8. F(3, -1), j?'(3, _ ^1), L.R. = I, directrix y =- 1^^. 9. F(l, 3), F{- 1, 3), L.R. = 8, directrix x = S. Page 136. — 1. y=x-l. 2. 3ic-4y + l=0. 3. 2y = x + 6,Sx+2y + 2 = 0. a r 3 ,-7 6. (1) (a, ±2 a), (2) (0, 0) and (3 a, 2aVB). 7. 2^ + W? + «a/- = 0- '0 'a 24. (1) y = A^a:, (2) kx/^ — y^ + 2ax = 0, (3) A;x = a, where A; is the constant. 26. y'^ = ax, where y^ _ 4 Qj/g is the given parabola. 27. y^ = a(2x—a). 28. y'^-2ax'-ky + 2ah = 0, (1) 1/2 = 2 a(a: - a), (2) y2 = 2ax, (3) 2/2 = 2 a(x - 4 a), (4) y* = 2 a(x + a). CHAPTER IX Page 142. —1. 2(x^ + y^)-Ux-ny + 12 = 0. 2. x2 + y^-ix-\-2y = 0. 3. ar2 + y2 + 8 a; + 20 2/ + 31 = 0. 4. x^ + y^ -ISx - y -i-lO = 0. 5. 2 a; + 3 y = ±2Vl3. 6. 3a;-22/ = 9±3\/l3. 't.x + 2y = 5. 8. 9a; -6?/ = 14. 9. 6« -42/ = 14. 10. (4,2), (2,-6). 11. (2,-4), (3, |). 12. (-3,2). Page 143. — 1. The point (x', y') is then inside the circle. 2. The product of the segments of any chord (or secant) drawn through the origin. 3. The ori- gin is outside, on, or inside the circle according as c is positive, zero, or negative. Page 146. — The products of the segments of the chords in examples 1 to 6 are : 1. 9, 37. 2. - 12, 7. 3. 15, - 4. 4. - IJ, 7J. . 5. 4, 25. 6. - 11, 94. 7. 5a; -6y + 4 = 0. 8. 16x + 8y = 17. 9. h{a-h)y = ac. 10. x-y = 0, VJ^(a + 6)2-4c. 12. (0,1). 13. (0,0). 14. y = m(a; - a) + 6, where (a, 6) is the centre. 15. One, viz. the line through the given point and the centre. 16. a;2+?/2-6a;-4?/=0. 17. x'^+y'^-Qx-^y+^-O. 18. x^+y^-\-Qx-Qy-\-^-0, a;2+?/2+30a;-30y-|-225=0. 19. 4(a;2+y2^cx_ci/) + c2=0. 20. 2x+5y=9. 2L a; + 5y + 13 = 0. 23. 9 x + 12 y = 29, 9 a; + 12 y + 71 = 0. 25.(2,-3). 26. (1) X + my = ± r vTTwi2, (2) /x±yVc2-r2 = cr, (3) x±y=±rV^. 27. 14 X - 12 y + 29 = 0. 28. ax - &y = a2. 29. (f f , ^j). 30. ( - 2 a, 2 6). 34. ci = C2 = cs, where ci, cs, cz are the constant terms in the equations of the circles. 35. 3(x2 + y2) _ 6x - By = 0, 1:V^^. 36. Imaginary. 37. 1 : \/3. 39. 2y = ±xV5, and 2y±x\/77 + 24 = 0. 42. 4(x2 + y2)_29x+ 12y + 25 = 0. 43. 2 X + y = 1. 44. The circle (x^ + y2) (\2 _ 1) _ X2(2 ax - a2 + r2) = 0, where (0, 0) is the fixed point, X the constant ratio, and (x — a)2 + y2 _ ^2 jg t^e fixed circle. The locus will be a straight line if X = 1. 47. (2, 0), (5, 0). 49. x2 + y^ = 2 r2. 50. x2 + y2 = y2 csc2 § • 53. The radius of the circle is J (a + 6 ± VoHP) . ANSWERS 257 CHAPTER X Page 150.-1. (^±fl%|-: = l, ^: + g±'-^ = 0, (-^%(l4^^ = l. 2. It is perpendicular to the polar of the point (x', y') with respect to the conic. Page 151.— 3. Four. Page 153. — 1. In the case of the ellipse the reflected rays would converge and meet in the other focus. In the hyperbola the reflected rays would diverge^ taking the directions of lines which meet in the other focus. 2. ae^, 0, a(e'*— 1); Page 154.— 1. |V2, (±\/2, 0), 2. 2. ^\/l3, (iVlS, 0), f._ 3. ^VS, . (0, ± V6), V2. 4. 2, (±2V3, 0), 6V3. 6. i, (l±iV3, -2), ^VS. 6. jV?, (-1, l±jV21),iV3. 7.^ + ^ = 1. 8- i + f = 1- 9. x^-2/^ = 8. 10. JV3, a;2+42/2=rt2. n. 3a;2-i/2=i2. 12. 3 xH 5 2/2= 32 ; 3^2-7x2=20. 13. i, 3x2 + 42/2 = 3a2. 2, 3 a;2 ^ 2,2 _ 3 ^2. 17. y = x±2y/b, y = xV3±2Vl3. Page 161. — 1. (- 3, - V3), 210?. 6. a;2 + ^2 = ^2. 7. The locus of § is a circle with centre at the other focus and radius 2 a. Page 167. — 1. 8x + 27 2/ = 0. 2. 5a; + 8y + 30 = 0. 3. y = a; + 3. Page 169 — 1. h'^x-\- ahj = 0, 62a; _ a^y = 0, 6'ic + a^y = 0, 6a; + ay = 0, where 62^2 + a2?/2 = a262 is the conic. 2. x-y = and a; + 2y = 0. 3. 32« + 9y = 0. 4. x + 3y = 0and 12x-25y = 0. 6. 2» + 3y = 5. Page 175.-3. (1) tan-i-, (2) tan-i^-^ (3) 46°. 6. 6x + aei/ = 0and 6ex-ay = 0. 6. 2x + 3y = and 6x + 5y=[0. 13. y = ±x± Va^ ± h\ [ , » ^- ). 16. ^ where Q is the angle the chord VVa2±62 V^-j-62y a2sin2e±62cos2^^ makes with the axis of the conic. 40. An ellipse with major axis equal to the semi-major axis, and minor axis equal to the semi-minor axis of the given ellipse. CHAPTER XI Page 187. —The standard forms of these equations may be written as follows : 1. t^-^x. 2. y2 = 3x. 3. f + f = 1. 4. f-f = 1. 6. f-^; = l. 94 6^36 468 Vl3 10. _^+_J^ = 4. 11. ^-1^ = 1. 12. y2 = J_a;. 13. (x-3y-l)2=0. ^ 5 + V6 h-Vh 2 3 V2 14. ya=__2_x. 15. — 2L_ + _J^ = 2. 16. y2 _ a;a -. 40, or xy = - 20. \/6 5-\/2 5 + V2 258 ANSWERS 17. ?! + y! = i. 18. ^-t. = l, 19. 9 w2- 16x2 = 202. 20. (5x-22/-2) 4 1 9 4 (5x-2y+3)=0. 21. —^ ?^ = 1. 22. (2a:-3y+l)(x+2?/-3)=0. V85+2 V85-2 2 23. 2/2__Lx. 24. (x + y + l)(cc-2y-2)=0. 25. ^2_a;2 = ioV2. 26. 2/2 = ^x. 27. x2 = 3y. 13^ EXAMPLES ON LOCI Page 188. — 2. A parabola whose focus is the centre of the fixed circle. 3. A hyperbola, an ellipse, or a circle. 4. A circle having the line joining the fixed point and the centre of the given circle for a diameter. 5. An ellipse whose axes are the fixed rods. 6. A hyperbola, with one focus at the rifle and the other at the target. 7. The two circles p = ± r sin ^, where r is the radius of the given circle, and 6 is the angle BOG. 8. A circle with its centre on the line AB. 9. A rectangular hyperbola with centre at 0. 10. A circle passing through the points B and C. 11. Two circles passing through the points A and B, and having their centres on the given circle. 12. A circle passing through 0. 14. A circle with centre at the centre of the given triangle. 15. A circle tangent to the two equal sides of the triangle at the ends of the base. MISCELLANEOUS PROBLEMS ON LOCI 2. A sinusoid. 3. A sinusoid. 4. This problem would be the same as No. 2, if the cylinder in No. 2 were an elliptic cylinder. 5. Let a = the distance the fly crawls in a unit of time, w = the angle through which the wheel turns in a unit of time, and t = the time. Then p = at and 6 ■= wt. Hence the polar equation of the locus is p = ( - ) ^. If w represents the number \w/ f a \ of revolutions the wheel makes in a unit of time, the equation is p = r — ^• \£i TTW/ 6. A series of parallel lines. 7. A sinusoid. 12. A rectangular hyperbola. 14. A series of confocal hyperbolas with foci at the centres of the waves. 15. The locus in the plane is a circle. In space the locus is a sphere. 16. A parabola with its axis vertical. 18, 19, 20, 21. If the axes are rectangular, these curves are all rectangular hyperbolas. 23. a =p{\ + ry, where a repre- sents the amount, p the principal, r the rate per annum, t the number of years. If the interest is calculated at n equal intervals each year and added to principal as soon as it is earned, this equation becomes a=j9 ( 1 + - 1 • If we put n = mr, so that when n becomes infinite m also becomes infinite, the equation may be written a=j9flH — j *"• If now n becomes infinite, we approach the con- dition in which the interest is added on continuously. The equation then becomes a =i)e'*, where e is the base of Naperian logarithms. This is known as the Compound Interest Law. ANSWERS 259 SOLID GEOMETRY CHAPTER XII Page 194. —1. x = 0, ?/ = 0, z = 0. z = 0, y = 0; x = 0, z = 0; x = 0, y = 0. 2. The planes bisectiug the angles between the coordinate planes. 3. The lines through the origin equally inclined to the coordinate axes. 6. bx = ay, cy = bz, az = ex. Page 199. — 1. ±-^, ±-^, ±-^. 2. 45° or 135°. 3. 60° or 120°. V3 \/3 V3 4. 0, m, n, where m^ + n^ = 1. Z, 0, w, where l^ -{■ n^ = l. I, m, 0, where Z2 + to2 = 1. 6. 1,0,0; 0,1,0; 0,0,1. 6. -|=, - -|=, i=. - h h ^^ Vli Vli Vli ^ xi -X2 yi -y2 y/ixi - x^y + iyi - 2/2)2 + (^;si - z.)^' Vixi - x^y + {y^ - y^y + (^i - z^Y Z\ — Zi V(Xl - X2)2 + (2/1 - 2/2)2 + (;^j _ ;j2)2 CHAPTER XIII Page 203. — 1. Straight lines. 2. Circles. 3. The ccy-contour is the circle jp2 ^ y2 _ (.2 . the other contours are all straight lines parallel to the ^-axis. The locus is a circular cylinder around the 2!-axis. Page 207. —4. (1) The sphere x^-\-y'^-\- z'^=c'^-a'^ ; (2) the plane 2ax = c^. 7. A sphere with centre at the centre of the cube. 8. If the ellipse — + ^ = 1 a^ b^ is revolved about its major axis, the equation of the generated surface is — 4- ^ +— = 1, if revolved about the minor axis the equation is— + ^ + — = 1. The equations of the surfaces generated by the hyperbola ^ — ^ = 1 are - — V- — — =:\ and — -f ^^ — — = 1. The equation of the surface generated by 02 62 ^,2 «2^a2 52 ^ ^ ^ revolving the parabola ?/2 = 4 ax about its axis is y2 _[. ^^a _ 4 q^^. 9. The equa- tion of the surface generated by revolving the parabola z^ = 4iax around the 2;-axis is 16 oP' (x2 + y^) = 2*. 10. The paraboloid of revolution y^ + z^=. lai^-a). U. (l)^ + | + |? = l,where6^ = a»-A (2) |-|!-|? = 1,. where 62 = c2 _ ^2. 12. y = xtan az, where a depends upon the number of turns the blade makes per unit of length. CHAPTER XTV Page 214. — The symmetric equations may be written as follows : 2 ^-y-S- g-2 3 a;_ y-4 _ g-3 . x + 2 _y _ z-l "2-5-3' 12 3 ' '31 -6* 260 ANSWERS 5. ? = yZlM: 2 3 M. 6. a line perpendicular to the 2!-axis; the plane e = 0. 8. « — 2 = y + 3 = 10. ? = 1^ = ^. 11. (0,6,3). 70° 32'. 8. ±-^-1 = ^-^1^ = =^:^. 10. cos-i-^ 1 1 -1 X — a _ y — b _z ~ ~ a_ y - h m z — c ~0~' z-\. 9. — c x-4- 1 ? = 1^ = -. Z m n Page 216. — 3. cos-^| = ar — l _ y — 4 _ g — 3 3 4. 60°. : 109° 28'. 3 5. cos perpendicular to y-3 _ g--2 -6 -2' 3 35° 16^ 2 4 Page 218. — 8. A plane. 13. If the equations of the two fixed lines are y + mx = 0, 2; + c = 0, and y — mx = 0,_ 2; - c = (§ 147), the equation of the locus is 2/2 _ wi%2 _ (^a _ 1) (2;2 - c2). 14. . Let the equations of the two fixed lines be taken as in Ex. 13, and let 2 1 be the constant length of the moving line. Then the equation of the surface is c\cy - mxzy^ + m\\cmx - yzY + m\c^ - P) (c^ - z'^Y = 0. The locus de- scribed by any fixed point on the line is found by putting z = k\n this equation, where k is any constant. 16. If the fixed lines are the same as in Ex. 13, the equation of the surface is y2! — cmx = 0. CHAPTER XV Page 222. — 1. The constant term D represents the square of the length of any tangent drawn from the origin to the sphere, or the product of the segmei4% into which the origin divides any chord passing through the origin. The origifi is outside, on, or inside the sphere, according as D is positive, zero, or negative. If nn = — d2, P is at the centre of the sphere. 2. Four, since there are four independent constants in the general equation. 8. The equation of a sphere through the four points (Xi, 2/1, zi), (xa, 2/2, «2)» («3, ys, «8), («4, 2/4, «4) may be written : + 2/2 +z^ X, J/, z, 1 Xi2 + 2/i2 + 012, xx, yu «i, 1 1 1 1 Xi^ + 2/2^ + ^2% «2, 2/2, ^2, = 0. Xs^ + ys^ + zs», xs, 2/3, Z8, Xi^ + yi^ + Zi% X4, 2/4, «4, 4. (1, 2, 3), r = 3, outside, t = VE; intercepts 1 ± 2V^ ; 2 ± V^ ; 1, 5. 6. (- 6, 12, 0), r = 13, on, « = ; in tercep ts 0, - 10 ; 0, 24 ; 0, 0. 6. (- 3, 4, - 1), r = 6, inside, t = V- 10 ; intercepts -3 ± VTd ; 4 ± V26 ; -1± VTT. 7. (2, -3, -5), r= \/38, on, t=0 ; intercepts 0, 4 ; 0, -6 ; 0, -10. 8. x^ + y^ + z^±2rx = 0. 9. (x ± r)2+ (2/ ± r)2+ (^ - c)2 = r2. 10. (x±r)2+(2/±r)2+(«±r)2=r8. - H. x2+2/H«2±2ax±2a2/±2c«+a2=0. 12. x2+2/2+«2^2ax±2a2^±2a«+a2=0. 13. x2+2/2+«2-4x-42/±4«+4=0, and x2+y2+«2-20x-202/±20«+100=0. ANSWERS 261 19. The radical axis of three spheres is the locus of all points from which tangents drawn to the spheres are equal. The radical centre of four spheres is the point from which taugents drawn to the four spheres are all equal. If 2)i, 2>2, i>3, 2>4 are the constant terms in the equations of the circles, the condition isDi=2>2=^3 = 2>4. Pag© 237. — 3. Sections of the two surfaces are equal, if their planes are parallel to the y^-plane, and the difference of the squares of their distance*; fr^m *^s^ plane is equal to a^. '. •_ . " :. , J'-j-ttii^f ^^,pKi(^BJK^j(icioB3inate ofthe centj^.*- - ^^ / APPENDIX '^^?' Page 246. — 1, 2, 8. The same a^the ayfxis. 4. The y-axis. 6.. The je-axis. 6. The lines y = x and 2^= — «, -fei^i^^O, 0) y = 2a;, at (1, 0) y =-J x:-}- 1, at (2, 0) y = 2 X - 4. / 8. At ,(()^:%V == 0, at (1, oj 2^ = 2 x - 2, at (^ 1, 0) 2/ =- 2x~ 2. 9.,.At (0, 0) V=5^'l5!^,.at (- 2, 0) y = lOx - 20, at (3, 0) 2/=Jl^x-45. 10. A:t.(.Q, 12,)'yi:^r^ilx + 12, at (1, 0) y = - 12:^'il), at(--8,())y = 28(x+5), at(4,0)2/ = 21(x-4). ' .S"^*" ,'^'- A*^''^ /$-. J'-?^'^ 14 DAY USE STATISTICS LIBRARY lORROWED This book is due on the last date stamped below, or on the date to which renewed. Renewed books are subject to immediate recall. O f;'! i7 loog 0Cj:«iJ964, fmr^'im jim=mm^ 70 H0\U^<>\95|1 LD 21-507n-a,'60 (B132l8l0)i76 General Library University of California Berkeley ^ ASO QKss\ i it I! mili i fliiijjiHiiiliiHii mm- liiliii