,0 ANALYTIC GEOMETRY •V><y^° ANALYTIC GEOMETRY FOR TECHNICAL SCHOOLS ANI) COLLEGES P. A. LAMHERT, M.A. INSTKUCTOli IX MATHEMATICS, LEHIGH UNIVERSITY THE MACMILLAN COMPANY LONDON: MACMILLAN & CO., Ltd. 1897 All rights reserved ,5 QK^ Copyright, 1897, By the MACMILLAN COMPANY. Norfaaati 19rfS3 .1. S. Cushins S; Co. Berwick & Smith Norwood Mass. U..S.A. PREFACE The object of this text-book is to furnish a natural but thorough introduction to the principles and applica- tions of Analytic Geometry for students who have a fair knowledge of Elementary Geometry, Algebra, and Trigonometry. The presentation is descriptive rather than formal. The numerous problems are mainly numerical, and are intended to give familiarity with the method of Analytic (ieometry, rather than to test the student's ingenuity in guessing riddles. Answers are not given, as it is thought better that the numerical results should be verified by actual measurement of figures carefully drawn on cross- section paper. Attention is called to the applications of Analytic Geometry in other branches of Mathematics and Physics. The important engineering curves are thoroughly dis- cussed. This is calculated to increase the interest of the student, aroused by the beautiful application the Analytic Geometry makes of his knowledge of Algebra. The historical notes are intended to combat tlie notion that a mathematical system in all its completeness issues Minerva-like from the brain of an individual. P. A. LAMBERT. 80()5,';4 TABLE OF CONTENTS ANALYTIC GEOMETRY OF TWO DIMENSIONS CHAPTER I Rectangi'lar Coordinates uniriE i-AOK 1. Introduction 1 2. Coordinates 1 3. The Point in a Straight Line 2 4. The Point in a Plane 3 5. Distance between Two I'oints ....... 7 6. Systems of Points in the Phme 8 CHAPTER II Equations of Geometric Figures 7. Tlie Straight Line . 13 8. The Circle . 15 0. Tlie Conic Sections . 15 10. The Ellipse . 18 11. The Hyperbola . 21 12. The Parabola . 24 CHAPTER HI Plottixc; ou Ai.gki'.uaic Equations 13. General Theory 14. Locus of First Degree I'^piation 15. Straight Line through a Point IG. Tangents .... CONTENTS 17. Points of Discontinuity . 18. Asymptotes 19. Miixiinum and Minimum Ordinates 20. Points of Inflection 21. Diametric Method of Plotting Equations 22. Summary of Properties of Loci . PAGE . 33 . 34 . 30 . 37 . 39 . 39 CHAPTER IV PLOTTiNr, OF Transcendental Equations Elementary Transcendental Functions 45 Exponential and Logaritlimic Functions 45 Circular and Inverse Circular Functions 47 Cycloids 54 Prolate and Curtate Cycloids 57 Epicycloids and Hypocycloids 58 Involute of Circle 59 CHAPTER V Tkansfokmation of Coordinates 30. Transformation to Parallel Axes . 31. From Uectangular Axes to Rectangular 32. <)bli(iue Axes 33. From Rectangular Axes to Oblique 34. General Transformation 35. The Problem of Transformation . CHAPTER VI Polar Coordinates 30. Polar Coordinates of a Point . 37. Polar Equations of Geometric Figures 38. Polar Equation of Straiglit Line . 39. Polar Equation of Circle 40. Polar Equations of the Conic Sections 41. Plotting of Polar Eciuations . 42. Transformation from Rectangular to Polar Coordinates CONTENTS CH AFTER VII Propertiks of the Straight Line AKTICI.E 43. Equations of the Straight Line . 41. Angle between Two Lines . 4;"). Distance from a Point to a Line . 4<;. E(iuations of Bisectors of Angles 47. Lines through Intersection of Given Lines 48. Three Points in a Straight Line . 40. Three Lines through a Point 50. Tangent to Curve of Second ( )rder PAGE 81 84 85 86 87 88 89 91 CHAPTER VIII Properties of the Circle 51. Equation of the Circle 93 52. Connnon Chord of Two Circles 94 53. Power of a Point 95 54. Coaxal Systems 97 55. Orthogonal Systems 98 56. Tangents to Circles 1*'^ 57. Poles and Polars • .102 58. Reciprocal Figures 194 59. Inversion 196 60. 61. 62. 63. 64. 65. 66. 67. CHAPTER IX Properties of the Conic Sections General Equation HI Tangents and Normals H"^ Conjugate Diameters 119 Supplementary Chords 122 Parameters 1-'* The Elliptic Compass 12<> Area of the Ellipse 127 Eccentric Angle of Ellipse 128 Eccentric Angle of the Hyperbola 130 CONTENTS CHAPTER X Second Degree Equation ARTICLE PAGE 69. Locus of Second Degree Equation 133 70. Second Degree Equation in Oblique Coordinates . . . 138 71. Conic Section through Five Points 141 72. Conic Sections Tangent to Given Lines 142 73. Similar Conic Sections 144 74. Coufocal Conic Sections 146 CHAPTER XI Line Coordinates 75. Coordinates of a Straight Line 149 76. Line Equations of the Conic Sections 151 77. Cross-ratio of Four Points 151 78. Second Degree Line Equations 152 79. Cross-ratio of a Pencil of Four Rays 153 80. Construction of Projective Ranges and Pencils .... 155 81. Conic Section through Five Points 157 CHAPTER XII Analytic Geoaikti;y of thi; Complkx Yari 82. Graphic Rcpioscntation of the Cuniiilex Variable 83. Arithmetic Operations applied to Vectors . 84. Algebraic Functions of the Complex Variable . 85. Generalized Transcendental Functions 160 162 165 168 ANALYTIC GEOMETRY OF THREE DIMENSIONS CH.VPTER XIII Point, Line, and Plane in Space 86. Rectilinear Space Coordinates 171 87. Polar Space Coordinates 173 88. Distance between Two Points 174 CONTENTS 89. Equations of Lines in Space 90. Equations of the Straight Line . 9L Angle between Two Straight Lines 92. The Plane 03. Distance from a Point to a Plane 94. Angle between Two Planes CHAPTER XIV Curved Sukfaces 95. 90. 97. 98. 99. 100. 101. 102. 103. 104. 105. 100. 107. 108. 100. 110. 111. 112, 113. Cylindrical Surfaces . Conical Surfaces Surfaces of Revolution The Ellipsoid The Hyperboloids The Paraboloids . The Conoid Equations in Three Variables CHAPTl Second Degree Equation in Three Variables Transformation of Coordinates Plane Section of a Quadric Center of Quadric Tangent Plane to Quadric . Reduction of General Equation of Quadric Surfaces of the First Class Surfaces of the Second Class Surfaces of the Third Class Quadrics as Ruled Surfaces Asymptotic Surfaces . Orthogonal Systems of (Juadrics R XV ANALYTIC GEOMETRY CHAPTER I EEOTANGULAK COORDINATES Art. 1. — Introduction The object of analytic* geometry is the study of geometric figures by tlie processes of algebraic analysis. The three fundamental problems of analytic geometry are: To find the equation of a geometric figure or the e(;[uations of its several parts from its geometric definition. To construct the geometric figure represented by a given equation. To find the relations existing between the geometric prop- erties of figures and the analytic properties of equations. Art. 2. — Coordinates Any scheme by means of which a geometric figure may be represented by an equation is called a system of coordinates. * The reasoning of pure geometry, the geometry of Euclid, is mainly synthetic, that is, starting from something known we pass from conse- (luence to consequence until something new results. The reasoning of algebra is analytic, that is, assuming what is to be demonstrated we pass from consequence to consequence until the relation between the unknown and the known is found. The term "analytic geometry" is therefore equivalent to algebraic geometry. The application of algebra to the de- termination of the properties of geometric figures was invented by Descartes (1596-1050), a French philosopher, and published in Leyden in 1G37. 2 ANALYTIC GEOMETRY The coordinates of a point are the quantities which deter- mine the position of the point. Along the line of a railroad the position of a station is determined by its distance and direction from a fixed station ; on our maps the position of a town is determined by its lati- tude and longitude, the distances and directions of the town from two fixed lines of the map ; the position of a point in a survey is determined by its distance and bearing from a fixed station. On these different methods of determining the position of a point are based different systems of coordinates. Akt. 3. — The Point in a Straight Line On a straight line a single quantity or coordinate is sufficient to determine the position of a point. Let be a fixed point -8 -7 .-6 -5 -4 -3-2-10 1 2 3 4 5 6 7 8 Fio. 1. in the line; adopt some length, such as 01, as the linear unit; call distances measured from towards the right positive, dis- tances measured from towards the left negative. Let a point of the line be represented by the number which ex- presses its distance and direction from the fixed point 0. Then to every real number, positive or negative, rational or irrational, there corresponds a definite point in the straight line, and to every point in the line there corresponds a definite real number. This fact is expressed by saying that there is a " one-to-one correspondence " between the points of the line and real numbers. The algebra of a single real variable finds a geometric inter- pretation in the straight line. Denoting by x the distance and direction of a point in the straight line from 0, that is letting X denote the coordinate of the point, the equation cc^ — 2a; — 8 = locates the two points (4), (— 2), in the straight line. UECT ANGULAR COORDINATES 3 Problems. — 1. Locate in the straight Hue the points 3; —2; 1^ ; -2.5; -5; f. 2. Locate VG ; -VS; VlO ; VT. Suggestion. — The numerical value of VS can be found only approximately. The hypotenuse of a rit;ht triangle /^ whose two sides about the right angle are 2 and 1, repre- '^/ sents v'5 exactly. I 3. rind the point midway between xj and x^. Fig. 2. 4. Find the point dividing the line from Xi to X'2 internally into seg- ments whose ratio is /•. 5. Find the puiiit dividing the lino from Xi to X2 externally into seg- ments wiinsi' ratio is r. 6. Locatt! the roots of a;2 + 2 a; - 8 = 0. 7. Locate the roots of xr — i x — i = 0. 8. Locate the roots of x^ - x- + 11 x - = 0. 9. Find the points dividing into three equal parts the line from 2 to 14. 10. Find the points dividing into three ecjual parts the line from X] to X.J. 11. Find the point dividing a line 8 feet long internally into segments in the ratio 3 : 4. 12. A uniform bar 10 feet long has a weight of 15 pounds at one end, of 25 pounds at the other end. Find the point of support for equilib- rium. Al^T. 4. — Thk Poikt in a Plane To determine the position of a point in a plane, assume two straight lines at right angles to each other to be fixed in the plane. These lines are called the one the A'-axis, the other the F-axis. The distance from a point, in llic plane to either axis is moasiired on a line ]):ira]l('l to tlie other axis; the direction of the point I'roui tlic axis is indicated by the alge- braic sii^n prchxcd to the nniiilicr expressing,' tlie distance from tlio axis. 4 ANALYTIC GEOMETRY Distances measured parallel to the X-axis to the right from the l''-axis are called positive ; those measured to the left from the F-axis are called negative. The distance and direction of a point from the T-axis is called the abscissa of the point, and is denoted by x. + Y ( r3,2) G,2) -X A + X ■ ( -3-2) (3-2) -Y Distances measured parallel to the F-axis upward from the X-axis are called positive; those measured downward from the X-axis are called negative. The distance and direction of a point from the X-axis is called the ordinate of the point, and is denoted by y. The axes of reference cut the plane into four parts. Calling the part +X^+F the first angle, +F^4~X the second angle, -XA~ Y the third angle, ~ F^l+X the fourth angle, it is seen that in the first angle ordinate and abscissa are both positive ; in the second angle the ordinate is positive, the abscissa nega- tive ; in the third angle ordinate and abscissa are both negative ; in the fourth angle the ordinate is negative, the abscissa posi- tive. The a1)scissa of a point determines a straight line parallel to the l''-axis in which the point must lie. For, by elementary geometry, the locus of all points on one side of a straight line RECTANG ULA li COORD IN A TES ami equidistant from the straight line is a straight line parallel to the given line. The ordinate of a point determines a straight line parallel to the X-axis in which the point must lie. If both ordinate and abscissa of a point are known, the point must lie in each of two straight lines at right angles to each other, and must, therefore, be the intersection of these lines. I lence ordinate and abscissa together determine a single point in the plane. Conversely, to a point in the plane there correspomls one ordinate and one abscissa. For through the point only one straight line parallel to the I'-axis can be drawn. This fact determines a single value for the abscissa of the point. Through the given point only one parallel to the X-axis can be drawn. This determines a single value for the ordinate of the i)oint. The abscissa and ordinate of a point as defined are together the rectangular* co- ordinates of the point. The point whose co- ordinates are x and y is spoken of as the point (.r, ?/). There is a " one-to-one corre- spondence " between the symbol (x, y) and the points of the Xl'-plane. Problems. — 1. Locate the point (3, — 4). Lay off ;] linear units on the X-axis to the right from the origin, and thrru is found the straight line parallel to the 3'-axis, in whicli the point must lie. On this line lay off 4 linear units downward from its intersec- tion with tlie J-axis, and the point (3, — 4) is located. * This method of representing a point in a plane was invented by Des- cartes. Hence these coordinates are also called Cartesian coordinates. Y A '' ( 3,-4) 1 6 ANA L YTIC GEOMETli Y 2. Locate (-3,0); (0,4); (1, -1); (-1,-1); (-7,5); (10,-7); (15, 20). 3. Locate (2i^3); (- 1, SJ); (% - 51)^(7.8, - 4.5). Locate (V2, \/5); (-Va, Vl7); (V50, V75). Construct the triangle whose vertices are (4, 5), ( — 2, 7), Find the point midway between (4, 7), (0, 5). Find the point midway between (x', y'), (a;", y"). Find the area of tlie triangle whose vertices are (0, 0), (0, 8), (0,0). Find the area ol the triangle whose vertices are (2, 1), (5, 4), (9, 2). 4. 1. -3, -6). Y (5 4) / \ / ^ ::^ (-97-2 1 > 1 "l Z L- - ■ a' " (2:o) (5 0) (9i0) Fio. 5. Suggestion. — The area of the triangle is the area of the trapezoid wliose vertices are (2, 1), (2, 0), (5, 4), (5, 0), plus the area of tlie trapezoid whose vertices are (5,4), (5, 0), (9, 2), (9, 0), minus the area of the trapezoid whose vertices are (2, 1), (2, 0), (9, 2), (9, 0). 10. Show that double the area of the triangle whose vertices are (xi, 2/i), (X2, 2/2), (X3, 2/3) is 2/1 (a-3 - .r2)+ 2/2(a'i - a^3)+ 2/3(^2 - .^i). 11. Show that double the area of the quadrilateral whose vertices are (.1-1, 2/1), (a^2, 2/2), (scs, 2/3), {Xi, 2/4) is yx{Xi - X2) + 2/2(xi - 0:3) + 2/3(0:2 - Xi) + 2/4(*'3 - a^i). 12. Show that double the area of the pentagon whose vertices are (a;i, 2/1), (3^2, 2/2), ('^3, 2/3), (X4, 2/4), (A-5, 2/5) is 2/1 (a-5 - 0:2) + 2/2(^1 - s's) + 2/3(^2 - 3:4) + 2/4(«3 - X5) + y^ixi - Xi). Notice that double the area of any polygon is the sum of the products of the ordinate of each vertex by the difference of the abscissas of the adjacent vertices, these differences being taken in the same direction, anti-clockwise, around the entire polygon. 13. Find the area of the triangle whose vertices are (12, - 5), (- 8, 7), (10, 15). 14. Find the" condition that (x, y) lie in the straight line through (x', 2/'). (*"' y")- RECTANGULAR COORDINATES 7 15. Show that the points (1, 4), (3, 2,), (- 3, 8) lie in a straight line. 16. The vertices of a pentagon are (-J, 3), (— 5, 8), (11, — 4), (0, 12), (14, 7). Plot the pentagon and find its area. 17. A piece of land is bounded by straight lines. From the survey the rectangular coordinates of the stations at the corners referred to a N. S. line and an E. W. line through station A are as follows, distances measured in chains : ^00 D 22.85 17.19 B 14.30 - 15.04 E 7.42 40.09 C 22.85 -4.18 F -8.29 29.80 Plot the survey and find the area of the piece of land. 18. Find tiie point which divides the line from (x', y') to (x", y") internally into segments whose ratio is r. 19. Find the point which divides the line from (x', y') to (x", y") externally into segments whose ratio is r. 20. Locate the points (2, — 9), (— fi, 5), and also the points dividing the line joining them internally and externally in the ratio 2 : 3. 21. Show that the points (x, y), (x, — y) are symmetrical with respect to the X-axis. 22. Show that the points (x, y), (— x,y) are symmetrical with respect to the r-axis. 23. Show that the points (x, y), (— x, ~ y) are symmetrical with respect to the origin. Art. 5. — Distance between Two Points The distance between the points (x', ?/'), (x", y") is the hypote- nuse of the right triangle whose two sides about the riglit angle are («' — x") and {y' — y"). Hence d = ^{x' -x"y + (y'-y"f. Problems. — 1. Find distance between the points (4, 2) , (7, 5) ; (-3,6), (4,-9); (0,8), (7,0); F,,;. r,. (15, -17), (8,2); (-4, -7), (-12, -19). 2. Derive formula for distance from (x', y') to the origin. 3. Find distance from origin to (5, 9) ; (7,-4); (12,-15); (■ Y ^ iKv ') ^ ^ (X," i") A X 9, 14). 8 ANALYTIC GEOMETRY 4. Find the lengths of the sides of the triangle whose vertices are (-3, -2), (7,8), (-5,0). 5. Tlie vertices of a triangle are (0, 0), (4, —5), (—2, 8). Find the lengths of the medians. 6. Find the distance between the middle points of tlie diagonals of the quadrilateral whose vertices are (2, 3), (—4, 5), (6, - 3), (U, 7). 7. Show that the points (6, 0), (1^, 15), (- 3, - 12), (- 7^, - 3) are the vertices of a parallelogram. 8. Find the center of the circle circumscribing the triangle whose ver- tices are (2, 2), (7, - 3), (2, - 8). 9. Find the equation which expresses the condition that the point (x, y) is equidistant from (4, - 5), (— 3, 7). 10. Find the equation which expresses the condition that the distance from the point (x, y) to the point ( — 3, 2) is 5. 11. Find the equation which locates the point (x, y) in the circum- ference of a circle whose radius is r, center (a, b). Akt. 6. — Systems of Points in the 1'lane If any two quantities, which may be called x and y, are so related that for certain values of x, the corresponding values of y are known, the different pairs of corresponding values of X and y may be represented by points in the XF-plane. Comparative statistics and experimental results can fre- quently be more concisely and more forcibly presented graphi- cally than by tabulating numerical values. In the diagram the abscissas represent the years from 1878 to 1891, the corre- sponding ordinates of the full and dotted lines the production of steel in hundred thousand long tons in the United States and Great Britain respectively.* The diagram exhibits graphically the information contained in the adjacent table, condensed from " Mineral Kesources," 1892. Observe that if the points are * In the figure the linear unit on the X-axis is 5 times the linear unit on the r-axis. It will be noticed that the essential feature of a system of coordinates, the "one-to-one correspondence" of the symbol (x, y) and the points of the A"l'-plane, is not disturbed by using different scales for oidiuates and abscissas. RECTANGULAR COORDINATES 9 inaccurately located the diagram becomes not only worthless, but misleading. 45i 1878 '79 '80 '81 '82 '83 'S-t Fm. '85 'SO '87 T. '88 '89 '90 '91 U.S. G. B. U.S. n. p.. 1878 7.3 10.6 1885 17.1 19.7 1879 9.3 10.9 1886 ■ 25.6 23.4 1880 12.5 13.7 1887 .33.4 31.5 1881 15.9 18.0 1888 29.0 34.0 1882 17.4 21.9 1889 33.8 36.7 1883 16.7 20.9 1890 42.8 36.8 1884 15.5 18.5 1801 39.0 32.5 The table furnishes a number of discrete points which in the figure are connected by straight lines to assist the eye. Problems. — Exhibit graphically the information contained in the fol- lowins? tables : 1894 Cost of steel rails per long ton in Penn.sylvania mills f rom 18(_ . (Mineral Resources.) 1867 $166.00 1874 $04.25 1881 $61.13 1888 $29.83 1868 158.50 1875 68.75 1882 48.50 1889 29.25 1869 132.25 1876 59.25 1883 37.75 1890 31.75 1870 106.75 1877 45.50 1884 30.75 1891 29.92 1871 102.50 1878 42.25 1885 28.50 1892 30.00 1872 112.00 1879 48.25 1886 34.50 1893 28.12 1873 120.50 1880 67.50 1887 37.08 1894 24.00 10 ANALYTIC GEOMETRY 2. Commercial value of one ounce gold in ounces silver from 1855 to 1894. (Report of Director of Mint.) 1855 15.38 1865 15.44 1875 16.59 1885 19.41 1850 15.38 1866 15.43 1876 17.88 1886 20.74 1857 15.27 1867 15.57 1877 17.22 1887 21.13 1858 15.38 1868 15.59 1878 17.94 1888 21.99 1859 15.19 1869 15.60 1879 18.40 1889 22.09 1860 15.29 1870 15.57 1880 18.05 1890 19.76 1861 15.50 1871 15.57 1881 18.16 1891 20.92 1862 15.35 1872 15.63 1882 18.19 1892 23.72 1863 15.37 1873 15.92 1883 18.64 1893 26.49 1864 15.37 1874 16.17 1884 18.57 1894 32.56 3. Expense of moving freight per ton mile on N.Y. C. & H.R. R.R. from 1866 to 1894. (Poor's Railway Manual.) 1866 <?'2.16 1873 J? 1.03 1880 ^54 1887 fM 1867 1.95 1874 .98 1881 .56 1888 .59 1808 1.80 1875 .90 1882 .60 1889 .57 1869 1.40 1876 .71 1883 .68 1890 .54 1870 1.15 1877 .70 1884 .62 1891 .57 1871 1.01 1878 .60 1885 .54 1892 .54 1872 1.13 1879 .55 1886 .53 1893 1894 .54 .57 4. Pressure of saturated steam in pounds per square inch at intervals of 9° from 32'^ to 428° Fahrenheit. (Based on Regnaulu's results.) 32° .085 lbs. 131° 2.27 lbs. 230° 20.80 lbs. 329° 101.9 lbs 41 .122 140 2.88 239 24.54 338 115.1 50 .173 149 3.02 248 28.83 347 129.8 59 .241 158 4.51 257 33.71 356 145.8 68 .333 167 5.58 260 39.25 305 163.3 77 .456 176 6.87 275 .45.49 374 182.4 86 .607 185 8.38 284 52.52 383 203.3 95 .800 194 10.16 293 00.40 392 225.9 104 1.06 203 12.20 302 09.21 401 250.3 113 1.38 212 14.70 311 79.03 410 276.9 122 1.78 221 17.53 320 89.80 419 428 305.5 336.3 RECTA NG ULA R COOR DIN A TES 11 In these problems it is evident that theoretically there corresponds a determinate value of the ordinate to every value of the abscissa. Hence the ordinate is called a function of the abscissa, even though it may be impossible to express the relation between ordinate and abscissa by a formula or analytic function. 5. Suppose a body falling freely under gravity down a vertical guide wire to have a pencil attached in such a manner that the pencil traces a line on a vertical sheet of paper moving horizontally from right to left with a uni- form velocity. To determine the relation between the distance the body falls and the time of falling.* Take the vertical and horizontal lines through the starting point as axes of refer- ence, and let 01, 12, 23, •••, be the equal distances through which the sheet of paper moves per second, the spaces 05, 510, •■•, on the vertical axis represent 5 feet. Then the ordinate of any point of the line traced by the pencil represents the distance the body has fallen during the time represented by the abscissa of the point. Careful meas- urements show that the distance varies as the square of the time. Calling the distance .s', the time t, the distance the body falls the first second \ g, where g is found by experiment to be 32.16 feet, the relation between ordinate and abscissa of the line traced by the pencil is expressed by the -1 r 1 I 3 4 5 \ 10 \ 15 ^ . 20 \ 25 \ 30 \ 35 40 45 50 55 60 65 3 proportion , which leads to the equa- tion .<? = \ gt'. The curve and the equation express the same physical law, tJK! one algebraically, the other geometrically. In this problem the ordinate is an analytic function of the abscissa, for the relation between the two is expressed by a formula. The ordinate is a continuous function of the abscissa ; that is, the difference between two ordinates can be made as small as we please by sufficiently diminishing the difference between the corresponding * This is the principle of Morin's apparatus for determining cxperi mentally the law of falling bodies. 12 ANALYTIC GEOMETRY 6. A body is thrown horizontally with a velocity of v feet per second. The only force disturbing the motion of the body taken into account is gravity. Find the position of the body t seconds after starting. Calling the starting point the origin, the horizontal and vertical lines through the origin the A'-axis and F-axis respectively, the coordinates of the body t seconds after starting are x — vt, y = — ^ gfi. Eliminating t, y = 2_ X-, an equation which expresses the relation existing between 2 v^ the coordinates of all points in the path of the body. CHAPTER II EQUATIONS OF GEOMETRIO EIGUEES Art. 7. — The Straight Line A point moving in a plane generates either a straight line or a plane curve. Frequently the geometric law governing tlie motion of the point can be directly expressed in the form of an equation between the coordinates of the point. This equa- tion is called the equation of the geometric figure generated by the point. Draw a straight line through the origin. By elementary geometry -^" = -^ = ^^ = • • •. This succession of equal ratios ..Irt Aa^ Aa.2 expresses a geometric property which characterizes points in the straight line ; for every point in the line furnishes one of these ratios, and no point not in the straight line furnishes one of these ratios. Calling the common value of these ratios m, and letting x and y denote the coordinates of any point in the line, the equation y = mx expresses the same geo- metric property as the succession of equal ratios. Hence if tlie point (x, y) is governed in its nurtion by the equation, it generates a straight line through the origin. Uy trigonometry m is the tangent of the angle through which the X-axis must be turned anti-clockwise to bring it into coincidence with the straight line. 13 Y / / 7 y r I >/ / A / X / 1 i" T 14 ANALYTIC GEOMETRY Y /^ / /^ // / ^-' / /^ y ■7- n y X— A This angle is called the angle which the line makes with the X-axis, and its tangent is called the slope of the line. Give the straight line y=^mx a motion of translation parallel to the y-axis upward through a distance n. The ordinate of every point in the line in the new position is n greater than the ordinate of the same point in the line through the origin. Hence the equation of the straight line, whose slope is m, and which intersects the l''-axis at a point n linear units above the origin, is 2/ = '"'-^ -f ^- ''^ is called the intercept of the line on the Y-axis, x and y are called the current coordi- nates of the straight line, m and n are called the parameters of the straight line. To every straight line there corresponds one pair of values of m and n ; for a straight line makes only one angle with the X-axis, and intersects the T-axis in only one point; conversely, to every pair of values of m and n there corresponds only one straight line. Problems. — 1. Write the equation of the line parallel to the F-axis at a distance of 5 linear units to the right of the F-axis. 2. Write the equation of the line parallel to tlie X-axis intersecting the F-axis 6 below the origin. 3. Write the equation of the straight line through the origin making an angle of 45° with the A'-axis. 4. Find the equation of the line making an angle of 135° with the X-axis, intersecting the I'-axis 5 above the origin. 5. Write the equation of the line whose slope is 2, intercept on F-axis — 5. 6. Find the equation of the path of a point moving in such a manner that it is always equidistant from (3, — 5), (— 3, 5). 7. Find the equation of the path of a point moving in such a manner that it is always equidistant from (4, 2), (- 3, 5). EQUATIONS OF GEOMETRIC FIGURES 15 8. Find the equation of the locus of the points equidistant from (7, 4), (-3, -5). 9. Find the equation of the straight line bisecting the line joining (2, — 5), (G, 3) at right angles. AiiT. 8. — The Circle According to the geometric definition of the circle the point (x, y) describes the circumference of a circle with radius r, center (a, b), if the point (x, y) moves in the XF-plane in such a manner that its distance from (a, h) is always r. This con- dition is expressed by the equation {x — a)- + (y — b)- = r, which is therefore the equation of a circle. Problems. — 1. Write the equation of the circle whose radius is 5, center (2, - 3). 2. Find the equation of the circle with center at origin, radius r. 3. Find equation of circle radius 5, center (5, 0). 4. Find equation of circle radhis 5, center (5, 5). 5. Find equation of circle radius 5, center (—5, 5). 6. Find equation of circle radius 5, center (—5, — 5). 7. Find equation of circle radius 5, center (0, — 5). 8. Find equation of circle radius 5, center (0, 5). Art. 9. — Thk Comc Sections After studying the straight line and circle, the old Greek mathematicians turned their attention to a new class of curves which they called conic sections, because these curves Avere originally obtained by intersecting a cone by a plane. Tt was soon discovered that these curves may be defined thus : A conic section i s a curve traced by a point muving in a plane iiisucli_a manner tliatthe ratio of the distances from the moving point to a fixed point and to a fixed line is constant. This definition will be used to construct these curves, to obtain their properties, and to find their equations. The fixed point is called the focus, the fixed line the directrix of the conic section. When the constant ratio, called the character- istic ratio and denoted by e, is less than unity, the curve is 16 ANALYTIC GEOMETRY called au ellipse ; when greater than unity, an hyperbola ; when equal to unity, a parabola.* The following proposition is due to Quetelet (1796-1874), a Belgian scholar : If a right circular cone is cut by a plane, and two spheres are inscribed in the cone tangent to the plane, the two points of contact are the foci of the section of the cone by the plane ; and the straight lines in which this plane is cut by the planes of the circles of contact of spheres and cone are the directrices corresponding to these two foci respectively. Let the plane cut all the elements of one sheet of the cone. F, F' are the points of contact of the spheres with the cutting plane; F any point in the intersection of plane and surface of cone ; T, T' the points of contact of element of cone through P with spheres. The plane of the elements Sa, Sa' is perpendicular to the cutting plane and the plane of the circles of contact. Since tangents from a point to a sphere are equal, PF= FT, FF = FT'. Hence FF + FF' = FT+ FT' = TT', Fig. 11. a constant. Through F draw DD' perpendicular to the parallels HH', KK'. From the simi- lar triangles FDT and FD'T', FT FT PD FD' hence r>y composition FF PD TT -, by interchanging means, 1)1/ ^ * * ' FD PF ^ PD FF' PD'' PF _ TV DD'' J^W" PD' P77" a constant. Similarly, ^^- = -^, i-^ ^'TT' DD'' PD' TT DD • Call the points * Cayley, in the article on Analytic Geometry in tlie Britannioa, niiiMi edition, calls this definition of conic sections the definition of Apolloiiius. ApoUonius, a Greek mathematician, about 203 b.c, wrote a treatise on Conic Sections. EQUATIoys OF GKOMICTUHJ FKiUUES 17 of intersection of the straight line FF' with the section of the cone V, V'. Since VF + VF = FF' + 2 VF = TT and VF + VF' - FF + 2 VF = TT', VF=VF' and T 'F + VF = VF' + 1 7^" = I ' V = 2'7^ Hence the constant ratio 77^ = 7777, i^^ l*^ss than unity, and the conic section is an ellipse. It is seen that the ellipse may also be defined as the locus of the points, the snni of whose distances from two fixed points, the foci, is constant. Let the plane cut both sheets of the cone. With the same notation as before, PF= PT PF' = PT' ; lienee PF- PF' = TV = a constant. , PT PD From the similar triangles PDT and PD'T , ^^,= 7777,; PP PD PF PI) , ' ^ ^ ^ hence — = -— . P.y division £±- = ±^- hence PF' PD' ^ TV DD' PF ^ TV PD DD' a constant. 18 ANALYTIC GEOMETRY Similarly, = is greater than unity, and the PD' DD'' DB' conic section is an hyperbola. The hyperbola may also be defined as the locus of points, the difference of whose distances from the foci is constant. Let the cutting plane and the element MN make the same angle 6 with a plane perpendicular to the axis of the cone. The intersections of planes through the element 3fN with the cutting plane are perpendicular to the intersection of cutting plane with plane of circle of contact. FF=: FT— MN= PD, and the conic section is a parabola, focus F, directrix HIl'. Art. 10. — The Ellipse, e<l Construction. — Let F be the focus, HH' the directrix. Througli F draw i^Vr perpendicular to HH', &nd on the perpen- dicular to FK through F take the points P and P' such that PF ^P'F^ FK FK ^' Through K and P, and through A" and P', draw straight lines. Draw any number of straight lines parallel to HH', intersecting KP and KP' in r?„ n^, n^, v^, •••, FK in ?)?j, m-j, vi^, •••. With F as center and mj?;, as radius describe an arc intersecting niit, in R. Then H / r' / / / / n / / r 'i ^ ^ "'"' y -~^ ^ T / .' . ' N K y / VF />1 r, ^ A [ ' \ v' K \ V. ni] / - — ,N ■. / \ ^ X ^ ,> < ', ^ \ 11 \ \ \ \ \ H \. TTliK A' FK ' EQUATIONS OF GEOMETRIC FIGURES l!» and Pi is a point in tlie ellipse. Similarly, an infinite number of points of the cnrve may be located. Definitions. — The perpendicular through the focus to the directrix is called the axis of the ellipse. The axis intersects the curve in the points V and V', dividing FK internally and externally into segments whose ratio is e. The points V and F' are called the vertices, the point A midway between V and V, the center of the ellipse. The finite line VV is the transverse axis or major diameter, denoted by 2a; the line PjPi perpendicular to VV at A and limited by the curve is the conjugate axis or minor diameter, denoted by 26; the finite line PP' is called the parameter of the ellipse, denoted by 2 p. The lines KP and KP' are called focal tangents. The ratio of the distance from the focus to the center to the semi-major diameter is called the eccentricity of the ellipse. Properties. — The foci F and F' are equidistant from the center A. By the definition of the ellipse VF = e • VK, V'F= e ' VK Subtracting, FF' = e- VV. Dividing by 2, AF = e • AV Hence e = ^^—~, that is, in the ellipse the o eccentricity equals the characteristic ratio. FP By definition ? = e, and by construction ^ AK ^ FP,= An, By definiti(m eccentricity e = =^ ^ ^ AV a From the figure, VF = AV— AF = a — ae = a (1 — e) ; VF = A V + vlP = a + ae = a(l + e). FPand VF' are called the focal distances. VT+ V'T _VF+ V'F_ ^^ 2 2 Hence ^17r=^. e ^ . ., AF Va-- a eccentricity e = = -b' 20 A NA L YTIC GEOMETli Y 15y clefinitiun ^ = e, hence FA^= "'^^~^^ ^^=e, hence V'K^''-^^-^^- e From the fiLnire i^/f = ^lA" - .li^ = 'i - ae = "'^^~^'^ ; F'A'= .1 A + AF' = 1^ + «e = li(l±i^. By cletiiiitiou --^= e, hence FK ' p — ail — e-)— a[ 1 -^ — = a— = — \ a- J a^ a Equation — Take the axis of the ellipse as X-axis, the perpendicular to the axis through the center as F-axis. Let P be any point of the curve, its coordinates x and y. Tlie problem is to express the definition PF=e- PH by means of an e(]uation between x and y. The definition is equivalent to PF''=e^-PH\ which is the same as pff + (AD + AFf = e\AK+ADy, which becomes y'^ + (x + aey — e^[ - + a; ] , reducincf to — + — — ^ = 1. a' a\l - e') Since the point (o, b) is in the curve, a\l — e^) = 6', and the equation finally becomes - -f-^= 1, Y (0,?J ) p H — — ^ ^ ■^- — — ~Z^ ■\ / ^ ^ \ K ^ («,n \, F A D 1 X \ / •^ ^ H EQUATIONS OF GEOMETUKJ FIGURES 21 Summary. — Collfrtiiig tlic results of the preceding pani- graphs, the fuudanieutal properties of the ellipse -- -f • , = 1 are : a- b'^ Distance from focus to extremity of conjugate diameter a Distance from center to directrix - e Distance from focus to center ae Distance from focus to near vertex Distance from focus to far vertex Distance from directrix to near focus Distance from directrix to far focus Distance from directrix to near vertex Distance from directrix to far vertex Eccentricity Square of semi-conjugate diameter Semi-parameter Art. 11. — The Hyperbola, e>l Construction. — Draw FK through the focus F perpendicular to the directrix ////'. On the perpendicular to FK through F take the i)oints P and P' such that -— - = — - = e. Through K ^ FA FK and P, and through K and P' draw straight lines. Draw any number of parallels to ////', and on these parallels locate points of the curve exactly as was done in the ellipse. The hyperbola consists of two infinite branches. The vertices Fand V divide FK internally and externally into segments whose ratio is e. The construction shows that the parallels to ////' l)etween V and V do not contain points of the curve. The notation is the same as for the ellipse. a(l-e) a{\ + e) a{\ - e2) e a(l -1- e2) e a(l-e) e a{\+e) e e- («'- h-^y^ e — O 62=«i(l - -e^) p = a{\-e .) = ^ a 22 ANAL VriC GEOMETli Y Properties. — From the definition of the hyi)erbohi VF— e • VK, V'F^e ■ V'K. Adding FF'=e- VV] dividing by 2, jLF^e-AV. eccentricity, that is in the hyperbola also the Hence AF characteristic ratio equals the eccentricity. \ \ / // \ \ \ H n. // \ \ ^ 'R \ \ / % s N^ P. /; \ s \ / 1/ F' v' A k\ / V. /'F vu, m / / \ \ / / / \ / / p-^ \ / / \ s / / / \ / / Hi \ From the figure VF=AF-AV= a{e - 1) ; V'F= AF -\- AV = a{e + 1). By definition YK==e, hence VK= ''^"-^^ ■, 1'^ = e, hence V'K= e FA a(e + 1) VK From the figure AK ^ AV — F/i = a — a From the figure e — 1 _ a^ e FK ^AF-AK = ''^'-% F'K AK-{-AF = aO- + 1) By definition pU= c, hence p = a{e- — 1). Equation. — To find the equation of the hyperbola take the axis of the curve as X-axis, the perpendicular to the. axis EQrATIO.XS OF GEOMKiniC FKJl'llKS 2o tlii'oai;li llic (•('iittn- as I'-axis. Let /' he any ituiiit of the curve, its coordinates x and //. The piubleiu is to express the delini- tiou FF=e- PH by lueaus ot an e(iuation between x and y. \ Y / \ H / \ / kp \ / ''/ \ \ / ' 1 \ / / f' A K F / |D X \ — / \ _ — / \ / \ / \ / / H' \ \ The definition is equivalent to PF' — e' • PII , Avliich is tlie same as PD' -it{AD — AFf = <i-{AD — AKf, which becomes y- + (.c — ac)' — e-( x — -] , reducing to "— -J- = 1. \ ej ' tt- a'-(e- — 1) IMaciug <i'{('' — 1)= b-, the ecjuation takes the form ^ — -., = 1. Since Ir = a-c^ — a-, it is seen from the figure that an arc described from the vertex as a center witli a radius e(iual to distance from focus to center intersects the F-axis at a distance b from the center. 2 6 is called the conjugate or minor diameter of the hyperbola. Summary. — Collecting the results of the preceding para- graphs, the fundamental properties of the hyperbola ^.2 = 1 an 24 A iVM L Y TIC GEOMETlt Y Distance from vertex to extremity uf coiijujjate diameter ae Distance from fucus to center Distance from focus to near vertex Distance from focus to far vertex Distance from directrix to near vertex Distance from directrix to far vertex Distance from directrix to near focus Distance from directrix to far focus Eccentricity Square of semi-conjugate diameter Semi-parameter ae a{t -1) ait + 1) a(e -1) e fl(e + 1) e a(e 2-1) e a(e 2+1) e e = (a2 + ^2)^ a 62. = rt2(e2_ 1) P = .,(e'-l)J-- AuT. 12. Y /)t2 H / n / P= / ^> ^ ^ <' P /- / / '/ K / V /f D \ \ %s X \ •• \ ~ — P' \ ^^ ^ ^ ^ \ \ ^ \. — ~K fix \ \ Pa ^. PiiE Parabola, e = l Through F, the foctis, draw FK perpendictilar to HH.\ the direc- trix. On the perpendicular to FK through F lay off FP = FF = FK. Draw the focal tangents KP and KP', draw a series of parallels to HH', and locate points of the pa- rabola as in the case of ellipse and hyperbola. From the figure, it is seen that the distance from vertex to focus is ^ p, distance from vertex to directrix is ^j), the parameter being 2 j^- To find the equation of the pa- rabola, take the axis of the curve as X-axis, the perpendicular to the axis at the vertex as F-axis. Let J*(x, ?/) be any point in the curve. EQUATIONS OF GEOMETIIW FIGURES 25 The problem is to express the deliiiitioii PF = I)K by means of an eqnation between x and y. The definition may bo written 1^'^ = UK', which is the same as PD' + TTf = ( VI> + VKf, wliich becomes if + i^c-^pf^ix + llif, re(bicing to y- = 2 jkv- A parabola whose focus and directrix are known may be generated mechanically as in- dicated in the figure. Problems. — 1. Construct the el- lipse wliosc pariviiK'ter is G, eccentri- city 2. 2. Construct tlie hyperbola whose panuueter is S, eccentricity J. 3. Construct the ellipse whose diameters are 10 and 8. Find tlie equation of the ellipse, its eccentri- city, and parameter. 4. Construct the hyperbola whose diameters are 8 and C>. Find Uw. e(iuation of the liyperbola, its eccentricity, and parameter. 5. Construct the parabola whose parameter is 12 and find its e(iuation. 6. Find the equation of the ellipse whose eccentricity is j^, major diameter 10. 7. The diameters of an hyperbola are 10 and 0. Find distances from center to focus and directri.K. 8. The distances from focus to vertices of an hyperbola are 10 and 2. Find diameters. H -==: ■-^ ^ ^ ^ ^ --, P^ >< k ^ •-. / / .... / / K \ F X - \ \ X \, 9. The parameter of a parabola is 1-2. Find distance from focus to point in curve who.se abscissa is 8. 10. Find diameters of the ellip.se whose parameter is 10, eccentricity I. 11. In an ellipse, the distance from vertex to dinctrix is 0, eccentri- city J. Find diameters and construct ellipse. 26 ANALYTIC GEOMETRY 12. In the ellipse — + ■'" = 1 show that the distances from the foci to a- h- the point (x, y) are r = a - ex, r' = a + ex. r and »•' are called the focal radii of the point (;c, //). The sum of the focal radii of the ellipse is con- stant and equal to 2 a. x" 13. In the hyperbola ^- = 1 show that the focal radii of the point 5- The constant difference of the focal (x, ?/) are r — ex - a, r = ex + a, radii of the hyperbola is 2 u. 14. Find the equation of the ellipse directly from the definition: The ellipse is the locus of the points the sum of whose distances from the foci equals 2 a. Take the line through the foci as A'-axis, the point midway between the foci as origin. When the point (x, y) is on the F-axis its distances from F and F' are each equal to a. Call AF — AF' = c, the distance of (x, y) when on the T-axis from the origin b. Then ci^ - c- = b'\ The geometric condition PF + PF' = 2 a is expressed by the equation - Y ^ ,P x?/) / l/ ,y ^ '\ \ / ^ >■ ^ l\ \ \ F' A D ' ) X \ / ^ _ ^ VyH (x-r)H V//-+(x + 0'-^ = 2 ff, — which reduces to = 1. The definition used in this problem suggests a very simple mechanical construction of the ellipse whose foci and major diameter are known. Fasten the ends of an inextonsible string of constant length 2 a at the foci F and F' . A pencil point guided in the plane by keeping the string stretched traces the ellipse. 15. Find the equation of the hyperbola directly from the definition : The hyperbola is the locus of the points the difference of whose distances from the foci is 2 a. Take the line through tlie foci as X-axis, the point midway between the foci as origin. Call AF = AF' = c, c^ - a" = U^. The condition PF' - PF=2a leads to the equation Vy'^ + (X + c)2 which reduces to \/y"- + (a 1. c)-^ FA^UATIONS OF dKOMFTIUC FKiUIiES 27 The inechiuiical fonstnictidu of tho hyperbola is effected as indicated in tlie figure. "" \ ^ \ >^^ ;;^- \ 5^ r \ ^ •^ =-^ ^^ X / \ r^ f-^^ ^ '" /, ifj) \ ^ r < --' / '/ ^ r ^ / / V t^ ^ F'/ A \ F / \ / \ / \ y \ y \ ^ Fic. 21. 16. Two pins fixed in a ruler are constrained to move in grooves at right anf,des to each other. Show that every point of the ruler describes an ellipse whose semi-diameters are the distances from the point to tho pins. Tliis device is called an elliptic compass. Tlio folldwint;- statcMneiits may help to form an idea of flio importance of the conic sections : The planets and asteroids move in ellipses with the sun at one focus. The eccentricity of the earth's orl)it is about j.\^. The eccentricity of the moon's elliplic }iath about the earth is about J^. Nearly all comets move in parabolas with the sun at the focus. Tho caljlo of a suspension bridLfe, if the load is uniformly distributed over the horizontal, takes the form of a ]iaral)ola. A projectile, iinless projected vei-tically, moves in a ])arabola, if the earth's attraction is the only distinl)ing forcu^ takcui into account. CHAPTER III PLOTTING or ALGEBEAIO EQUATIONS AuT. 13. — General Theory The locus of the points (x, y) whose coordinates are the pairs of real values of x and y satisfying the equation f(;x, y) = is called the graph or locus of the equation. Constructing the graph of an equation is called plotting the equation, or sketching the locus of the equation. An equation f{x, y) = is an algebraic equation, and ?/ an algebraic function of x, when only the operations addition, sub- traction, multiplication, division, involution, and evolution occur in the equation, and each of these only a finite number of times. When the equation has the form y =f(x), y is called an explicit function of x; when the equation has the form f(x, ?/) =0, y is called an implicit function of .a*. The locus represented by an equation f(x, y) — depends on the relative values of the coefficients of the equation. For mf{x, y) = 0, wliere m is any constant, is satisfied by all the pairs of values of x and y which satisfy f(x, ?/) = 0, and by no others. If the graphs of two equations /i(.)-, y) = 0, /.(.r, y) =0 an' constructed, the coordinates of the points of intersection of these graphs are the pairs of real values of x and y which satisfy /i {x, y) = and f, {x, y)—0 simultaneously. Occasionally it is possible to obtain the geometric definition of a locus directly from its equation, and then construct the locus mechanically. The equation x^ -^-y"^ = 25 is at once seen 28 PLOTTING OF ALGEBRA W EQUATIONS 20 to represent a circle with center at oritijin, radius 5. In general it is necessary to locate point after point of the locus by assign- ing arbitrary values to one variable, and computing the corrc- si)onding values of the other from the equation. Akt. 14. — Locus OF First Degukk Equation The locus of the general first degree equation Itotween two variables x and y, Ax -\- B>/ + C = 0, is the locus of ^ b"^ B Moving the locus represented by this equation parallel to the I'-axis upward through a distance -'^, increases each ordinate fi ^ ... l)y — . Hence the equation of the locus in the new position is ?/ = — — .r, which represents a straight line through the origin, since the ordinate is proportional to the abscissa. The equa- tion Ax -\- By + C = () therefore represents a straight line whose slope is — --, and whoso intercept on the F-axis is — — . The intercept of this line on the X-axis, found by placing y equal to zero in the equation and solving for x, is — -• The straight line represented by a lirst degre(> c(pi:iii(>u may be constructed by determining the point of intersection of th(( line with the F-axis and the slope of the line, by dcici'mining any point of the line and the slope of tlie line, by determining the points of intersection of the line with tlie coordinate axes, by locating any two points of the line. Problems. — Construct by the different methods the linos repn^sentcd by the equations. 1. 2x + 32/ = 6. 3. \x-\y = \. 5. •%;( = 1- 2. j/ = x-5. 4. ix-4?/ = 2. 6. ^+^-1. 30 ANALYTIC GEOMETRY 7. Show that ?/- — 2 .r>j — 8 x- = represents two straight lines through the origin. 8. Show that a homogeneous equation of the ?ith degree between x and y represents n straight lines through the origin. 9. Construct the straight line ^ 4- ^ = 1 and the circle x- + 2/" = 25 and compute the coordinates of the points of intersection. Verify by measurement. AiiT. 15. — Stuaigiit Ltxe through a Point Tliroiigh the fixed point (Xf,. an an.ufle « with the X-axis. yo) draw a straight line making Let (x, y) be any point of this line, d the distance of (;r, ?/) from (.r,|, ?/(,). From the figure X — a'o = d cos a, y — ?/o --= d sin a, whence x = .r,, 4- d> cos a, y = y^^ + d sin a, an d y — ?/„ = tan a (x — .r„). That is, if a i)oint (a-, y) is governed in its motion by the ecpiation Fro. 22. 1/ ~ 1/0 = tan a (x — x,|), it generates a straight line through (a'o, ?/o), making an angle a with the X-axis, and the coordinates of the point in this line at a distance d from (.r,„ ?/„) are x = Xo-\- d cos a, y = yQ-\- d sin a. The distance, d, is posi- tive when measured from (xq, y^) in the direction of the side of the angle a through (x„, 7/,,) ; negative Avhen measured in the opposite direction. Y ■^ d {X }!> (X ,^ < ^ ^ — A X siraiglit lino terms of the Problems. — 1. Express the coordinates of a point in tl through (2, ".) making an angle of 30' with tlio A'-axis i distance from (2, 3) to the point. 2. On the straight line through (o, — 2) making an angle of 00" with the X-axis, find the coordinates of the points whose distances from (?., - 2) are 10 and - 10. riOTTINa OF ALaEBRAW ^JQUAriONS 31 3. Write the oiiuation of the s(rai,i;ht line tlimu-li ( - 2, 5) and making an angle of 45 with the A'-axis. 4. Write the equation of the straight line thniugh (4, - 1) wiiose slope is J. 5. Find the distances from the point (2, ;5) to the points of intersection of the line through this point, making an angle of 30° with the A'-axis and the circle x^ + y'^ = 25. The coordinates of any point of the given line are x = 2 + (?cos.OO\ y = 3 + d sin 30°. These values of x and y substituted in the equation of the circle x^ + ?/ = 25 give the equation d^ + (4 cos 30° + 6 sin 30°)rt = 12, which determines the values of d for the points of intersection. AiJT. 16. — Taxcexts To plot a numerical algebraic equation involving; two vari- ables, put it into the form y=f(x) if possible. Comi)uto the values of y for different values of x, and locate the points whose coordinates are the pairs of corresponding real values of x and ?/. Connect the successive points by straight lines, and observe the form towards which the broken line tends, as the nunilicr of points locaiod is indofinitoly increased. This limit of tlu^ broken line is tlic locus of tlie eqnatiim. ExAiNrPLK. — Plot ?/- + a-- = 9. Here y = ± V'.) — x-, a- = d .T = — 4 —3 ji = ± V^^ +1 ±3 • ±2V2 or extracting the roots r = - ;; - 2 ,/ = ± 2.2;;7 ±2. SI'S // has two iMimerically equal values for each value of x. Hence the locus is symmetrical w^ith respect to the X-axis. For a like reason the locus is symmetrical with respect to the : V'.) - y-. — 2 -1 ± VT) ±2V2 -f 2 +3 ± Vr. + 4 10 -f 1 js ± ;; ± 2.siis + 2 ± 2.2;m 32 ANAL YTIC GEOMETR Y Y I'-axis. For values of a- > + 3 and for values of a;< — 3, y is imaginary. Hence the curve lies between the lines a; = +3, a; = — 3. The curve also lies between the lines ?/= +3, y=—o. Locating points of the locus and connecting them by straight lines, the figure formed apjj roaches a circle more and more closely as the number of points located is increased. The form of the equation shows at once that the locus is a circle whose radius is 3, center the origin. Through a point (.t,,, ?/„) of the circle an infinite number of straight lines may be drawn. The coordinates of any point of the straight line y — ?/o = tan a{x — a;,,) through (.?(„ ?/(,), making an angle a with the X-axis are .t = ccq + dcos, a, y — ?/„ + f' sin n. The point (x, y) is a point of the circle x"^ -{-y^ = 9 wlien (x„ + d cos ay + (,Vo + (I sin a)' = 9, that is, when (1) (.i-,2 + 7/,/ - 9) + 2 (cos a • a-o + sin a ■ y„)d + (1~ = 0. Equation (1) determines two values of d, and to each of these values of d there corresponds one point of intersection of line and circle. Since the point (xo, ?/o) is in the circle x^ -i- y'^ = 9, the first term of equation (I) is zero, hence the equation has two roots equal to zero when cos n • .T„ + sin a • v/,, = 0, that is, when tan « = .Vo Y \ /^ ..'■o .'/o) / \ V ^. A N k^ \ / \ \. ^ To d = there corresponds the point (xo, ?/„), and when both roots of PLOTTING OF AIj; hlUlA KJ ICijUATIONS :;:j eqiialiuu (1) ;iic! zero, Hit- two jH»iiil„s ol' iiitL'rsi'clitiii of the straight line // — y,, = t;ui <« (,f — .r„) and the eircle x'- + j/- — \) coincide at {j\„ //„), and the line is the tangent to the circle (.?•„, _?/„). Hence the eqnation of the tangent to the circle x--\-y-=*J at the point {x„, y^ is which reduces to xxy + yy^ — 9. A tangent to any curve is defined as a secant having two points of intersection with the curve coincident.* \\y the direction of the curve at any point is meant the direction of the tangent to the curve at the point. The circle x? 4- ?/- = i) at the point (.r,,, //,,) makes, with the X-axis, tan~'( — " " ). At the points corresponding to .*; = U the angles are tan-'( ^ )= loS^'oT' and tan-'f -^^- I — 41'' 2.';'. Problems. — 1. Sliuw that -"'-" + ••'"=1 is tanj'ent to the ellipse •|2 + p = l at(:r,„2,,). 2. Show that ^^-" - y'-'^ = 1 is tani^oiit to the hyperbola orr «2 ^2-ft^=l at(.ro, yo). 3. Show that yun - p{x + re) is tangent to the parabola if- - '2 px at (■'•0, 2/0) • Ai;t. 17. — I'oix'is (JF Discontinuity KXAMI'LK. — Plot >f = ^^-^^^■ X — 'J a; = -cc •••-;■> -4 -,S -2-1 (» +1 -f-lj +:; + -| ... -f ^ y = 4- 1 ... 4-i +'j, +1 +\ -t -2 Tco +4 +l>i-... +1 * The secant definition of a tangent is due to Descartes and Fermat. D 34 ANAL YTIC GEO ME TR Y From X = to x- = + 2, y is negative and iiu-reases iiuleli- nitely in numerical valne as x approaches 2. From x — + 2 to x — + cTj, y is positive and diminishes from + qk> to + 1. y is negative, and decreases numerically from — i to wliile X passes from to — 1. y is positive and in- creases to + 1 from x = —1 to a; := — Oj . The curve meets each of the two straight lines x = 2 and II = 1 at two points infi- nitely distant from the origin. The point corresponding to X = 2 is a point of dis- continnity of the curve. J,',,;. 25. I'^^Ji" if two abscissas are taken, one less than 2, tlie other greater than two, the difference between the corre- sponding ordinates approaches infinity Avhen the difference between the abscissas is indefinitely diminished, while the definition of continuity requires that the difference between two ordinates may be made less than any assignable quantity by sufficiently diminishing the difference between tlie corre- sponding abscissas. Y \ ~ \ \ \ ^ ^^ — — — A \ ^ X \ ~ - — \ \ — AkT. 18. ASYMI'TOTES Example. — Plot / — x"- = 4. Here // — ± V.ir + 4. a; = -co 4 -3 -2 -1 0+1 +2 + 3 +4 ...+CO Z/ = ±oo-.. ±4.47 ±3.r>l ±2.83 ±2.24 ±2 ±2.24 ±2.83 ±3.61 ±4.47"- ± ^ PLOTTli\G OF ALUKIIRAIC EQUATIONS 1/ lias two iiiiiiici'ically ('(pial iH'al values with opposiit; si^i's fur every value of x. Tlie values of // iuereuse iudetiuitely in numerical value as x in- creases indeHnilely in iniuieri- cal value. It now l)ec(Mnes important to determine whether, as was the case in Art. 17, a strai.n'ht line can be drawn whieh meets the eurve in two points iniinitely distant from the origin. The [xjints of in- tt'rseetion of the straight line // — iiix -f "■ iiiiil the locus of y- — x- = 4 are found by making F"^- '-''• these equations sinudtaneous. Eliminating y, there results the equation in x, (?/r — 1) x' + 2 mnx + n- — 4 == 0. The problem is so to determine m and n that this equation has two infinite roots. An equation has two infinite roots when the coefficients of the two highest powers of the unknown quantity are zero.* Hence y = mx-\-n meets y"^ — x^ = 4: at two points infinitely distant from the origin when nv — 1 = 0, 2 mn = 0, whence 7/1 = ± 1, n = 0. There are, therefore, two straight lines y = x and ?/ = — X, each of which meets the locus of y- — x- = 4 at two points infinitely distant from the origin. These lines are called asymptotes to the curve. \ ^ Y / ^ \ y r/ 1 \ V y / \ / \/ / \ X / \ ^ \ \ ^ L ^ \ /^ L_ ^\l Problem. — Show that y ±-x are asymptotes to the liyperbola X- y^ , a- 62 * Place x = - in (1) ax" + hx"^ + cx""-^ + •■■ + h-x^ + Ix + m = 0. There results (2) a + bz + cr^ + ... + kz" ^ + Iz" • + mz" = 0. Eiinatioii (2) has two zero roots wlit'ii a = 0, b - 0. Hence L'(iualion (1) h;i,s two infinite roots when a = 0, b = 0. 36 ANA L VTIC GEOMETli Y y = AiiT. 19. — Maximum and Minimum Okdinatks EXAMI'LK. l^lut // = X'' — 1 X -\- 7. ^-y,... -4 -3 -2 -1 +1 +1\ +2 +3.--+^ ..-29 +1 +13 +13 +7 +1 -i +1 +13. ..+00 For .T = + 1, _v = + 1 ; fol- ic =+ 2, ?/ = + 1 ; for a; = 11 y= — \. Hence between x=l and a: — 2 the curve passes below the X-axis, turns and again passes above the X-axis. At the turning point the ordi- nate has a niinimuni value ; that is, a value less than the ordinates of the points of the curve just before reaching and just after passing the turning point. The point generating the curve moves upward from a; = to .T = — 1, but some- where between x =— 1 and X — — 2 the point turns and starts moving towards the X-axis. At this turning point the ordinate is a maximum; that is, greater than the ordinates of the points next the turn- ing point on either side. To determine the exact position of the turning points, let x' be the abscissa, ?/' the ordinate of the turning point. Let h be a very small quantity, y, the value of y corresponding to X = x' ± h. Then ?/i - ?/' must be positive when y' is a mini- mum, negative when >/' is a maximum. Now ,/, -y' = (3 x' - 7) ( ± //) + .'5 -v ( ± J'Y + ( ± /')'• h may be taken so small that the lowest power of h deter- Y ^ r : T t \ \ \ I T 1 r J^ f^ 1 A X Fig. 27. PLOTTING OF ALCKHnAW Ei^UATlONti 87 mines the si!j,ii of //, — //'.* //, — //' (-an tlicrciore have the same sign for ± h only when the coefficient of the lirst power of h vanishes. This gives 3 x*^— 7 = 0, whence x = ± V^. x = + V|, rendering y^ — y' positive for ± li, corresponds to a minimum ordinate; a; = — V|, rendering //i — ,v' negative for ± //, corre- sponds to a maximum ordinate. _ ANhen x = + V|, .'/ = 7 - V" V^I = - .2 ; wlien x = - Vf, y^7 +i^V21 = U.2.t The values of a; which make // — are the roots of the ctpui- tion ur' — 7 .f + 7 = 0. These values of x aiv. tlie abscissas of the points wliere the locus of y = x^ — 7x + 7 intersects the X-axis. X' — 7 .« + 7 = 0, therefore, has two roots between -|- 1 and -f 'J, and a negative root between — ."> and — 4. AUT. 20. I'olXTS OK IXFLECTIOX ExAMi'LE. — Plot y + x-y — x = 0. Here // — X ^-cc 3 - 2 - 1 +1 +2 + 3 ... -f X) If (x, y) is a point of the locus, (— x,— y) is also a point of the locus. Hence the origin is a center of symmetry of the locus. A line may be drawn through the origin intersecting the curve in the symmetrical points /-• and P'. If this line is * Let s = ah^ + hh^ + civ' + (IW' + ••• be an infinite scries with finite coefficients, and let li be greater numerically than the largest of the co- efficients b, c, <1, ••-. Tlien hh + eh- + dh^ + ••• < /t — and \ — h s = h\a + hh + '•//■•! -h ,1h^ ...) = /(,3(rt i ^1), when A<h — • 1 — ft When h is indefinitely diminished, h — '- — diminishes indefinitely. Con- 1 — h sequently A becomes less than the finite quantity a, and s has the sign of a/i*. t This method of examining fur maxima and minima was invented by Fermat (1590-l(iG;3). 38 ANALYTIC G EOMETR Y turned about xi until P coincides with A, F' must also coincide with A. The line through A now becomes a tangent to the curve, but this tangent intersects the curve. From the figure it is seen that the coincidence of three points of intersection Y_^ at the point of tangency, and the consequent intersection of the curve by the tangent, is caused by the fact that at the origin the curve changes from concave up to convex up. Such a point of the curve is called a point of inflection. To find the analytic condition which determines a point of inflection, let (a-,,, yo) be any point of // + x'-i/ — x = 0. The coordinates of any point on a line through (.i-,„ ?/„) are X — Xq + d cos a, !i = //„ + d sin a. The points of intersection of line and curve correspond to the values of d satisfying the equation O/o + -V^/o — -^'o) + (sin « — cos « + 2 cos a ■ a;,?/,, + sin a • .r,,^) d -f (cos^ a • ?/o^ + 2 sin a cos a • x^y) d- + cos" a sin a • d'^ — 0. The first term of this equation vanishes by hypothesis, and if the coefticients of d and d- also vanish, the straight line and curve have three coincident points of intersection at (Xf,, ?/o)- The simultaneous vanishing of the coefticients of d and d'^ requires that the equations sin a — cos a -\- 2 cos a • aVi,'/o + sin a • xj^ — and cos- « • %- + 2 sin a cos a ■ .r,, = determine the same value for tan «. This gives the equation l-2av/„ . ; reducing to ^/^) — o x^fi/^ + 2 .(\, = 0, which to- PLOTTING OF ALGEURAKJ EQUATIONS 81) gcther with //„- + x^{y„ — a;o = determines the three puiuts of iutiection (0, 0), (V3, ^V3), (- V3, - \Vo). Art. 21. — Diametukj Mictuod of Tlotting Equations ExAMi'LE. — riot y- — 2 xy -\-'S.xr — Hj x — 0. 1 1 ere y — x ± Vl(J x — 2 x'. Draw the strai.^•ht line y = x. Adding to and subtracting from the ordinate of this line, corre- sponding to any abscissa x, the cpiantity VlO x — 2 x^, the corre- sponding ordinates of the re- qnired locus are obtained. This locus intersects the line y = X when VlGx—2xr=0, that is when x = and x = 8. y is real only for values of x from to 8. The curve intersects the X-axis when x = VlG x — 2 a?, '''"^- '"'*• that is Avhen x=^. Points of the curve are located by the table, x = +1 +2 +3 Vl6 x-2 x' = ±Vl4 ±2V6 ±V3() + 4 -1-5 +6 +7 +8 ±4V2 ±V30 ±2VG ±VT4 Y / /- \ / / / / / / / / / / / / 1 / / / / A / / X / / \ / V / ' AUT. 22. SuMMAliV OK FliOl'KUTlKS OF LoOI From the discussions in the i)receding articles, the following conclusions are ol)tained : 1. If the absolute term of an eijuation is zero, the origin is a point of the hicus of tlu' equation. 40 ANALYTIC GEOMETIIY 2. To iind wlu've the locus of an equation intersects the X-axis, pkice y = in the equation and solve for x ; to find where the locus intersects the F-axis, place x= and solve for y. 3. The abscissas of the points of intersection of the locus of ij = f{x) with the X-axis are the real roots of the equation 4. If the equation contains only even powers of y, the locus is symmetrical with respect to the X-axis ; if the equation con- tains only even powers of x, the locus is symmetrical with respect to the Y-axis. The origin is a center of symmetry of the locus when (— x, — y) satisfies the equation, because {x, y) does. 5. Tlie points of intersection of the straight line 2/ — ^u^ tan«(:« — a-o) with the locus of J\x, ?/)= are the points {x, y) correspond- ing to the values of d Avhich are the roots of the equation obtained by substituting x = x^ + d cos a, y — y^-}- d sin a in f{x, y)— 0. The number of points of intersection is equal to the degree of the equation, and is called the order of the curve. 6. The distances from any point {xq, y^ to the points of in- tersection of the straight line y — ?/„ = tan a(x. — x^ Avith the locus of fix, y)—0 are the values of d which are the roots of the equation obtained by substituting x = .Vo -f- d cos a, y — ?/,, + d sin a in J\x, y) — 0. 7. The tangent to /(.r, y) = at (x*,,, ?/„) in the locus is found by substituting x= .»■„ + d cos«, y = ?/„ + d sin a in /(.t, _?/)= 0, equating to zero the coefficient of the first power of d, and solving for tan a. This value of tan u makes y-y, = timu(x- x^) the equation of the tangent tof(x, _?/)= at (xq, y^). 8. If the curve f(x, y) = has infinite branches, the values of 'tn. and u found by substituting mx + n for // in the equation PLOTTING OF ALGEIiUAW EQUATIONS 41 f{x,ij)=0, and tNiuating to zero the coetticients of tlio two liighest powers of x in the resultini^ equation, deteruiine tlie line // = mx -\- u which meets the curve at two points at in- finity; that is, the asymptote. 9. To examine the locus of ?/=/(.»•) for maximum and mini- mum ordinates form /(.r ± //) — /(.!•). Equate to zero the co- efficient of the first power of h, and solve for x. The values of X which make the coefficient of the second power of h positive correspond to minimum, those which make this coefficient negative correspond to maximum ordinates. 10. To determine the points of inflection of f(x, y)= 0, sul)- stitute X — .To -f d cos a, y = yQ-\-d sin cc in f(x, y) = 0. In the resulting equation place the coefficients of d and d'- ecpial to zero, and equate the values of tan a obtained from these equa- tions. The resulting equation, together with the equation of the curve, determines the points of inflection. Problems. — I'lot tlie numerical algebraic equations: 1. 2x + :],j~7. 5. (X- 2)0/ + 2)= 7. 10. yi = -\Ox. 2 ^^U=\ 6- ■'■^ + y'' = -^- 11- ^ ■'■'- + ^>J^ = 36. '48' 7. a:-2-?/2 = 25. 12. 4x^-9>/ = m. 3- a-// = 4. 8. X- - ?/2 = - 25. 13. 4 x- - ?/2 = - 30. 4. (x-2)ij = [,. 9. if=z\Ox. 14. ?/-i = 10.C-X2. 15. 2/- = X- - 10 r. 26. ?/ = x- - 4 .c + 4. 16. x^ + 10 ./•// + >/i ^ 25. 27. 2/2 = (a: + 2) (X - 3). 17. .t2 + 10 x)/ + >/ + 25 = 0. 28. 2/2 = x2 - 2 x - 8. 18. 2/"^ = 8 x2 - x^ + 7. 29. 2/" = x2 - 4 x + 4. 19. x2 + 2 xy + y2 ^ 25. 30. 2/ = (x - 1 )(x - 2) (x - 3) . 20. .r2 + 10 .r.v + f = 0. 31. y^ = x'^ - G x"- + \\ x - G. 21. I/- = .rt - x2. 32. y = x* - 5 x:- + 4. 22. 2/2 = x2 - X*. 33. 2/- = x« - 5 x2 + 4. 23. 2/- = x^ - x\ 34. y = x» -I- 2x3 - 3x2 - 4 X -I- 4. 24. 2/ = (x + 2)(x-3). 35. y=_^L_ 2/ = x2-2x-8. -^ l-x'J 42 ANALYTIC GEOMETRY 2x-l 3 X + 5' y + 5 3 - x' 4-3x 5x — 6 39. ?/ - 2 x?/ - 2 = 0. 40. y- + 2 x^ + 3 x"^ - 4 X = 0. 41. m2 = x3-2x-^-8x. 36. y 37. y 38. 2/ 42. ?/ = x3-9x2 + 24x + 3. 43. ?/=(3x-5)(2x + 9). 44. 2/'i = x3 - 2 x2. 45. 2/2 + 2x2/- 3x2 + 4x = 0. 46. 2/^ -2x2/ + x2 + X = 0. 47. 2/^ + 4 X2/ + 4 x2 - 4 = 0. 48. 2/"-^ - 2 X2/ + 2 X'- - 2 X = 0. 49. 2/--2x2/ + 2x2+22/ + x + 3 = 0. 50. 2/" - 2 X2/ + x- - 4 2/ + X + 4 = 0. 1. x2 + 2x- 15 = 0. 5. x^ - 7 X + 7 = 0. 2. x3-3x- 10 = 0. 6. x'^ - 7 X - 7 = 0. 3. x2-4x + 4 = 0. 7. x3 + 7 X + 7 = 0. 4. x2 - 5 X + 9 = 0. 8. x3-5x + 2 = 0. riot the real roots of the following equations : 9. X* - 5 x'- + 4 = 0. 10. xHa^'Hx-+x+l=0. 11. x^ + X- + X + 1 = 0. 12. x^ - X- + X + 1 = 0. riot the real roots of the following pairs of simultaneous equations : 1. 2/^ = 10x, x^ + 2/2 = 25. Plot the.se equations to the same axes. The coordinates of the points of intersection of the loci are the pairs of real values of x and y which satisfy each of the given equations. The points of intersection are (2.07,4.42), (2.07, -4.42). By the angle of intersection be- tween two curves is meant the angle between the tangents to the curves at their intersection. Hence the angle between two curves is the differ- ence between the angles the tangents to the curves at their intersection make with the X-axis. Calling the angles the tangents to 2/^ = 10 x and x2 + 2/2 = 25 at the point of intersec- tion (xo, 2/0) make with tlie A'-axis a' and a respectively, tan a' -y ^" v^ z / \ y 7 V t t \ A X V ^t ^ \ \ L ^ X z ^- -< ^\ PLOTTING OF ALGEllUAIC EQUATIONS 43 Evaluating for a-o = 2.07, |/n = 4.48, tan a' = 1.10, tan a = — .47 ; whence a' = 48° 4', a = 154° 50', ami the anglo between the curves is 104° 52'. 2. 2 X + 3 y = 5, ?/ = i .'c + 3. 8. /- + ;/- = 25, ij- = 10 x - x^. 3. y = 3 X + 5, .T^ + !/-^ = 25. 9. 3 .c^ + 2 y-^ = 7, ?/ - 2 x = 0. 4. X- + J/'- = 9, */2 = 10 X — x^. 10. y- = 4 X, 2/ — X = 0. 5. 2/'^ = 10 X, 4 x2 - y^ = ;](!. 11. 2 x^ - r' = 14, x^ + 2/^ = 4. 6. 2 x- — IJ- = 14, X- + y'^ = !). 12. x- -f ?/- = 25, x'^ — )/- = 4. 7. >/ = x^ — 7 X + 7, ?/ — X = 0. Solve the following equations graphically 1. x2 — X - (5 = 0. Plot // = X- and y = x + C> to the same axes. For the points of intersection of the loci x- = x + d ; that is, x- — x — C = 0. Hence the abscissas of the jioints of intersection of y = x^ and y — x + are the real roots of x- — x — = 0. For all quadratic equations, X- + ax + h = 0, the curve ;/ = x- is the same, and the roots are the abscissas of the points of intersection of the straight line y = — ax — h with this curve. In like manner the real roots of any trinomial equation x" + rtx4-/;=0 ai-e the abscissas of the points of in- tersection of y=x" and y + ax-\-b=0. 2. x--3x + 2 = 0. 3. x"- + 5 X + -- 4. .r2 -4=0. 5. :,•;:_ Ox -10: 0. 0. \ Y 1 \ I/. — I A \ / \ / \ / \ / / / \ / / \ / / \ / \ / X A Fk! .•?!. 6. .r2-4x- 15 = 0. 7. 3x2-12x + 2=0. 8. x'-i+5x + 10 = 0. 9. X-- 5x + 5 = 0. 10. x3 - 7 X + 7 = 0. 11. x^ + 7 X + 7 = 0. 12. x^ + 7 X - 7 = 0. 13. .x«-7x-7 = 0. 14. x'»-10x+ 15 = 0. 15. a:»-10x-15 = 0. 44 ANALYTIC GEOMETRY Sketch the following literal algebraic ecjuations : ?/2 = a;3 1. ?/2 _ 3.3 _ (5 _ c)x^ — bcx. Here y — ± vx(x — 6) (x + c). Unless numerical values are assigned to b and c, it is impossible to plot the equa- tion by the location of points. How- ever, the general nature of the locus may be determined by discussing the equa- tion. The A'-axis is an axis of sym- metry, the origin a point of the locus. For < X < 6, y ifi imaginary ; when X = b, y = ; for x > b, y has two nu- merically equal real values with op- posite signs, increasing indefinitely in numerical value with x. For > x > - r, y has two numerically equal values with opposite signs ; for x = — c, y = ; for ^ < — <"» 2/ is imaginary. Sketching a curve in accordance witli these condi- tions, a locus of the nature shown in the figure is obtained. ili = 1. X2 Pig. 32. = 1. 3. ?/ = ax. 6. 4. ?/ = ax + b. 7. »/- = -J, px. X- + y- — a-. b-^ b-^ = 1. 13. (x-rt)(2/- b) = m. 14. y = (x-a)(x-b)ix-c). 15. 2/2 = (x-a)(x-6)(x-0. 16. V2-(X a)-^^-^ 17. y-x =4 a- (2 a - x). 18. ?/- = (x - a) (x -I- /;) (x - c). 11. ?/ =(x - a)(x -f ?;). 12. ?/- =(x- ffl)(x-(- ft). 19. ^y-=x^, the semi-cubic parabola.* 20. rr// = x^, the cubic parabola. ?/'- = (x — f'i)(x — P2)(x — Cs), ei, real, Ci, cs, conjugate imaginarios. 2/- = (x - Pi) (x - eo) (x - es), <'i, ^1;, '';!, real, ^ > r.2 > ^3- y' = (x — (?i)(x — <'2)(x - P3), fii, ^25 Cti real, C] = r.:>, ei > pj. ?/- = (x - ei)(x — r2)(x - Pz), P\, e-2, Cs, real, ^1 > ^2, ^2 = ea- ?/2 =: (X - Pi) (x - r.,) (x - fs), Ci = e-2 = es- 21 25 * The rifling of a cannon, wlion the bore is rolled out on a plane, technically " (Icvclopeil," is a srmi-cul)ic paralmla. CHAPTER IV PLOTTING OF TRANSCENDENTAL EQUATIONS Art. 23. — Elemkntauv Tiianscendkntal Fltnctions Transoendental e(]u<ati()iis are e(iuat.it»ns involvinj;- ti'aiiscon- dental functions. The elementary transcendental functions are the exiioiion- tial, logarithmic, circular or trigonometric, and inverse circular functions. The expression of ti-anscendental Cnnctioiis by means of tlu' fundamental operations of algehra is possil)k^ only hy means of infinite series. AliT. 24. ExroXKXTIAL and LoOAUrTHMIC FirXCTIONS The general type of the exponential function is y = h-a", where a is called the base of the exponential function and is always positive. To plot the exponential func- tion numerically, suppose /> = 1 , c = 1, (( = U. Then y = 2'' and a;= — CO 4 — 3 — 2 — 1 11= 0... i, I. \ I 1 2 3 4...r>D. 1 2 4 S 1 ('.... X. Vov all values of a the locus of y = a' contains the point (0, J) and indefinitely approaches the X-axis. Increasing the value of a causes the locus to recede more 45 _ Y _ I , ^ — 46 ANALYTIC GEOMETRY rapidly from the X-axis for x > 0, and to approach the X-axis more rapidly for x<(). When « = !, the locus is a straight line parallel to the X-axis. When a < 1, the locus approaches the X-axis for x>l, and recedes from the X-axis for x < 0. When c is not unity the function y = a" may be Avritten y =(cfy, and the base of the exponential function becomes a". When b differs from unity, each ordinate of y = h - a" is the corresponding ordinate of ?/ = a" multiplied by b. To plot the exponential function y = b • a" graphically, com- pute ?/n and ?/i , the values of y corresponding to x = and X = a'l, where x^ is any number not zero. Adopt the following notation for corresponding values of x and y. x= 4a-, -'Sxi -2x, -x, x, 2x, ?.x, 4.x,--- y = — 2/- 4 Vz y-2 y-i ih Vx y-2 y-s yi---- Then •lsl = l^ = '!h^'!h = yi=.h=... ««,. On two intersect- y 2 y-i 2/o yi v-i .Vs ing straight lines take OA — yo, OB = yi. Join .1 and B, B x./ \A. O K A C E C. Fin. .'54. draw BC making angle OBC = angle OAB. Then draw CD, DE, EF, •••, parallel to AB and BC alternately ; AH, II K, KL, • ••, parallel to BC and AB alternately. From similar triangles 0K_ OII^ OA ^OB^OC^OD^ OE ()L~ 0K~ OII~ OA OB OC 01) Hence, if 0.4 = ?/o, OB = yi, it follows that OL = y_:„ OK PLOTTING OF TIlANSCENDENTAL EQUATIONS 47 = V .,. on = i/^x, OC — y.2, OD = II .^ ; that is, the ordi nates cor- resi)onding iox = — ox^, — 2:Ci, — x^, 2a-,, Sifj become known and the points of the curve can be located. The logarithmic function ex = log„ {by) is equivalent to tlie exponential function ?/ = a". When y = log a.- is plotted, tlie logarithm of the product of any two numbers is the sum of the ordinates of the abscissas which represent the nund)ers, and the product itself is the aV)Scissa corresponding to this sum of the ordinates. The slide rule is based on this principle. In the slide rule the ordinates of the logarithmic curve are laid off on a straight line from a common point and the ends marked by the corre- sponding abscissas. Art. 25. — Circular and Inverse Circular Functions Ty\ definition, am AOP PD OP P'D' OP' angle ^lOP in circular measure is Hence, if the radius OA' is the linear unit, the line P'D' is a geometric rep- resentative of sinvlOP, the arc A'P' a geometric representative of the angle AOP. The measure of the angle ylOPis 1 when arc AP= OP; that is, the unit of circular measure is the angle at the center which in- tercej^ts on the circumference an arc equal to the radius. The unit of circular measure is callcil the radi of four riglit angles, or .".r.0°, is ' and the value arc A P ai of the c.I'P' OP OP radian is ecpiivalent to 360'' r°.3 -. 48 ANALYTIC GEOMETRY Calling angles generated by the anti-clockwise motion of OA positive, angles generated by the clockwise motion of OA negative, there corresponds to every value of the abstract num- ber X a determinate angle. Unless otherwise specified, angles are expressed in circular measure. When an arc is spoken of without qualiiieation, an arc to radius unity is always understood. In tables of trigonometric functions angles are generally ex- pressed in degrees. Hence, to plot y = sin x numerically, assign arbitrary values to x, find the valne of the correspond- ing angle in degrees, and take from the tables the numerical value of sin x. Y \ / X \ \ ^ ^^ / A \^ - 200° 32' ?/= ••• + .350 3. - ,S5° 57 - .997 \ 1 28° 39' 57° 18' .479 .841 3 171° 53' .141 -4 - 143° 14' - .598 -114° 35 - .909 -1 -57° 18' - .841 -28° 39' -.479 85° 57' 114° 35' 143° 14' .997 .909 .598 3 .... 171' .141 In itvactical problems the ecpiation frequently occurs in the form y = sine (ttx). Were PLOTTING OF TRANSCENDENTAL EQUATIONS 4'J ?/= Wli 1 W'2 -^V2 -1 jV2 1 jV2 -|V2 -1 -^-V2 ^Vii Y lAtbz:: To plot ^ = sin X grapliically, draw a circle witli radius unity, divide the circiunfereuce into any number of equal parts, and placing the origin of arcs at the origin of coordinates, roll the circle along the X-axis, marking on the A'-axis the points of division of the circumference 0, 1, 2, .'5, 4, 5, (>, •••. Througli the points of division of the circumference draw perpendicu- lars to the diameter through the origin of arcs 00, 11, 22, «*>.'», 44, 55, GG, •■•. On the perpendiculars to the X-axis at the points 0, 1, 2, o, 4, 5, G, •••, lay off the distances 00, 11, 22, ',y,\, 44, 55, GG, •••, respectivt'ly. lu this inauucr any nundjer of points of y = sin x may be located. 50 ANALYTIC GEOMETRY On account of the perioilicity of sin.T, the locus of ?/ = sin x consists of an infinite number of repetitions of the curve obtained from a; = to a; = 2 tt. The locus has maximum or- dinates y — -{-1 corresponding to x = (4n + 1)^, minimum ordinates y = — 1 corresponding to x = (4 m + 3)^, where n is any integer. The locus crosses the a>axis when x = mr. To plot y—o sin a.", it is only necessary to multiply each ordinate oi y — sin x by 3. This is effected graphically by drawing a pair of concentric circles, one with radius luiity, the other with radius 3. Since OP' is the linear unit, F'D' rep- resents sin X, and PD represents 3 sin x, *"^' '^^' while X is represented by the arc A'P'. To plot v/=3 sin a-+sin (2 x), plot ?/i=3 sin a; and ?/2=sin (2 a-) on the same axes. The ordinate of ?/ = 3 sin x + sin (2 .}•) cor- N ~ 'h ^ / ■\; ; ■\ \ / % 1/ v> \>, / \ \ '''^ \ / '• \ A I' \ \ v' ^^ I'/ \ X \^ /j \^ \ /j ' \ V ij \ i S /) \ y \ \, \ '/ V, ' v responding to any value of x is the sum of t])o ordinates of yi — 3 siiiiv and y., = sin (2.r) corresponding to the same value of X. When the sine-function occurs in the form y — a sin (wt + 6), where w is uniform angular velocity in radians, t time in sec- onds, a is called the amplitude, 6 the epoch angle. The periodic PLOTTING OF THANSCEN DENTAL EOUATIOXS hi time is t = - — Tlie construction of the curve is indicated in the iigure. The jtrojcction of unirorni motion in the circum- I'ci'ent'e of a circle on a diamctei' is caUcd harmoiuc motion. f=2 t = ] 1=0 !J A r\ / ^-N \ / \ / \ \ / \ \ / \ t \ / \ \ 1 \ \ 1 \ \ k/ 1 — — \ To add gi'apliically two sine-functions of equal periods ?/, = «i sin (co^ + ^,), ?/^ = ao (sin w^ + ^o), draw a pair of con- centric circles with rndii «, and a~y Let 1\0D and P.,OD be oj^ + $1 and (ot -f d., corresponding to the same value of (. Then 52 ANAL YTIC GEOMETU F I\()D - P.,OD = ^1 - e.,. The ijarallelogram on 01\ and 01\ for different values of t is the same j)arallelogram in different positions. This parallelogram has the same angular motion as OPi and 01\. Now y^ = PiAj 2/2 = P^D.,, hence p/;-PiA + P.A = 2/i + 2/.. and the sum of the sine-functions corresponding to the circular motions of Pi and P., is the sine-function corresponding to the circular motion of P. The resultant sine-function has the same period as the component sine-functions, its amplitude is OP, its epoch angle the angle POD corresponding to the position of P for t = 0. The resultant sine-function is y = a s\n{wt-\-6), where a^ = a^^ + a./ -\-2 ttia^ cos {61 — 62), a I sin $1 + €(,2 sin O2 * cij cos 61 + a.^ cos 62 (1) ij = sm~^x is equivalent to x tan 6 . is equivalent to x ■ sin ^; (o) y = sin o sin y ; (2) // = 3 sin~' x is equivalent to a; = 3 sin:;; (4) /y = 3 sii is equivalent to x — 2 sin graphic interpretation of ccpuitions (1), (2), (3), (4) is shown in figures (a), (b), (c), (d), which also indicate the nu^nncr of plotting the equations graphically. * A jointed paralk'lograin is used for conipoundiiig harmonic motions of different periods in Lord Kelvin's tidal clock. PLOTTING OF TRANSCENDENTAL EQUATIONS 53 The reiiuiining circular and inverse circular fuuc^tions are l)k)ttcd in a manner entirely analogous to that employed in l)lotting y = s'lnx and y = sin"' x. Problems. — riot 1. )j = 2^. 2. */ = lO-'. 3. y=(\y. 4. y=(.l)^ 5. i/ = 2-^. 6. .v = 5-2^. 7. y=3^\ 8. .v=:e^* Q. y = e-\ 10. ij = \(e^ + e-^). This function is called the hyperbolic cosine, and is written y = cosh a;. 11. y — - (e^ + e-^^) or y = c cosh x. This is the equation of the catenary.t the form assumed by a perfectly flexible, homogeneous chain whose ends are fastened cat two points not in the same vertical. 12. y — I (e-^ — e""^). This is the hyperbolic sine, and is written sinh r. 13. y = K 14. y = logiox. 19. ;K-2 = logi„(y + 5). 20. r + 5 = logio (^ - 2). 1. y = logo X. 17. 2/ = ;]log.,x. ;. , = iog^,^_x. 18. 2x = logio2/. 26. »/ = .3sinx. 27. )/ = sin (4x). 28. y = sm(lw + ^x). 29. rj — 3 + sin x. 30. 2/ = 3 + sin(7r + 2x). 31. J/ = sinx + 2 sin -■ 22. ^ ^ = logio(y+l). X + a 23 y = sin ^^' ■ ^ 2 32. 2/ = 2 sin (2 x) + 3 sin (3 x) . 24. ?/ = .sin(2x). 33. ?/ = 3sin(2 + 2x) + fisin(l + 4x). 25. ;/ = sin (x + .[ tt). 34. y = cosx. 35. »/ = 5 cosx. 39. ?/ = secx. 43. y = versx. 36. y -2 cos (1 + Gx). 40. y = 2 secx. 44. y = covers x. 37. ?/ = tanx. 41. ?/ = cosecx. . , ,, 45. X = sm-i ^• 38. y = 2 + tan(l + x). 42. y = cosec (x - 1). 2 * e represents the base of the Napierian sy.stem of logarithms, a tran.scen- dental number, whose value to nine places is 2.718281828. y — e" may be plotted graphically by computing the values of y corresponding to any two values of X ; numerically by writing the function in the form x = log,.»/ and using a table of Napierian logarithms. t The catenary was invented by Jolni and James nernouUi. The center of gravity of the catenary is lower than for any other position of the same chain with the same fixed points. 54 ANALYTIC GEOMETRY 46. 2 M = cos-i X. -r b2. x — 2 — shr^ y. 50. w = 3 cos-i -• 47. 2/ = J taii-i x. 3 53. x+3 = siu-i (i/-2). 48. 2/ = sec-i(x-3). ^^ y = 5sin-i?. ^*- ^ = cos-i (?/ - 1). 49. ?/ = 2 + vers-i X. 4 55. 2/+2=:cos-i(a;-2). 56. y = sill (] nt). b1. y = sin (] ivt + \ v). 58. y — sill (.] wt + 2 t) + .siii(^ tt^ + v). 59. ;/ = sin (I nt) + sin (} irt + ^ tt). 60. y = sin (|: 7r( + 1 tt) + sin {\ -ret + i tt). The elementary transcendental functions are of great impor- tance in mathematical physics. For instance, if a steady electric current /flows through a circuit, the strength i of the current t seconds after the removal of the electromotive force is given by the exponential function i= le ^, where R and L are constants of the circuit. The quantity of light that penetrates different thicknesses of glass is a logarithmic function of the thickness. The sine-function is the element by whose composition any single-valued periodic function may be formed. Vibratory motion and wave motion are periodic. The sine-function or, as it is also called, the simple harmonic function, thus becomes of fundamental importance in the mathematical treatment of heat, light, sound, and electricity. Art. 26. — Cycloids A circle rolls along a fixed straight line. The curve traced by a point fixed in the circumference of the circle is called a cycloid.* The fixed line is called the base, the point whosr distance from the base is the diameter of the generating circli^ the vertex, the perpendicular from the vertex to the base th<' * Curves generated by a point fixed in the plane of a curve which \\A[a along some fixed curve are called by the general name "roulettes.'' Cycloids are a special class of roulettes. a XI in P LOTTING OF TRANSCENDENTAL EQCATlONH .55 mS of tlicryi'loiil. The courtliiiates of any inuiitof the cycloid ay be expressed as trausceudental i'uiictions cd' a variable angle. Take the base ol' the cyck)id as X-axis, the perpendicular to the base where the cycloid meets the base as I'-axis, and call the angle made by the radius of the generating circle to the tracing point with the vertical diameter 6. l>y the nature of the cycloid AK= arc PK = rO, y = PD = OK - OL ^ r - r cos d, X = AD = AK — DK=^ i-e — r sin 9. Kence (1) x = rO - r sin 0, y = )• — r cos 9 for every value of 9 determine a point of the cycloid. The equation of the cycloid between x and y is obtained either directly from the figure, X = AK — DK — arc PK — PL = r vers"'* — Vli ni r or by eliminating 9 between ecpuitions (1), T V . V -5 ry - y-, hence Vi/ ry Now r vers~^-^ has for the same value of y an infinite number of values differing by 2tt, and V2ry — // is a two-valued func- tion which is real only for values of y from to -f 2 r. Hence the equation determines an infinite number of values of x for every value of y between and + 2 r. This agrees with the nature of the curve as determined by its generation. 56 A NA L YTIC GEOMETR Y By observing that tlie center of the generating circle is always in the line parallel to the base at a distance equal to the radius of the generating circle, the generating circle may readily be placed in position for locating any point of the cycloid. At the instant the point P is being located the generating circle is revolving about K, hence the generating point P tends at that instant to move in the circumference of a circle whose center is A' and radius the chord KP. The tan- gent to the cycloid at P is therefore the perpendicular to the chord KP at P, that is the tangent is the chord PII of the generating circle.* The perpendicular to the tangent to a curve at the point of tangency is called the normal to the curve at that point. Hence the chord KP is the normal to the cycloid at P. Take the axis of the cycloid as X-axis, the tangent at the vertex as F-axis. By the nature of the cycloid 3//ir=arcP/i, MN— semi-circumference KPH. X =AD = HL=OH-OL = r-r cos 0, 7j = PD = LD+PL=(MN-Miq-\-PL = arc IIP + PL= rO + r sin 0. That is, (1) X: y rd -\- r sin i determine for every value of ^ a point of the cycloid. The equation between x and y is found either directly from the figure, y =z LD + PL = r vers"' - + V2 rx — x- ; This method of drawing a tangent to the cycloid is due to Descartes. V LOTTING OF THAN SiCEN DENTAL EQUATIONS 57 or by elimiiiatini;- 6 between eiiuations (1), ^=:cos-/l --")=: vers-' ^, .6^^V2' hence y=r vers"' - + V2 rx r Art. 27. — Prolate and Curtate Cycloids When the genercating point instead of being on the eircuni- ference is a point fixed in the pkme of the rolling circle, the curve generated is called the prolate cycloid when the point is within the circumference, the curtate cycloid when the point is without the circumference. Let a be the distance from the center to the generating point. From the figures the equations of these curves are readily seen to be x = r6 — a sin 0, y = r — a cos 6. Fio. 40 * If the cycloid is concave up and the tanfjcnt at. the vertex horizontal, the time required by a particle sliding down the cycloid, suiJjiosed friction- less, to reach the vertex is independent of the starting point. On account of this property, discovered by Huygens in 1G73, the cycloid is called the tautochrone. The frictionless curve along which a body must slide to pass from one point to another in the shortest time is a cycloid. On account of this property, discovered by John Bernoulli in 1G9G, the cycloid is called the brachistochrone. 58 ANALYTIC GEOMETRY Art. Epicycloids and Hypocycloids If a circle rolls along the circumference of a fixed circle, the curve generated by a point fixed in the circumference of the rolling circle is called an epicycloid if the circle rolls along the outside, an hypocycloid if the circle rolls along the inside of the circumference of the fixed circle. By the nature of Oi ~U D X the epicycloid arc HO = arc HP, that is E-6^r- (f>. From the figure X = AD = AL + DL Fiu. 47. =(R + r)cos + r cos CPM. ... Ji + r. „ _ 180°-: Hence x = (R + r)cos PD = CL R + r CM = (R + r)sin^ — r sin By the nature of the hypocycloid R • 6 = r • <^, hence ^ = ^-6. x = AD r X = AL - PM = (R - r)cos e- 7- cos CPM. NowCPJ/=lSO°-</, + ^ R- x=(R — r)cos 6 + r cos 180° R- Hence PD = CL - CM = (R- r)s\n O-r sin Epicycloids and hypocycloids are used in constructing gear teeth. PLOTTING OF TRANSCENDENTAL EQUATIONS f)!) AiiT. 29. — Involutk of Cikcli': A string whose length is the circumference of a circle is wound about the circumference. One end is fastened at and the string unwound. If the string is kept stretched, its free end traces a curve which is called the involute of the circle. From the nature of the involute, IIP is tangent to the fixed circle and equals the arc HO, which equals Ji$. X = AD = AL + KP = 11 cos + BO sin 6, y=PD= IIL - IIK = n sin 6 - 116 cos 6* * The invohite is also used in cnnstructing gear teeth. CHAPTER V TEANSrOEMATION Or COOKDINATES Art. 30. — Transformation to Parallel Axes Let P be any point in the plane. Keferred to the axes X, Y the point P is represented by {x, y) ; referred to the parallel axes Xj, Yi, the point P is represented by (x^, y-^. Let the origin A^ be {m, n) when referred to the axes X, Y. — Xi Prom the figure x = m + x^, y = n -{- yi. Since (x, y) and — X (xi, ;Vi) represent the same point P, if f{x, ?/) = is the equation of a certain geometric figure when interpreted with refer- ence to the axes X, Y, f(m + Xi, n -|- ?/i) = is the equation of the same geometric figure when interpreted with reference to the axes Xj, Y^. Example. — The eqiiation of the circle whose radius is 5, center (2, 3) is (1) (a; - 2)2 + (y - 3)^ = 25. Draw a set of axes Xj, Yi parallel to X, Y through (2, 3). Then x = 2 + x„ y = ?y -\- ?/,. Substituting in equation (1), there results (2) x^- + y,^ = 25. Notice that the equation of a geo- metric figure depends on the position of the geometric figure with respect to the axes. GO TltANSFOIiMATION OF COOHDINATES Gl AUT. 31. — FUOM KeCTANGULAU AxKS to JiK( TAN(i[M.AIl Let (x, y) represent any point in the plane referred to the axes X, F; (x„ y^ the same point referred to the axes X,, \\, where Xj, Yi are obtained by turning A', Y about A througli the angle a. Now x=AD = An-KD' = x'l cos a — .'/i sin a, y=. P1) = D'H+PK — .Xisin a + yi cos a. Since (.x*, .7) and (.i-,, yO rep- resent the same point 7*, /(x, ?/) = interpreted on the A", Y axes and r><i. 52. /(a'l cos a — ?/i sin «, a;, sin « + ?/, cos «) = interpreted on the Xj, Fj axes represent the same geometric figure. Example. — y = a- + 4 is the equation of a straight line. To find a set of rectangular axes, the origin remaining the same, to which when this line is referred its equation takes the form yi=n, substitute in the given equation X = Xi cos a — yi sin a, y = Xi sin a + y^ cos a. There results Xy (sin a — cos«) + ?/i (sin a + cos «) = 4, and this equation takes the re- quired form when sin a — cos a = 0, that is, when a = 45°. Substituting this value of a, the trans- formed equation becomes ?/, = 2 V2. ANALYTIC GEOMETRY Art. 32. — Oblique Axes Hitherto the axes of reference have been perpendicular to each other. The position of a point in the plane can be equally well determined when the axes are oblique. The ordinate of the point F referred to the / / / oblique axes X, Yis the distance and direction of the point from the X-axis, the distance being measured on a parallel to the I''-axis, the side of the X-axis on which the point lies being indicated b}^ the algebraic sign prefixed " '"' ""' to the number express- ing this distance. Similarly the abscissa of the point P is the distance and direction of the point from the F-axis, the dis- tance being measured on a parallel to the X-axis, the alge- braic sign prefixed to this distance denoting on what side of the F-axis the point lies. Problems. — Tlie angle between the oblique axes being 45' : 1. Locate the points (3, - 2); (- 5, 4); (0, 8); (- 4, - 7); (2i, - 3); Observe that the geometric figure represented by an equa- tion depends on the system of coordinates used in plotting the equation. 4. Find the equation of a straight line referred to oblique axes including an angle /3. The method used to find the equation of a straiafht lino referred to ( v/5, 2. -V7); (-21, V Plot 2/ = 3 X ; ?/ = '10) 3x + 5; y = -2x 3. Plot x^ + y^^ IG ; !/■ = 4x . x^ ' 9 -! = TRAysFOnMAriON OF COOIiDrXATES 03 rcctancjular axes sliows tliat the equation of a strai,i,'lit line referred to oblique axes is y = vix + «, where in — of the line on the I'-axis. sni (/S — a) and n is the intercept 5. Show that V (x' - x")'^ + (y' - y")'^ + 2(3;' - x") {y' - y") cos /3 is the distance between the points (z', ?/'), (x", y") when the angle between the axes is /3. 6. Find the equation of the circle whose radius is Ji, center (m, 7i), when the angle between the axes is /3. 7. Show that double the area of the triangle whose vertices are (3^1, yO, (3-2, !/2), (a^s, 2/3) is {yi(^3 - X2) + yo(xi - T:i) + y3(x-2 - xi)}sin /3. AuT. 33. — From Rectangular Axes to Oi-.lique It is sometimes desirable to find tlie equation of a £;eometric figure referred to oblique axes when the equation of this figure referred to rectangular axes is known. This manner of ob- taining the equation of a figure referred to oblique axes is fre- quently a simpler problem than to obtain the equation directly. To accomplish the transformation, the rectangular coordinates of a point must be expressed in terms of the oblifjue coordinates of the Fio. 5g. same point. From the figure a: = AD = AII+ D'K= .r, cos a + ,v, cos a', y = PD= D'II+ PK= .r, sin a + //, sin u'. ExAMi'LE. — To find the equation of the h_viHTl)i)l;i rcfcvvcd to its asymptotes from the common C(piatit)n of the hyperbola, x^ _ jf _ 1 64 ANALYTIC GEOMETin b The asymptotes of the hyperbola, y of the rectangle on the axes. Hence b ± -X, are the diagonals a cos a — Va- + b- and the transformation for- mulas become " (.-^-i + z/O, Va^ + b' b Va^ + 6- Substituting in the common equation of the hyperbola and reducing, a\y^ = '-, 4 the equation of the hyperbola referred to its asymptotes. The formulas for passing from oblique axes to rectangular, 'y the origin remaining the same, are x = AD = AH-D'K _ y, sin (/3 — a) _ ?/, cos ((3 — a) sin/? sin/3 = PD = D'll + PK _ Xi sin a ?/i cos n sin /3 sin (3 Art. 34. — General Transformation The general formulas for transforming from one set of recti- linear axes to another set of rectilinear axes, the origin of the second set when referred to the first set being (m, n), are TUANSF(>i;MAril)i\ OF I'OOUDl X ATKS 05 x = AD=An + A/r-\-J)'h', _ .r I s i 11 (/3 - u) //i sill {13 -a') -'"+ sm(3 ^ sm(3 ' 2/ = PL* = .l,7t* + D'T + I'K, .r, sill u , Vi sill a' From tliesc L;eiieral formulas all tlie ijrecediiiy formulas may bo derived by substituting for ?/;, v, fS, a, a' tlieir values in each special case. However, if it is observed that in every case the figure used in deriving the transformation formulas is constructed by drawing the coordinates of any ])oint P referred to the original axes, and the coordinates of the same point referred to the new axes, then through the foot of the new ordinate parallels to the original axes, it is simpler to derive these formulas directly from the figure, whenever they are needed. Art. 35. — Thk ruouLEM of Tiia\sfoi;i\i.\tion An examination of the transformation formulas shows that the values of the rectilinear coordinates of any point in terms of any other rectilinear coordinates of the same ])oint are of the first degree. Hence transformation from one set of recti- 66 ANALYIUC GEOMETRY linear coordinates to auotlier rectilinear set does not change the degree of the equation of the geometric figure. Two classes of problems are solved by the transformation of coordinates : I. Having given the equation of a geometric figure referred to one set of axes, to find the equation of the same geometric figure referred to another set of axes. II. Having given the equation of a geometric figure referred to one set of axes, to find a second set of axes to which when the geometric figure is referred its equation takes a required form. Problems. —Transform to parallel axes, given the coordinates of the new origin referred to the original axes. 1. 2/ - 2 = 0(x + 5), origin (- 5, 2). 2. (x - 3) (?/ - 4) = 5, origin (3, 4). 3. y = 2x + 5, origin (0, 5). 4. a;2 + ?/2 4- 2 X + 4 ?/ = 4, origin ( - 1, - 2). 5. a;2 + 2/2 + fi »/ = 7, origin (0, - 3). 6. X- + 2/2 - G X = 16, origin (3, 0). 7. 2/^ + 4 2/ - G X = 4, origin (0, - 2). 8. 25(2/ + 4)2 + lC(x - 5)2 = 400, origin (5, - 4). 9. ,;2 + f. ^ 25, origin ( - 5, 0). ^^ ^J + ?^ = 1, origin (0, - h). 10. x2 + 2/2 = 25, origin (0, - 5). "' ''' 11. x2 + 2/2 = 25, origin (-5, -5). '^- | + g = 1, origin (- a, - 6). 12. ^^t^ 1, origin (- a, 0). 15. ^-^- = 1, origin (a, 0). a- 62 a^ h^ Transform from one rectangular set to a second rectangular set, the second set being obtained by turning the first about the origin tlirough 45'^. 16. x2 + 2/- = 4. 18. y + x = 5. 21. y'^-3ry + x«,= 0. 17. x2 - 2/2 = 4. 19. 2/2 ^ jy _ a:2 = G. 22. ?/" + 3 ry - .r' = 0. 20. if' + 4 xy + x2 = 8. Notice that to plot equations 21 and 22 directly requires tlie sohition of a cubic equation, whereas the transformed equations are plotted by the solution of a quadratic equation. TliANSFORMATION OF COORDINATES (37 In the following problems the first equation is the equation of a geo- metric figure referred to rectangular axes. The origin of a parallel set of axes is to be ftiuml to which when the geometric figure is ri'ferred its equation is tlie second eiiuation given. 23. .'/ + 2 = 4(x - :J) ; y = -1 •^■- 26. if - X- - lU x = ; x- - ij- = 25. 24. (.<• + 1) (// + 5) = 4 ; xij ^ 1. 27. U' - 1U(..: -|- 5) = ; >f = 10 x. 25. f + x^ + lOx=0; x:- + y'=25. 28. if + x;^ + -i >j--2 x=l\ ; x- + f = YG. In the following problems the first equation is tlu; equation of a geo- metric figure referred to rectangular axes; find the inclination of a secoiul set of rectangular axes to the given, origin remaining the same, to which when the geometric figaire is referred its equation is the second eiiuation given. 29. y = x + 4; y = 2V2. ^^ „o ^ 2n ■ ,-2 4- y2 = 2:,. 32. ^ - 2g = 1 ; :r// 30. y- 33. //■' - 3 axy + .';' = ; y^ - y 3V2aa;' d' 4- }i~ 2x3 2x -|-;j\/2a Construct the locus of tlie first etiuation in the following problems by drawing tlie axes A'l, I'l and plotting the second eipiation. 34. 11-2 = log(x -f- ;:i) ; ?/i = log Xi. 36. ?/ -f 3 == 2^ t •» ; y.^ = 2a. 35. 2/ = 3 sin(x + 5); 2/1 = 3 sin Xi. 37. y + 5 = tau(x - 3); yi = tan Xi. 1, obtain 3.2 ,,2 From the common eciuation of the hyperbola,-— — -- the equation of the hyperbola referred to oblique axes through the center, 1,2 such that tan a tan a' = — a- The transformation formulas are x = Xi cosa -f- 2/1 cosa', y = Xisina -1- yi sin a' ; the transformed equation /cos^a sin-a\ , [—2 ^p' + 2f "°^°' sm g sm g' The condition tan a tan a the equation of the hyperbola referred to the oblique axes becomes Fic. f>i». renders the coefficient of Xij/i zero, and 68 ANALYTIC GEOMETRY .siii-c l.'/r = 1- Since only values uf a and a' less than 180° need be considered, the con- dition tan a tan a' = - shows that a and a' are either both less than 90° or «" h b both greater than 90", and that if tana <-, tana' >-. Since the equa- tions of the asymptotes of the hyperbola are y = ±~x, it follows that if the Xi-axis intersects the hyperbola, the Ti-axis cannot intersect it. Calling the intercepts of the hyperbola on the Xi- and I'l-axis respectively ai and biV— 1, the equation referred to the oblique axes becomes 39. From the common equation of the ellipse, ~ -|- - = 1, obtain the 62 62 P P (m, n) is y=~(x + 7n), tano'=— . n n equation of the ellipse referred to oblique axes such that tan a tan a' 40. From the common equation of the parabola y^ = 2px obtain the equation of the parabola referred to oblique axes, origin (m, n) on the parabola, the AVaxis parallel to the axis of the parabola, the JVaxis tangent to the parabola. n- = 2 pm, a = 0, and, since the equation of the tangent to y"^ = 2px at The transformation formulas be- come X = m + Xi + ?/i cos a', ?/ = n + ?/i sin a', and the transformed equa- rfl -|- /i'.2 tion reduces to yi^ = 2 .ti, or yi~ ■= 2(p + 2 m)xi. 41. To determine a set of oblique axes, with the origin at the center, to which when the ellipse is referred, its equation has the same form as the common equation of the ellipse -^ + ra = 1- The substitution of X = a:i cosa + ?/iC0Sa', 2/ = Xi sin a -1- yi sin a' transforms the equation -2+72 = 1 into /Cos2a V rt2 + ft2 TRANSFOh'MATIO.Y OF COORDINATES G9 'I'lio problem requires thai the coefficient of Xij/i bo zero, hence tan a tan a.' = '- d- Tlie problem is indeterminate, since the etiualion between a and a' admits an infinite number of solutions. Let a and a' in the figure represent one solution, then (^ + ^^^^^-3 + ^^' + ^li^^y,. ^ 1 is the equa- tion of the ellipse referred to the axes A'l, I'l. Call the intercepts of the ellipse on the axes Xi and I'l respectively ai and hi , and the equation becomes ^ + f^ = 1. When the equation of the ellipse referred to a pair of lines through the center contains only the squares of the unknown quantities, these lines are called conjugate diameters of the ellipse. The condition of conjugate diameters of the ellipse is tan a tan a' = — — . a' 42. Determine a set of oblique axes, with the origin at the center, to which, when the hyperbola is referred, its equation takes the same form as the common equation of the hyperbola. The result, tanatana' = — , shows that the problem is indeterminate. jfl «'- tana tan a' = — is the condition of conjugate diameters of the hyperbola. d^ 43. Determine a set of oblique axes, origin at center, to which, when the hyperbola is referred, its equation takes the form xij — c. 44. Determine origin and direction of a set of oblique axes to which, when the parabola is referred, its equation has the same form as the common equation of the parabola. 45. Show that the equation of the parabola y'^ = 2pz when referred to its focal tangents becomes x^ + y^ — a^, where a is the distance from the new origin to the points of tangency. CHAPTER VI POLAR OOOEDINATES Art. 36. — Tolau Cooudinates of a Point In the plane, suppose the point A and the straight line AX through A fixed. A is called the pole, AX the polar axis, p ,/ The angle which a line AP makes ^'"''^^ with AX is denoted by 6. is positive when the angle is con- ceived to be generated by a line X starting from coincidence with AX turning about A anti-clockwise ; 9 is negative when generated by a line turning about A clockwise. When 6 is given, a line through A is determined. On this line a point is determined by giving the di-stance and direction of the point from A. . The direction from A is indicated by calling distances measured from A in the direction AP of the side of the angle positive, those measured in the opposite direction negative. The point P is denoted by the symbol (r, ^), the point P' by the symbol (-r, e). The symbols (r, ^ + 27r?i), (-'', ^ +(2?i + l)7r), where n is any integer, denote the same point. To every sym- bol (r, &) there corresponds one point of the plane ; to every point of the plane there corresponds an infinite number of symbols (r, d). Under the restriction that r and 6 are positive, and that the values of B can differ only by less than 2 tt, there exists a one-to-one correspondence between the symbol (/•, Q) 70 POLAR coon DIN ATES 71 aud the points of the plane, the pole only excepted, r and are called the polar coordinates of the point. 1. Locate the points whose polar coordinates are (2, 0) ; (-3,0); (3,l,r); (-2,7r); (4,f,r); (-4,l7r); (4,-|7r); (1,1); (-2,1); (-1,0); (1,180°); (-4,45°); (4,225°); (-4,405°); (0, 0) ; (0, 45°) ; (0, 225°). 2. Show that r'- + r"- - 2 r'r" cos (0' - 0") is the distance be- tween (?•', 6'), (r", 0"). 3. Find the distances between the following pairs of points, (4,l7r), (3,7r); (8,J-7r), (6,f,r); (2V2, -^7^), (l,i7r);(0,0), (10, 45°); (5, 45°), (10, 90°) ; (-0, 120°), (- 8, 30°). AuT. 37. — PoLAK Equations ok CiEoiiETiiio Fiuujies The conditions to be satisfied by a moving point can some- times be more readily expressed in polar coordinates than in rectilinear coordinates. If a point moves in the XF-plane in such a manner that its distance from the origin varies directly as the angle included by the X-axis and the line from the origin to the moving point, the rectangular equation of the locus is Va.'- -f y- — a tan~^-,, the polar equation r = a9. Desired information about a curve is often obtained more directly from the polar equation than from the rectilinear equation of the curve. This is especially the case when the distances from a fixed point to various points of the curve are required. Thus if the orbit of a comet is a parabola with the sun at the focus, the comet's distance from the sun at any time is obtained directly from the polar equation of the parabola. AuT. 38. — Polar Equation of Stuakjut Link A straight line is determined when tlie lengtli of tlic perpen- dicular from the pole to the line and the angle included by this perpendicular and the polar axis are given. Call the per- ANAL YTIC GEOMETR Y peiuliciilar p, the angle a, and let (r, 6) be any point of the line. The equation Fig. C3. \ ; for 6 = a. cos {p — a) expresses a relation satisfied by the coordinates r, of every point of the straight line and by the coordinates of no other point; that is, this is the equation of the straight line. For 6 — 0, from ^ = to ^ = 90° + «, r is cos « ' positive ; from = 00° + a to = 270° + a, r is negative ; from 6 = 270° + a to 6 = 3G0°, r is again positive. For 6 = 90° + a and for 6 = 270° + «, r = ± oo . These results ob- tained from the equation agree with facts observed from the figure. A straight line is also determined by its intercept on the polar axis and the angle the line makes witii the polar axis. Call the intercept b, the v'^-' angle a, and let (r, 6) be any point in the line. Then 6 sin a . ^, r =—. — 7 ;rr IS thc cqua- sin(« — ^) tion of the line. For ^ = 0, r = b; ?• is positive from ^ = to 6 = a; negative from ^ = a to 6 = 180° + a; ^"'- ^- again positive from 6 = 180° + atoe = 360°. ¥ov 0=a and 6 = 180° + «, r = ± oo. These results may be obtained from the equation or from the figure. Akt. 39. — Polar Equation of Circle The equation of a circle whose radius is E when the pole is at the center, the polar axis a diameter, is r = E. POLAR COOUDl NATES 73 AVheu tlic pole is on tlio circuinfeieuce, tlio polar axis a diameter, the equation of the circle is r = 2 li cos 0. r is positive from ^ = 0° to ^ = 90°, negative from $ = 90° to 6 = 270°, and again positive from 6 = 270° to ^ = 300°. The entire circumference is traced from ^ = 0° to 6 = 180°, and traced a second time from ^ = 180° to ^ = ,'>60°. The polar equation of a circle radius R, center (>•', 6'), cur- rent coordinates r, 6, is »-^ — 2 r'r cos (6 — 6')= lir — r'-, whence r = r' cos (6 - $') ± -y/R- - r'- sm\e - J'), r is real and has two unequal values when sin-(^ — ^')< -y,; that is, wIkmi R ■ R ^ <sin(^ — ^')< — ; these values of r become equal, and r r jy the radius vector tangent to the circle, when sin(^ — ^')= ± — ; R- ^ r is imaginary when sin^(^ — 6')> —- ^1 Art. 40. — Polar Equations of the Conic Sectioxs Take the focus as pole, the axis of the conic section as polar axis. From the definition of a conic section r = e- DE = e (DA + AE) =e['- + r cos d Hence r—j^ + ercofiO, r 1 — e cos 6 74 ANALYTIC GEOMETRY Since in the parabola e = 1, the polar equation of the parab- Ola IS r = :; 1,- In the ellipse and hyper- bola the numerical value of the semi-parameter p is a(l-e^); hence the polar equation of ellipse and hyperbola is 1 — e cos In the ellipse e is less than unity, and r is therefore Fig. 07. . -^ always positive. For ^ = 0, r — a(l + e), showing that the pole is at the left-hand focus. In the hyperbola e is greater than unity, and r is positive from 6 = to 6 = cos~^ - in the first quadrant, negative from 1 ^ 1 ,-1^ ;r, 4-1.0 fi,.of r.,no,iT.o,.f fr> fl — f-og-'- in the fourth cos~' - in the first quadrant to e quadrant, again positive from rant to ^ = 3G0' ,^il cos^^ - in the fourth quad- e 1 r becomes infinite for 6 = cos ^ - ; hence lines ' 1 through the focus making angles whose cosine is - with the axis of the hyperbola are parallel to the asymptotes of the hyperbola. Problems. — 1. The length of the perpendicular from the pole to a straight line is 5 ; this perpendicular makes with the polar axis an angle of 45°. Find the equation of the line and discuss it. 2. Derive and discuss the polar equation of the straight line parallel to the polar axis and 8 above it. 3. Derive and discuss the equation of the straight line at right angles to the polar axis, and intersecting the polar axis 4 to the right of the pole. 4. Derive and discuss the equation of the circle, radius 5, center (10, iTT). POLAR COORDINATES 75 5. Derive and discuss Uie equation of tlie circle, radius 10, center (5, Iw). 6. Derive and discuss tlic equation of the circle, radius 8, center (10, ]7r). 7. Derive and discuss the equation of the circle (15, tt). 8. Derive and discuss the equation of the circle. (10, ^tt). 9. Find the polar equation of the parabola whose parameter is 12. Find the polar equation of the ellipse whose axes are 8 and 6. Find the polar equation of the ellipse, parameter 10, eccentricity \ Find the polar equation of the ellipse, transverse axis 10, eccen radius 10, center radius 10, center 10. 11. 12. tricity 13. 14. 15. 4 c2,-2 co^l , Find the polar equation of the hyperbola whose axes arc 8 and G. Find polar equation of hyperbola, transverse axis 12, parameter 0. Find polar equation of hyperbola, transverse axis 8, distance be- tween foci 10. 16. Find the eijuation of the locus of a point moving in such a manner that the product of the distances of the point from two fixed points is always the scjuare of the half distance between the fixed points. This curve is called the lemniscate of Bernoulli. By definition ViV^ = c^. From the figure rx^ = r- + c^-2cr cos 6, r^ = r2 -f- c2 + 2 cj- cos 9, hence rrro- — r* + 2 c^r'^ + <: and r2 = 2 ^2 (2 cos2 ^ - 1), r^ = 2 c^ cos (2 0). Corresponding pairs of values of Vi and r2 may be found by drawing a circle with radius r, to this circle a tangent whose length is c. The dis- tances from the end of the tangent to the points of intersections of the straight lines through the end of the tangent with the circumference are corresponding values of r^ and j-o, for 7\S ■ Tli = c'. The inter.sections of arcs struck off from Fi and Fo as centers with radii TS and Tli deter- mine points of the lemniscate. 76 ANALYTIC GEOMETRY 17. A bar turns around and slides on a fixed pin in such a manner tliat a constant lengtli projects beyond a fixed straiglit line. Find the equation of the curve traced by the end of the bar. This curve is called the conchoid of Nicomedes. ^^-^ Y -^^ ^ ' — a, / — m b A / A X Take the fixed point A as focus, the line ^X, parallel to the fixed line mn, as polar axis. Call the distance from the i^ole to the fixed line h, the constant length projecting beyond the fixed line a. Then The conchoid is used to trisect an angle graphically. Let GAH be the angle. From any point B in one side of the angle draw a perpen- dicular mn to the other. With vertex of angle as fixed point, mn as fixed line, and FG = 2 BA as constant distance, construct a conchoid. At B erect perpen- dicular BC to mn, and join its A point of intersection with con- "'■ ' ■ choid C and J. by a straight line. GAC is \ CtAII, for, drawing through D, the middle point of BC, a parallel to mn and joining B and E, the triangles ABE and BEC are isosceles. Kence BAG = BE A = 2BCA = 2 GAC. ^ C G -^ ^^^ ^^ ^ \ D \ Ae "^ m \ n E F Art. 41. — Plottinct of Polar Equations Example. — Plot r = 10 cos 0. e = () ^TT ItT fir TT fTT §77 | TT 2,r r = l() nV2 -5V2 -10 -5VL> - SV^ 10 POL A R COORBINA TES 77 If tlie iiuinl.or oi' points located I'roiu ^ = to ^ = 2 TT is indefinitely increased, the polygon formed by joining tlu? successive points ap- proaches the circumfer- ence of a circle as its limit. Tlie form of the equation shows at once that the locus is a circle whose radius is 5. Example. — Plot r = aO. e = -4: -3 -2 -1 r = — 4a — 3 a — 2 a — a 12 3 4 a 2 a 3 a 4 a The curve is called the spiral of Archimedes. In rectangular coordi- nates the equation of this spiral is transcen- dental. Example. — Plot r = $ = cos-> f r = — 2 T 00 3 — T) cos 6 ±cx> 78 ANALYTIC GEOMETIIY From 6 = to 6 = cos~^ f , r varies continuovisly from — 2 to — cc ; from ^ = cos~^f to = TT, r decreases contimi- ously from + co to -{- -h ; from 6 = Tr to 6 = cos~^ | in the fonrtli qxiadrant, r in- creases continuously from ^j to + CO ; from 6 = cos~^f in the fourth quadrant to 6 = 2 TT, r increases from — CO to — 2. r is discon- tinuous for ^=cos"'|. This equation represents an h}'- perbola whose less focal distance is i, greater focal distance 2, semi-parameter A, eccen- tricity f Example. — Plot /-^ = 8 cos (2 9). = 0° 22.^° 45° 135° 157i° 180° r = ± 2.828 ± 2.378 imaginary ± 2.37 ± 2.828 From 180° to 300° the curve is traced a second time. The pole is a cen- ter of symmetry of the curve. ' Problems. — Plot COS cos (6 — J tt) 3. r = ncnsi{3 0). 4. )• = 2 cos 0. 5. r= asm (2 0). 6. r = acos(o0). 7. r = asin(Se). 8. »' = rtsin(4^). 9. r = a sin (5 6). 10. r = ^, the reciprocal spiral. 11. r — a", the logaritlimic spiral. POLAR COOIiT)TNATES 79 12. , the lituus 13. 2 ' 1 - cos 11 5 2 - 3 cos » 15 4 ' 3 - 2 cos » IB 10 1 + cos e 17. ?• = « (1 + cos e), the cardioid. 18. J- = 4(1 -COS0). 19. r = 5 + 2 sin 0. 20. r — 2p cot ^ cosec ff. 21. ^. _ 4 COS ^ 1 + 3 sin2 e 22. 4 COS e 1-5 sin- 23. 3 sin e cos sin^ d + cos-'' ^ 24. r = rt(sin2e + cos2<?). 25. r- cos (2 e) - 4. 26. )-2sin(2e)=8. 27. r- cos 1 e = 2. 28. )•- = 10 sin (2^). Art. 42. — Transformation from Rectangular to Polar Coordinates If the rectangular equation of a geometric figure is given, and the polar equation is desired, find the values of the rectan- gular coordinates x and y of any point in terms of the polar coordinates r and of the same point; substitute in the rec- tangular equation f(x,y)=0, and the resulting equation F(r, 6)=0 is the polar equation of the figure. Let the pole A' referred to the rectangular coordinates be (m, n), 6' the angle made by the polar axis with the X-axis. Then X = AD = m + r cos (6 + 6'), y = PD = n + r sin (d + 6'). When the pole is at the origin, and the polar axis coincides with the X-axis, these formulas be- come X = r cos 6, y — r sin $. If the polar equation of a geometric figure is given and the rectangular equation is desired, find the values of the polar coordinates r and 6 of any point in terms of the rectan- 80 ANALYTIC GEOMETRY gular coordinates x and y of the same point ; substitute in the polar equsition F(r, 6) = 0, and the resulting equation /(a;, ?/)=:0 is the rectangular equation of the curve. From the figure X — m r = y/{x - my +{y- w)^ cos (6 + 6') = sin(^ + ^')- •'~'" ^{x - mf + (2/ - ?0' V(.x'-m/+(y-n)^ When the origin is at the pole, and the X-axis coincides with the polar axis, these formulas become r — V.V- + ]j\ cos Q = ^ sin 9 = • VaT- + y- Var + y- Problems. — Transform from rectangular coordinates to polar, pole at origin, polar axis coinciding with A'-axis, and plot the locus from both equations. 1. x2 + ?/^ = 25. _ 6. 2/2 = i(4.T-a;2). 2. X- + 2/2 - 10 X = 0. 7. ?/ = - 1 (4 X - x^). 3. y^ = 2j-)X. 8. if -Sx>j + x^ = 0. 4. a;2 - 2/2 = 25. 9. (x^ + 2/-)'^ = «" i^' - f')- 5. x2/ = 9. Transform from polar coordinates to rectangular coordinates, X-axis coinciding with polar axis. 10. r = «, origin at pole. ^^ ,. ^ 9 ^ p^j^ ^^ ^^^ ^^ 11. r = 10 cos e, origin at pole. 4 - 5 cos ^ 12. 9-2 = «2 cos (2 61) , origin at pole. le. r = , pole at (4, 0). 5 — 4cos& 13. )•- cos .', — 2, origin at pole. , cos f 2 0^ . • . , - ' ° ^ 17. r- = '- ' , origm at pole. „ cos* e 14. ,. = i^ , poleat(ip, 0). , .. n •• . , 1 - cos 18. r2cos'*^ = 1, origni at pole. CHAPTER VII PROPERTIES OF THE STRAIGHT LINE AuT. 43, — Equations of the Straight Line The various conditions determining a straight line give rise to different forms of the equation of a straight line. I. The equation of the straight line determined by tlie two l.oints (x', ?/'), (x", y"). The similarity of the triangles rP'D and P'P"D' is the geometric condition which locates the point P{x, y) on the straight line through P'ix', y') and P"(.^•", ?/")• Tl>is condition leads to the equation y-y ^ '^, — ^ x' — x" tancrular coordinates ^(x — o:'). In rec- Fio. 77. tan a, where « IS the an,<rl( the line y -?/ _ x' — x" makes with the X-axis. In oblique coordinate! sin a -, where /3 is the angle between the axes, a x' — x" sin (/3 — a) the angle the line makes with the X-axis. II. The equation of a straight line through a given point (x', y') and making a given angle « with the X-axis is y — y' = tan «(a; — x'). If the point {x', y') is the intersection (0, n) of the line with the X-axis and tan « = m, the equation becomes y = 7nx + n, the slope equation of a straight line. On the straight line y — y' — tan a{x — x') the coordinates of the point whose distance from (.«', ?/') is (7, are x = x' -f d cos a, y = y' -\- d sin a. a 81 82 ANALYTIC GEOMETRY III. The equation of the straight line whose intercepts on the axes are a and 6. Let {x, y) be any point in the line. From the figure a—x _y a b 1, the which reduces to - + - a h intercept equation of a straight Fig. 78. Iji^g. IV. When the length p and the inclination « to the X-axis of the perpendicular from the origin to the straight line are given. Let {x, y) be any point in the straight line. From the figure, AB+BC=p, hence X cos (i + y sin « = p. This is the normal equa- tion of a straight line. The different forms of the equation of a straight line can be obtained from the general first degree equation in two variables Ax -\- By -\- C = 0, which always represents a straight line. (a) Suppose the two points (x', y'), (x", y") -to lie in the line represented by the equation ^x + i>?/ -|-C= 0. The elimina- tion of A, B, C from (1) Ax+By+C=0, (2) Ax'+By'+C=0, (3) Ax" -f By" + C = by subtracting (2) from (1) and (3) from (1), and dividing the resulting equations gives y-y = y' — y' x' - x' (x-x'). (b) Callin; X-axis ((, on : the intercept of the line Ax -\- By +C —0 on the C the F-axis b, for y = 0, x = - = a, for x — 0, y — — b. Substituting in the equation Ax -f By + C = 0, there results --f-?^=l. a h I'liOriCliTIES OF THE STliAKniT LINE 83 (f) Tlie equation .l.i- + B;/ +C= U may be written ^ b'^ b' wliicli is of the form y = vix + n. (d) Let Ax + Bij + C = and x cos a + y sin a =]> repre- sent the same line. There must exist a constant factor m such that VI Ax + vi By + i>iC = and x cos « + y sin « — p = are identical. I'rom tliis identity mA = cos a, mB — sin a, -,itC = —2>- The iirst two equations give nrA^ + vt^B'- — 1, hence iii — — - That is, Vvl- + B' ^A' + B' VA' + B' y/A' + B' is tlie normal form of the e(]uation of the straight line repre- sented by Ax + By + C = 0. The nature of the problem generally indicates what form of the equation of the straight line it is expedient to use. Problems. — 1. AVrite the tuiuation of the straight line through the points (2,3), (-1, 4). 2. Write tlie eiiuation of the straight line througli (- 2, 3), (0, 4). 3. Write the intercept eijuation of tlie straight line through (4, 0), (0, 3). 4. Write the equation of the straight line whose perpendicular dis- tance from the origin is 5, this i)erpentlicular malting an angle of 30° with the A'-axis. 5. Write the e(iuation ^ + | = 1 in the slope form. 6. Write the equation 2 x - 3 ?/ = 5 in the normal form. 7. Write the equation of the straight line through (4, -3), making an angle of 135° with the A'-axis. 8. On the straight line through (- 2, 3), making an angle of 30° with the A'-axi-s, find the coordinates of the point whose distance from ( - 2, 3) is 0. 9. The vertices of a triangle are (3, 7), (5, - 1), (-3, 5). Write equations of meilians. 84 ANALYTIC GEOMETRY Art. 44. — Angle between Two Lines Let V be the angle between the straight lines y — mx + n, y = m'x + n'. From the figure V= u — a', hence . Tr_ tan a — tan a' 1 + tan a tan a' Since tan « = m, tan «' = 7/1' — tanF= When the 1 + ?«?u' lines are parallel, V=0, which requires that m — m'. When the lines are perpendicular, F=00°, Avhich requires that 1 -\-mm' = 0. or m' = m If the e<|uations of the lines are written in the form Ax + By+C=0, A'x + ]^y+a = 0, tanF^^^-||. The lines are parallel when A'B — AB' = 0, perpendicular when AA' + BB' = 0. The equation of the straight line through (x', y') perpen- dicular to y = mx + H is y — y' = (x — x'). m The equation of the straight line through (x', y') parallel to y = mx + n is y — v' — ta (x — x'). Let the straight line y — y' = tan a'(x — x') through the point (x', ?/') make an angle 6 with the line y — mx -f n. From the figure, «' = ^ + «. Hence , ,_ tan^+tan« _ ta,n6 + m 1 — tan ^ tan a 1—m tan 6' since tan a = m. Therefore the equation of the line through (x', y') Fic. 81. making an angle 6 with the line y — mx-\-n is tan 6 + m 1 — 7)1 tan ( y-y .(x-x'). PEOPEPiTIES OF THE STliAiailT LINE Problems. — 1. Find the angle the line 85 •^ = 1 makes with the 3 A'-axis. 2. Find the angle between the lines 2x + 3y = 1, lx+ lij = 1. 3. Find the e(iuation of the line through (4, -2) parallel to 5x-7i/ = 10. 4. Find the equation of the line through (1, 3) parallel to the line through (2, 1), (-3, 2). 5. Find the equation of the line through the origin perpendicular to 3x-?/ = 5. 6. Find the equation of the line through (2, - 3) perpendicular to |a;-.\y = l. 7. Find the ecpation of the line through (0, - 5) perpendicular to the line through (4, 5), (2, 0). 8. The vertices of a triangle are (4, 0), (5, 7), (-0, 3). Find the equations of the perpendiculars from the vertices to the opposite sides. 9. The vertices of a triangle are (3, 5), (7, 2), (- 5, - 4). Find the equations of the perpendiculars to the sides at their middle points. 10. Write equation of line through (2, 5), making angle of 45' with 2x-3i/ = G. Akt. 45. — Distance from a Point to a Line Write the equation of the given line in the normal form cccos« + ?/sin«-i:» = 0. Through the given point P{x', y') draw a line parallel to the given line. The normal equation of this xiarallel line is X cos a + y sill (t = AP'. Since (x', y') is in this line, ■ x' cos a 4- y' sin « = AP'. ^^^ ^, Subtracting p = AD', there results x'cosa + y' sin a- p=PD; that is, the perpendicular distance from the point {x',y') to the line x cos a + y sin a -p=0 86 ANALYTIC GEOMETRY is x' cos « + y' sin « — p. The manner of obtaining this result shows that the perpendicular FD is positive when the point P and the origin of coordinates lie on different sides of the given line ; negative when the point P and the origin lie on the same side of the given line. The perpendicular distance from {x\ y') to Ax + J5^ + C'= is found by writing this equation in the normal form V^' + B" ^'A' + B" V^- + B- j{x' 4- Bii' + C and api)lving the former result to be PD = ' „ ' 3= — i L ^ ^ V.l- + B' This formula determines the length of the perpendicular; the algebraic sign to be prefixed, which indicates the relative posi- tions of origin, point, and line, must be determined as before. Problems. — 1. Find distance from (-2,3) to 3 x + 5 y = 15. 2. Find distance from origin to | x — | ?/ = 7. 3. Find distance from (4, - 5) to line through (2, 1), (-3, 5). 4. Find distance from (3, 7) to ^^_^ = IjL=li^. 5. The vertices of a triangle ar6 (3,2), (-4,2), (5, -7). Find lengths of perpendiculars from vertices to opposite sides. 6. The sides of a triangle are ?/ = 2 x + 5, 3 - ^ = !> 4 x - 7 y = 12. Find lengths of perpendiculars from vertices to opposite sides. 7. The sides of a triangle are 2/ = 2x + 3, 2/ = -|x + 2, y -x-b. Find area of triangle. Art. 46. — Equations of Bisectors of Angles Let the sides of the angles be Ax+By+C=0, A'x+Bhj^-G'=0. The bisector ah is the locus of all points equidistant from the given lines such that the points and the origin lie either on the I' HOP Eli TIES OF THE STRAIGHT LL\E 87 same side of each of the two ; of each of the two given Hues. In either case the perpendicu- lars from any point {x, y) uf the bisector to the given lines have the same sign, and the equation of the bisector is Ax -f Bif + C _ A'x + B'n + C" riven lines or on diiferent sides V^- + B' VA" + B" The bisector cd is the locus of all points equidistant from ^'"- ^^• the given lines and situated on the same side of one of the given lines with the origin, while the other line lies between the points of the bisector and the origin. The perpendiculars from any point (x, ?/) of the bisector cd to the given lines are therefore numerically equal but with opposite signs, and the n ,, T ^ ,. Ax + By + C A'x + B'jf + C equation of the bisector m is .' _ VA' + B' -VA" + B' Problems. — 1. Find the bisectors of the angles whose sides are 3 .X + 4 (/ = 5, Hx - 1 >j = 2. 2. Find the bisectors of the angles whose sides are ^x — ly = 1, 2/ = 2x-3. 3. Find locus of all points cciuidistant from the lines 2x + 7 y = 10, 8 a; — 5y = 15. 4. The sides of a triangle are 5a: + 3 ?/ = 9, l x + I y = I, y = d x - 10. Find the bisectors of the angles. 5. The sides of a triangle are 7x + 5i/ = 14, lOx — 15y = 21, y = 3x + 7. Find the center of the inscribed circle. AuT. 47. — Lines thuough Ixtekskctiox of Givk.v Lines Let (1) Ax + B>/+C.= and (2) A'x + B'>/ + C = Ije the given lines. Then (3) Ax + By + C -\-k (A'x + B'y + C) = 0, where k is an arbitrary constant, represents a straight line 88 ANALYTIC GEOMETRY tlirougli the point of intersection of (1) and (2). For equation (3) is of tlie first degree, hence it represents a straight line. Equation (3) is satistied when (1) and (2) are satisfied simul- taneously, hence the line represented by (3) contains the point of intersection of the lines represented by equations (1) and (2). If the line Ax + B>j+C + Jc (A'x + B'y + C") = is to contain the point (x\ y'), k becomes - ^,^, _^ J^y _^ ^r Hence is the equation of the line through (x', y'), and the intersection of (1) and (2). If the equations of the given lines are written in the normal form, (1) X cos a + ?/ sin « — p = 0, (2) x cos «'+ ?/ sin «' — i''= '*? the A; of the line through their point of intersection (3) X cos u + y sin a — 2> + k i^' cos a' + y sin a' — 2>') = -,. , ^ ■ ■ i. ^ i- 7 a;cos«+?/sin«— /> has a direct geometric interpretation. A; = — ■■ — -. ; :, a;cos« +?/sin« — ^> that is, A; is the negative ratio of the distances from any point (x, y) of the line (3) to the lines (1) and (2). Problems. — 1. Find the equation of the line through the origin and the point of intersection of 3x - 4?/ = 5 and 2 x + 5?/ = 8. 2. Find the equation of the locus of the points whose distances from the lines i; x - 5 y + 2 = 0, - - ?^ = 1 are in the ratio of 2 to 3. 3 6 3. Find the equation of the line through (- 2, 3) and the intersection of the lines 8 x - 5 ?/ = 15, 3 x + 10 ?/ = 8. AiiT. 48. — TuuEE Points in a Straight Line Let the three points (x', ?/'), (a-*", y"), {x"\ V'") lie in a straight line. The equation of the straight line through the first two points is y -y' = -[, ~-'„ {x-x'). By hypothesis the point x — X PnOPEliTlES OF THE STRAIGUT LINE 8'J ?/' — ?/" (x'", y'") lies in tliis liue, hence y"' —y' = \, —^,(^"' ~ ^')- Simplifying-, (1) x'y'" - x"y"' + x"y' - x"'y' + x'y" - x"'y" = 0. When this equation is satisfied the three points lie in a straight line, whether the coordinates are rectangular or oblique. Notice that (1) expresses the condition that the area of the triangle whose vertices are (x', y'), (x", y"), (x'", y'") is zero. Problenls. — 1. In a parallelogram each of the two sides through a vertex is prolonged a distance equal to the length of the other side. Prove that the opposite vertex of the parallelogram and the ends of the produced sides lie in a straight line. 2. In a jointed parallelogram on two sides through a common vertex two points are taken in a straight line with the opposite vertex. Show that these three points are in a straight line however the parallelogram is distorted. Art. 49. — Tiiiiek Links through a Point Let the three lines Ax + % + C'= 0, A'x + B'y + C" = 0, A"x-j- B"y -\- C" = pass through a common iwint. IMako the first two of these equations simultaneous, solve for x and y, and substitute the values found in the third equation. There results AB'C" + A'B"C+A"BC' - A"B'C-A'BC" - AB"(J' = 0, wliieh is the condition necessary for the intersection of the given lines. The three lines necessarily have aconiiudu ])oiut if constants K,, K.,, K;, can be found such that ki(Ax -\- By + C)+ k2{A'x + J^'ll + C")+ K^{A"x + B"y + 6'")= is identically satisfied. For the values of x and y which satisfy Ax -\- By -\- C =0, and A'x -}- B'y -f- C" = simultaneously must then also satisfy A"x + B"y + C" = ; that is, the point of intersection of the first two lines lies in the third line. The second criterion is frequently more convenient of appli- cation than the first. 90 ANALYTIC GEOMETRY Problems. — 1. The bisectors of the angles of a triangle pass through a common point. Let the normal equations of the three sides of the triangle be a;cosa + 2/sina-i)i=0, xcosj3 + 2/sin/3-i)2=0, x cos 7 +2/ sin 7-^93=0. Denote the left-hand members of these \ equations by a, /3, 7. Then = 0, |3 = 0, 7 = represent the sides of the triangle, and a, /3, 7 evaluated for the coordinates of any point (x, y) are the distances from this point to the sides of the triangle. Hence the equations of the bisectors of the angles are a- p =0, ^-7=0, 7-0 = 0. The sum of the equations of the bi- sectors is identically zero, therefore the bisectors pass through a common point. 2. The medians of a triangle pass through a common point. For every point in the median through C, hence Simi- sin B sin A a sin A- ^sinB = is the equation of the median through C. larly the equation of the median through B is found to be 7 sin C — a sin ^4 = 0; of the median through A, /3 sin B -y sin C = 0. The sum of these equa- tions vanishes identically. 3. The perpendiculars from the vertices of a triangle to the opposite sides pass tlirough a common point. The equation of the perpendicular through C is aros A — yScos J5 = ; through B, 7 cos C - a cos .1 = 0; througli A, & cos 27 - 7 cos C = 0. riiOPERTIES OF THE STRAIGHT LINE 91 Akt. 50. — Tangent to Cukvk ok Skcond Okdkr The general equation of the curve of the second order is ax- + 2bxy + cy- -\-2dx + 2e>/ +/=(). Let (x„, y^ be any point iu the curve. The equation y — ?/„ = tan a{x — Xq) repre- sents any line through {xg, y^. The line cuts the curve of the second order in two points and is a tangent when the two points coincide. The coordinates of any point in the straight line are x = Xq-{-1 cos a, y = y^ -\- 1 sin «. The points of inter- section of straight line and curve of second order are the points corresponding to the values of I satisfying the equation {ax,; + 2 6.tv/o + cy,; + 2 dx, + 2 ey,, +/) + (2 a.rii cos « + 2 hx.;^ sin « + 2 hy^ cos « + 2 cyo sin « + 2 r/ cos « + 2 e sin a)l + (a cos- « -f 2 6 cos « sin « + c sin- «)Z^ = 0. Since (.i-q, .Vo) is in the curve, the absolute term of the equation vanishes. If the coefficient of the first poAver of I also van- ishes, the equation has two roots equal to zero, that is the two points of intersection 6f y — ?/„ = tan a{x — a-,,) with the curve coincide at (xq, ?/o) when ax,^ cos a + hx^ sin « + hy^^ cos « + ry,, sin a -\- d cos a-\-c sin a = 0. The equation of the tangent is found by eliminating cos a and sin« from the three equations x = x^ -\- I cor a, ?/ = ?/„ + Z sin «, (ixq cos « +■ bxo sin a + by„ cos a + cy^ sin a + fZ cos a-\- e sin « = 0. This elimination is best effected by multiplying the third equation by I, then substituting from the first two equations, I cos a=x—Xo, I sin a^y—y^. The resulting equation reduces to axxo + b(xy, + x„y) + ryy, + r/(.r + .^•,.) + K.'/ + ?/..) + /= 0. The law of formation of the equation of the tangent from the equation of the curve is manifest. Problems. — 1. Write the equation of tlie tangent to x- + y"^ — r"^ at (;^o, yo). 92 ANALYTIC GEOMETRY 2. Write the equation of the tangent to — + f^ = 1 at (xo, ?/o). d^ 0- 3. Write the equation of the tangent to •'- = 1 at {xq, yo). a- b'^ 4. Write the equation of the tangent to y- = 2px at (xo, yo). 5. Find the equation of the tangents to 4 x^ + y- = 30 at the points wliere x~\. 6. At what point of x^ — ?/2 = 1 nnist a tangent be drawn to make an angle of 45° with the X-axis ? 7. Find the angle under which the line y = hx — b cuts the circle x2 + ?/2 = 49. 8. Find tlie angle between the curves y'^ = C a-, 9 2/2 + 4 ^2 = 30. 9. Find the equations of the normals to the ellipse, hyperbola, and parabola at the point {xo, yo) of the curve. The normal to a curve at any point is the perpendicular through the point to the tangent to tlie curve at the point. 10. Wliere must the normal to ^" + ^ = 1 be drawn to make an angle of 135'" with the X-axis ? 11. Find the equation of the normal to ?/2 = 10 x at (10, 10). 12. Find equations of focal tangents to ellipse ^ + f- = 1. a2 b^ CHAPTER VIII PKOPERTIES or THE CIRCLE AuT. 51. — Equation of tiik Cikclr The equation of the circle referred to rectansjfuhir axes, radius E, center (a, b), is (x — af +{y — h)- = R'\ This equa- tion represents all circles in the Xl'-plane. The equation expanded becomes x^ -\-y~ — 2ax — 2 by -f «- + ?>- — K' = 0. an equation of the second degree lacking the term in .17/, and hav- ing the coefficients of x^ and y^ equal. Conversely, every second degree equation lacking the term in xy, and having the coefficients of x^ and y- equal, represents a circle when interpreted in rectangular coordinates. Such an equation has the form x- -f ?/■ — 2 ax — 2 by + c = 0, whicli when written in the form (x — ay -\-(y — b)' — (V + ^- — c, is seen to represent a circle of radius ((t- + /r' — c)-, with center at («, b). a, b,c are called the parameters of tlie circle, aiul the circle is spoken of as the circle (a, b, 0). When the center is at the origin, a = 0, b =^ 0, and the eipia- tion of the circle becomes x^ + y^= R'- When the X-axis is a diameter, the F-axis a tangent at the end of this diameter, the circle lying on the ]>ositive side of the F-axis, a = R, b = (), and the ecjuation of the circle be- comes 2/^ = 2 Rx — x". Problems. — Write tlie cciuafions of tlic fnllmvinn; circles: 1. Center (-2, 1), radius 5. 2. Center (- 5, 5), radins 5. 3. Center (- 10, 15), radius 5. 4. Center (0, 0), radius 5. 0:J 94 ANALYTIC GEOMETBY 5. Find equation of circle througli (0, 0), (4, 0), (0, 4). 6. Find center and radius of circle through (2, — 1), (— 2, 1), (4, 5). 7. Find center and radius of circle x"^ -{■ y'^ -\- i x — Id y = 1 . 8. Find center and radius of circle x- -\- y- + l(i x = 11. 9. Does the line 3 x — 5 y = 12 intersect the circle a;2 + j/2 - 8 X + 10 2/ = 50 ? 10. Find the points of intersection of the circles x2 + 2/2-10x + 6y = 20, x2 + 2/2 + 4x- 15y = 25. Art. 52. — Common Chord op Two Circles The coordinates of the points of intersection of the circles a? -{-y'^ -2ax -2hy -\- c = Q, x^ + y^ -2a'x -2 b'y -\-c' = satisfy the equation (a;2 + ?/2 - 2 ax - 2 by + c) - (x^ + y^-2 a'x - 2 b'y + c') = 0, which reduces to (a - a')x + (b - b')y +(<■• - r)= 0. This is the equation of the straight line through the points of intersection of the circles, that is the equation of the common chord of the circles. The intersections of two circles may be a pair of real points, distinct or coincident, or a pair of conjugate imaginary points. Since the equation of the straight line through the points of intersection is in all cases real, it follows that the straight line through a pair of conjugate imaginary points is real. Problems. — Write the equations of the common chords of the pairs of circles : 1. X- + y- -Gx + 4y = 12, x" + y^ - 'ix + 6y = 12. 2. x2 + ?y2 - lOx - 6?/= 15, x^ + ?/2 + lOx + 6?/ = 15. 3. x" + y- + lx + Sy = 20, x- -y y- + 4x - \0y = 18. PROI'EirriES OF THE CIRCLE 95 AkT. 53. POWKR OF A POIXT Let (;c', y') be any point in the plane of the circle The eqnation of any straight line throngh (x', y') is y — y' = tan a (x — x'), and on this line the point at a distance d from (x', y') has for coordinates x = x' + d cos a, yz=y'-\-d sin «. The distances from (x', y') to the points of intersection of line and circle are the values of d found by solving the equation |;(.^' _ ay + iy' - by - R'] + [2 (x' - a) cos a + 2(y' - h) sin «] d + d- = 0. Since the product of the roots of an equation equals numeri- cally the absolute term of the equation, it follows that the product of the distances from the point (x', ?/') to the points of intersection oi y — y' = tan «(x — x') with the circle (.^ _ ay + (y - by = E"^ is (x' - o)- + (y' - by - /?-. This product is independent of a ; that is, it is the same for all lines through (x', ?/'). This constant product is called the power of the point (x', ?/') with respect to the circle. The expression {x' — a)--f (?/' — by — Er is the square of the distance from {x\ y') to the center (a, b) minus the square of the radius. This difference, when the point (x', ?/') is without the circle, is the square of the tangent from the point to the circle ; when the point (.!•', )/') is within the circle, this difference is the square of half the least chord through the point. Let *S' represent the left-hand member of the equation xr -\- y- — 2 a S = is the equation of the circle, and S evaluated for the co- Fir,. S". 2 by + c = 0. Then 96 ANALYTIC GEOMETRY ordinates of any point (x, y) is the power of that point with respect to the circle. Let /i5i = and So = represent two given circles. aS'i = S2 is the equation of the locus of the points whose powers with respect to /S'l = and /S'2 = are equal. This equation, Avhich may be written Si — S-, = 0, represents a straight line called the radical axis of the two circles. The radical axis of two circles is their common chord. If three circles are given, Si — 0, S., = 0, S-^= 0, the radi- cal axes of these circles taken two and two are Si — S^— 0, S2 — Ss = 0, S3 — Si=: 0. The sum of these three equations is identically zero, showing that the radical axes of three circles taken two and two pass through a common point. This point is called the radical center of the three circles. Problems. — 1. Find the locus of the points from which tangents to the circles x- + 7/- -i- 4x - 8y = b, x^ + 2/2 - 6x = 7 are equal. 2. Find the point from which tangents drawn to the three circles a;2 + y2 _ 2 x = 8, x2 + 2/2 + 4 y - 12, x2 + ?/- + 4 x + 8 ?/ = 5 are equal. 3. Find the length of the tangent from (— 3, 2) to the circle (x - 7)2 + (y- 10)2 = 9. 4. Find the length of tlie tangent from (10, 15) to the circle x2-|- ?/2-4x + C?/- 12. 5. Find tlie length of the shortest chord of the circle x2 + ?/2 - X + 4 ?/ = 3 through the point (—4, 3). 6. Find the equation of the radical axis of x2 + ?/2 + 5.r - 7 y = 15, X- + 2/2 - 3 a: + 8 2/ = 10. 7. Find the radical center of x- + y- - ".x - 5, xr + y" - 4x + y - 8, a;- + y^ + 7 2/ = 9. 8. Find the point of intersection of tlic tln-ee common chords of the circles x2 + 2/" - 4x - 2 2/ = 0, x2 + ?/2 + 2x + 2 2/ = ll, x2-|- 2/"-6x + 42/ = 17 taken in pairs. I'JlorKin'lKS OF TIIK CIliCLIC 97 AuT. 54. — Coaxal Svstkms Let Si — and iS.2 = re[)i't'sent two circles. Then ,S', - kS, = 0, for all values of the parameter A;, represents a circle tlirough the intersections of Si = 0, aS^ = 0. The equation ^i — kS., — 0, interpreted geometrically, gives the proposition, the locus of all the points vi^hose powers with respect to two circles *Si = 0, ^2 = are in a constant ratio is a circle through the points of intersection of the given circles. Si — kS^ = 0, by assigning to k all possible values, represents the entire system of circles such that the radical axis of any pair of circles of the system is the radical axis of *S, = and S., = 0. If tlie parameters of *S'i = and aS'2 = are a', h', r' and a", b", c" respectively, the parameters of Si — kSo=^ are a' — k(i" h' — kh" c' — kc" 1-/0 ' 1-k ' 1-k ' Let aS' = represent a circle, L = a straight lino. Then S —kL = represents the system of circles through the points of intersection of circle and line. The commnii radical axis of this system of circles is the line L = 0. Circles having a common radical axis are called a coaxal system of circles. Problems. — 1. Write the equation of tlie system of circles tlirougli the points of intensection of x^ + y- — 2 x + G ?/ = 10 and x- + y- — 4y = 8. 2. Find the equation of the circle through the points of intersection of X- + y^-2x + Gy = 0, X- ■+ y"^ -4y = 8, and the point (4, - 2). 3. Find the equation of the circle through the points of intersection of oc^ + 2/2 4. 10 7/ = 6, I X - 1 2/ = 3, and the point (4, 5). 4. Find the equation of the locus of all the points which have etiual powers with respect to all circles of the coaxal system determined by the circles x^ + j/2 - 3x + 7 y = 15 and x^ + y'^ + [>x - iy = 12. 98 A NA L YTIC GEO MET R Y Art. 55. — Okthogonal Systems Two circles a? + if — '^ a'x ~-2b'y + c' — 0, x^ + if — 2 a"x — 2 b"y + c" = intersect at riglit angles when the square of the distance be- tween their centers equals the sum of the squares of their radii ; that is, when {a' — a")-+{b' — b"f ^a"+b'--c' + a"'+b"''-c", or 2a'a" + 2b'b" -c' -c" ^0. If the circle (oj, bi, Cj) cuts each of the two circles (a', b', c'), (a", b", c") orthogonally, it cuts every one of the circles ka" b' - kb" c' - kc 1-k 1-k 1-k of the coaxal system orthogonally. For the hypothesis is expressed by the equations 2a'a, + 2 b% - c'- c^ = 0, 2a"a, + 2b"bi -c" -c, = 0; the conclusion by the equation 9 g' — ka" c b' — ^'^'\ ''l-k^'l-k ' c' - kc' 1-k 0, which is a direct consequence of the equations of the hypothesis. The condition that the circle (aj, &i, Cj) cuts the circles (a', b', c') and («", b", c") orthogonally, is expressed by two equations between the three parameters ttj, &„ Ci. These equa- tions have an infinite number of solutions, showing that an infinite number of circles can be drawn, cutting the given circles orthogonally. Let Oi, bi, Ci and a.^, bi, c^ be the parameters of any two circles aSi = 0, aSj = cutting aS' = and S" = orthogonally. Then PliOPEliTIES OF THE CIRCLE 99 all circles of the coaxal system Si - kyS., = cut orthogonally all circles of the coaxal system S' - k'S" = 0. For the equa- tions 2a'((i + - f>'fJi — c' — c, = 0, 2 a"a, + '2 b"h, - c" -(\ = 0, 2a'a., + 2b'b2-c' -c., = 0, 2 a"a., + 2 b"b2 - c" - c.,= 0, have as consequence r,a' — k'a" a, — k,cu , r,b' — k'b" b^ — k^b.2 c' — k (1) (2) (3) (4) Ci — kiC2 _ 1-k' 1-/h ■" 1-k' 1-ki 1-k' 1-ki Subtracting (2) from (1), and (3) from (1), there results 2 (a' - a") ai + 2 (b' - b") b,-c' + c" = 0, (5) 2 (a, - a.) a' + 2 (6i - b.^ 6' - Ci + C2 = 0. (G) Equation (5) shows that the centers of the orthogonal system Si - kySo = lie in the radical axis of the sys- tem S'—k'S" = 0; equa- tion (6) shows that the centers of the system S' - k'S" = lie in the radical axis of the sys- tem Si - kiS2 = 0. Take for X-axis the line of centers of the system S' — k'S", for y-axis the radical axis of this system. Then the equation of any circle of the system becomes x-+y^-2a'x+c' = 0. (a) Since by hypothesis fiQ. 89. 100 ANALYTIC GEOMETRY tlie power of (0, ?/') is the same for all circles of the system, c' must be a fixed constant. In like manner it is found that the equations of the orthogonal system Si — k^S^ — have the form ar+^?/^-2 6i^ + Ci=0, {(3) where Cj is a fixed constant. The condition for the orthogonal intersection of two circles when applied to («) and (/3) becomes Ci = — c'. Hence the equations of two orthogonal systems of circles, when the radical axes of the systems are taken as reference axes, are X- + y^ — 2 a'x + c' = 0, x'^ + 11'— l>'u — c' = 0, where «' and h' are parameters, c' a fixed constant. The radii of the circles of the two orthogonal systems are given by the equations r^ — a^'^ — c\ r'- = h'" -\- c' respectively. When r and r' become zero the circles become points, called the point circles of the system. In every case one of the or- thogonal systems has a pair of real, the other a pair of imagi- nary, point circles.* Problems. — 1. Find the equation of the locus of the centers of the circles which cut orthogonally the circles a;'- + y- — 4 .x + y = 15, x2 -I- 1/2 -f 5 X - 8 2/ = 20. 2. Find the equation of the circle through the point (2, — 3) and cut- ting orthogonally the circles x--\-y'^-]-^ x — 7 2/ = 18, x'^-l-y^ — 2 x—iy~\2. 3. Find the equation of the circle cutting orthogonally x'^+y'^ — \0 x = 9, 3.2 4. 2/2 = 25, x2 + y-i-8y = IG. * Through every point of the plane there passes one circle of each of the orthogonal systems. The point in the plane is determined by giving the two circles on which it lies. This leads to a system of bicircular coordinates. If heat enters an infinite plane disc at one point at a uniform rate, and leaves the disc at another point at the same uniform rate, when the temperature conditions of the disc have become permanent, the lines of equal temperature, the isothermal lines, and the lines of flow of heat are systems of orthogonal circles. The points where the heat enters and Jeaves the disc are the point circles of the isothermal system. rnOPEHTIKS OF THE CIliC'LlS' '^"^ " 11)1 4. Find the equation of the system of circles cutting orthogonally the coaxal system detennined by x'^-\-if+ix + 6y-\5, x^ + y-+2 z-S y = l2. 5. Write the equation of the two orthogonal systems of circles whose real point circles are (0, 4), (0, — 4). Art. 56. — Takgents to Circles The equation of a tangent to the circle x^ + -if = r'^ at the point (a*o, ?/o) of the circumference is xxq + yy^ = r\ Let (xi, yi) be any point in the plane of the circle x--\-y'=i~, (x', y'), (x", y"), the points of contact of tangents from (x^, y^) to the circle. Then (.r,, y{) must lie in each of the lines xx' + yy' = 7", xx" + yy" = r-; that is, x^x' + y^y' = r, and Xix" + yiy" = rl Hence the equation of the chord of contact is xxi + yyi = r^. The distance from the center of the circle to the chord of contact is -, which is less than, equal to, or greater (a.V + 2/i-)' than r, according as the point (a-j, y^ lies without the circum- ference, on the circumference, or within the circumference. In the first case the points of contact of the tangents from (.Xi, r/i) to the circle are real and distinct, in the second case real and coincident, in the third case imaginary. In all cases the chord of contact is real. In the equation y = mx + n let m be a fixed constant, n a parameter. The equation represents a system of parallel straight lines. The value of n is to be determined so that the line represented by y = mx + n is tangent to the circle x^ + y~ = r^. The line is tangent to the circle when the per- pendicular from the center of the circle to the line equals the radius; that is, when = 7% ?i= ±rVl + m-. There- fore, the equations of tangents to v? ■{- y'^ = r parallel to y = mx -f n are y = mx ± r Vl + m'. 102 ANALYTIC GEOMETRY Problems. — 1. Find the equations of the tangents to x- -\- y- — 25 at x = 3. 2. Find the chord of contact of tangents from (2, —3) to x- + y"^ = 1. 3. Find the points of contact of tangents from (5, 7) to x- + y- — 9. 4. Find the equations of tangents to x- + ?/- = 16, making angles of 45° with the A'"-axis. 5. Find the equations of tangents to x^ +?/'- = 25 parallel to ?/=3 x+5. 6. Find the equations of tangents to x^ + y'- = 25 perpendicular to 2/ = 3 X + 5. 7. Find the slopes of tangents to x- + ?/- = 9 through (4, 5). 8. Find the equations of the tangents to x^ + ?/2 = \Q through (5, 7). 9. The chord of contact of a pair of tangents to x' + ?/2 = 25 is 2 X + 3 // = 5. Find the intersection of the tangents. 10. Find equation of tangent to (x — a)^ + (y — b)- — f- at (x', y') of circumference. Art. 57. — Poles and Polars Since it is awkward to speak of the chord of contact or the point of intersection of a pair of imaginary tangents, the point (xi, ?/i) is called the pole of the straight line xx^ + yi/i — i^ with respect to the circle s? -\- y- = r, and xx^ + yy^ — r^ is called the polar of the point (xj, y^). (x^, ?/,), which may be any point of the plane, determines uniquely the line xxi + yyi — r- ; and conversely, xxi -\- yyi = r, which may be any straight line of the plane, deter- mines uniquely the point (iCj, y^. The relation between pole and polar therefore establishes a one-to-one correspondence between the points of the plane and the straight lines of the plane. The polar of (.rj, ?/j) with respect to ar-|- ?/-=?- is xxi+yy^=r-, the line through the center of the circle and {x^, ?/i) is ?/ = ' '.t". Hence the line through the pole and the center is perpen- PliOrEllTIES OF THE (JIRCLK 103 dicular to the polar, and the angle included by lines fiom the center to any two points equals the ant,de included l)y the polars of the two points. The distance from the center of the circle .r +//- = r to the l)olar of (.1-,, ?/i) is that is, the radius is the geomet- ric mean between the distances of the center from pole and polar. The i)olar of (.).•„ y^), with respect to the circle x- + U' = t", is constructed geometrically by draAving a perpendicular to the line joining (.t„ ?/i) and the center of the circle at the point whose distance from the center is the third proportional to the distance from (a*,, ?/,) to the center and the radius of the circle. The pole of any line, with respect to the circle x- -\- y- = r, is constructed geometrically by laying off from the center on the perpendicular from the center to the line the third propor- tional to distance from center to line and the radius of the circle. The polar of (.r„ y{) Avith respect to the circle x- + y- = r- is xxi + yyi = ?•-, the polar of (x.,, y.,) is xx^ + yy-j = r- The condi- tion which causes (a'l,?/,) to lie in the polar of (X2,y2)is x^Xo+yiy2^r^\ this is also the condi- tion which causes the polar of (.r„ //i) to con- tain (x.,, y-_^. Hence the polars of all points in a straight line pass through the pole of the line, and the poles of all lines through a point lie in the polar of that point. 104 ANALYTIC GEOMETRY Problems. — 1. Write the equation of the polar of (2, :]) with respect to x~ + y- = 10. 2. Find the point whose polar with respect to x- + y- = d is S x + 7 y = 18. 3. Find distance from center of circle x- + y- = 26 to polar of (3, 4). 4. Find equation of polar of {x', y') with respect to circle (x - a)'^ + (2/ - '0- = '•-• 5. Find polar of (0, 0) with respect to x- + y- = r^. Art. 58. — Reciprocal Figures If a geometric figure is generated by the continuous motion of a point, the polar of the generating point takes consecutive positions enveloping a geometric figure. To every point in the first figure there corresponds a tangent to the second figure ; to points of the first figure in a straight line there correspond tangents to the second figure through a point; to a multiple point of the first figure there corresponds a multiple tangent in the second. If two points of intersection of a secant of the first figure become coincident, in which case the secant becomes a tangent, the pole of the secant at the same time must become the point of intersection of two consecutive tan- gents of the second figure, that is a point of the second figure. Hence the first figure is also the envelope of the polars of the points of the second figure. For this reason these figures are called reciprocal figures. Reciprocation leads to the principle of duality in geometry.* Problems. — 1. To find the reciprocal of the circle C with respect to the circle 0, x- + y- = r^. * The principle of duality was developed by Poncelet (1822) and Ger- gonne (1817-18) as a consequence of reciprocation, independently of reciprocation by Mobius and Gergonne, pnoPEirriEs of the circle 105 The line nir{xxx + mn = f-) is tlie polar of the center C{xu 2/i) with respect to the circle ; p (Xa, 2/2) is the pole of any tangent PT{xx.z + ijih = r^) to the circle C. Then OC = (:'•!- + 2/1-)', pK CP X1X2 + yiyo - r- X\Xi + Op^(x2-+y2~)l Hence OC-pK= CP- Op, or ^ = ^'. pK CP oc —^ is constant, and therefore p nmst generate a conic section whose focus 00 is O, directrix ////', eccentricity -— . This conic section is an ellipse when O is within the circumference of the circle C, a parabola when is on the circumference, an hyperbola when is without the circumference. 2. Find the reciprocal of a given triangle. Call the vertices of the given triangle A{xx, y{), B{xn, 2/2), C(xz, yz)- The polars of these vertices with respect to x^ + y~ = r'^ are bc(xxx + 2/2/1 = '•^)' ac{xx2 + 2/2/2 = »•-), a5(a:a:3 + 2/2/3 = »•')• Triangles such that the ver- tices of the one are the poles of the sides of the other are called conjugate triangles. The conjugate triangle of the triangle cir- cumscribed about a circle with respect to that circle is the triangle formed by joining the points of con- tact. 106 ANALYTIC GEOMETRY 3. The straight lines joining the corresponding vertices of a pair of conjugate triangles intersect in a common point. The equations of the lines through the corresponding vertices are Aa, (rciX3 + 2/12/3 - r'^) {xx^ + yy^ - r~) - (a;iX2 + 2/12/2 - r'^){xxz + 2/2/3 - r-) = ; Bh, {xiXo + 2/12/2 - r^){xxz + 2/2/3 - r-) - {x^xz + 2/22/3 - r^) {xx^ + 2/^1 - V'-) = ; Cc, {XiXs + yzys - r^) (xxi + 2/2/1 - r^) -(X1X3 + 2/12/3 - r'^) {xx2 + 2/2/2 - »■") = 0. The sum of these equations is identically zero, therefore the lines Aa, Bb, Cc, pass through a common point. 4. Show that if a triangle is circumscribed about a circle the straight lines joining the vertices with the points of contact of the opposite sides pass through a common point. 5. Reciprocate problem 3. The figure formed by the conjugate triangles ABC, ahc is its own reciprocal. The poles of the lines joining the corresponding vertices of ABC and abc are the points of intersection of the corresponding sides of ABC and abc. Hence the reciprocal of problem 3 is, the points of intersection of the corresponding sides of a pair of conjugate triangles lie in a straight line. 6. Reciprocate problem 4. The reciprocal of the circle is a conic section, the reciprocals of the points of contact of the sides of the triangle are tangents of the conic sec- tion, the reciprocals of the vertices of the triangle are the chords of the conic section joining the points of tangency, hence the poles of the lines from the vertices to the points of contact of the opposite sides in the given figure are the points of intersection of the sides of the triangle inscribed in the conic section with the tangents to the conic section at the opposite vertices of the triangle. These three points of intersection must lie in a straight line. Art. 59. — Inversion* Let P'(.T„ ?/i) be any point in the plane of the circle x- + y-=r", P(x, 11) the intersection of the polar of P', (1) xx^ + ?///i = i~, and * The value of inversion in geometric investigation was shown by Pliicker in 1831. The value of inversion in the theory of potential was shown by Lord Kelvin in 1845. PUOPERTIES OF THE CIRCLE 107 the diameter through P', (2) y = -Av. Then OP • OP' = r'-, that ^Vheu r becomes is (r - PA)(r + P'A)= r-, ^vheiice J 1_^1 PA FA V infinite the circle becomes a straight line, PA = P'A, and P and P' become symmetrical points with respect to the line. P is said to be obtained from P' by inver- sion, by the transformation by reciprocal radii vectors, or by symmetry with res})ect to the circle. This transformation establishes a one-to-one corre- spondence between the points within the circle and the points without the circle. The coordinates of P are obtained in terms of the coordinates of P' by making (1) and (2) simultaneous and solving for x and ?/. There results x = — — -'— y, y = -—-—• xl + y,- Xi- -\- y^ ilarly. t^x 2/1 r'y X' + y- X- + y- If the point (.t„ ?/,) describes a circle x,\-\-y:--2aj\-2hy, + c = 0, the inverse point (.r, ?/) traces a curve whose equation is r\-c- 4- r'^v" 2 arx 2 hr>i (x- + iff x^ + y- X- + y- which reduces to + c = 0, o , o sar- ^■" + v ;« c 2 hr y + the erpiation of a, circle. Hence iuvcrsiou Iransrorms llie circle (a, b, c) into the circle ("alliu''- the radius of tlie 108 ANALYTIC GEOMETRY given circle R, the radius of the transformed circle R', R- = ^+^J^. That is, c. c- R = -R. When c = 0, R' = cc; that is, the transformed circle becomes a straight line, c = is the condition which causes the center of the inversion circle, which has been taken at the origin of co- ordinates, to lie in the circumference of the given circle (a, 6, c). The inverse of a geometric figure may be constructed mechan- ically by means of an apparatus called Peaucellier's inversor. The apparatus consists of six rods, four of equal length h forming a rhombus, and two others of equal length a con- necting diagonally opposite vertices of the rhombus with a lixed point 0. The rods are fastened together by Fig. 95. . i. n r 4. pins so as to allow perfect freedom of rotation about the pins. If P is made to follow a given curve, P' traces the inverse, the center of inversion being O and the radius of inversion (p? -f If)^. For OP=a cos 6 — bcos 6', OP =^acos6 + b cos 6', a sin 0=1 sin 6>'. Hence OP • OP' = a- cos- d - Ir cos^ 6', o? sin- 6 -h- sin- 6' = 0, and by addition OP • OP = cr - Jr. If the point P describes the circumference of a circle passing through 0, P must move in a straight line. Therefore the inversor transforms the circular motion of P about 0', mid- way between and P, as center, into the rectilinear motion of P'. rUorKllTlKS OF THE CIRCLE 109 The cosine of the angle between two circles {a, h, c), (a', h', c') is found from the ecjuation (rt _ a'f + {b - b'f = r- + r" - 2 rr' cos 6 . 2aa' + 2 hh' - c — d to be 2Va^ +7>2 - c V^2 ^ in _ ^ The circles obtained by inverting the given circles are Calling their included angle $', cos ^' 2ffa'>-'' 2 ^j?/r^ _ ?;; _ r^ cc' cc' c c' which reduces to cos 9' = 2 «a' + 2 6/y - c - c' 2 Va^ + ?/ _ c V«'' + 6'2 - c' Hence the angle between two circles is not altered by inversion. For this reason inversion is called an equiangular or conformal transformation. If two orthogonal systems of circles are inverted, taking for center of inversion one of the points of intersection of that 110 ANALYTIC GEOMETRY system of circles which has real points of intersection, one of the systems of circles transforms into a system of straight lines through a point. Hence the other system of circles must trans- form into a system of concentric circles whose common center is this point. CHAPTER IX PROPERTIES OF THE CONIO SECTIONS Akt. 60. — General, Equation A point governed in its motion by the law — the ratio of the distances from the moving point to a fixed point and to a fixed line is constant — generates a conic section. To express this definition by an eqnation between the coordinates of the moving point, let the moving point be (x, y), the fixed point F, the focus (in, n), the fixed line UH'. the directrix a; cos a + y sin « — ^^ — 0. Calling the constant ratio e, the defini- tion is expressed by the equation PF' = e^ • PD^, which becomes {m - xf + (n - yf = e- (x cos « 4- 2/ sin a — py. a is the angle which the axis of the conic section makes with the X-axis, 1^ the distance from the origin to the directrix. By assigning to m, n, e, a, p their proper values in any special case, this general eqnation becomes the equation of any conic section in any position whatever in the XF-plane. For example, to obtain the common equation of the ellipse, which is the equation of the ellipse referred to its axes, make m = ae, n = 0, a — 0, p = "^, 1 — e^ = ';,• The general equation a -, e 112 A NA L YTIC GEOMETR Y becomes (ae — .^•)- -\- y- = {ex — a)-. Expanding and collecting terms, / + (1 — e-) X' = a- (1 — e-), or ^ + ^ = 1. To obtain the equation of the hyperbola referred to its axis and the tangent at the left-hand vertex, make m — a(l + e), „ = 0, « = 0, p='^Sl+-^, l-e' = --- The general eqna- e a- tion becomes (a + ae — a-)- + f = (ex - a — ae)-. Expanding and collecting terms, ?/- = (1 — e-) (2 ax — x"), or ?/" ^4(2rtx--x-). Problems. — From the general equation of a conic section referred to rectangular axes, obtain : 1. The common equation of the hyperbola. 2. The common equation of the parabola. 3. The equation of the ellipse referred to its axis and the tangent at the left-hand vertex. 4. The equation of the ellipse referred to its axis and the tangent at the right-hand vertex. 5. The equation of the hyperbola referred to its axis and the tangent at the right-hand vertex. 6. The equation of the parabola referred to its axis and the perpen- dicular to the axis through the focus. 7. The equation of the ellipse referred to its axis and the perpendicu- lar to the axis through the focus. 8. The equation of the hyperbola referred to its axis and the directrix. PUOPEUTIES OF THE CONIC SECTIONS 113 9. 'I'hc c'liuation of tlir panibola referred to its axis and the dirrctrix. 10. Show that in the hyperbolas ^"-'^"=1, ■'/' _ ^ = i tlie traiis- a- b'^ b- u- verse axis of the first is the conjugate axis of tlie second, and vice versa. Such hyperbolas are called a pair of conjugate hyperbolas. 11. Derive from the general equation of a conic section the equation fc2 of the hyperbola conjugate to — m = 0, n - be, a = 90^ p = '\ 1 - e^ = - ?^ e b- 12. Show that the straight lines tj =±-x are the conunon asymptotes a of the pair of conjugate hyperbolas — — ''- = 1, — — •'' = — 1. «- b- a- b- 13. Find the equation of the ellipse focus (—3,2), eccentricity |, major axis 10, the axis of the ellipse making an angle of 45° with the A'-axis. 14. Find the equation of the ellipse whose focal distances are 2 and 8, center (5, 7), axes parallel to axes of reference. 15. Find the equation of the hyperbola whose axes are 10 and 8, cen- ter (3, — 2), axis of curve parallel to X-axis. 16. Find the equation of the parabola whose parameter is 0, vertex (2, — 3), axis of parabola parallel to A'-axis. Art. 61. — Tangents and Nokmals Using the common equations of ellipse, hyperbola, and parab- ola, the equations of tangents to these curves at the point (.Tu, y/o) of the curve are tt- "^ V ' a- lr~ ' yjhi=2){x-\-x^, respectively. The slopes of these tangents are for the ellipse for the hyperbola — ■'-, for 114 ANALVriC GEOMETRY the parabola — Calling the intercepts of the tangent on the 1ft) 2 7 2 X-axis X, on the F-axis Y, for the ellipse X = —, Y=—, for the hyperbola X = ^, Y= , for the parabola X — — Xo, y= i_?/||. X and y may in each case be determined geometri- cally, and the tangent drawn as indicated in the figure. Suppose the point (,<■„ ?/,) to be any point in the plane of the ellipse "-T, + •— = 1. Let {x', y'), (x", y") be the points of con- tact of tangents from (a-j, y{) to the ellipse. Then must (xi, y^) ... , „ , ,. xx' ?/?/' . xx" ?/w" ^ ,, , . ,, he m each of the hues -^ -f ^4- = 1, -^ + '^ — 1 5 that is, the a- ¥ a- 0- XyV ?/,?/ equations — 5- + ' ' a- b' 1 ^^ + -M! ' a- b- 1 must be true. Hence the points of contact lie in the line +f=i, diich therefore the chord of contact. Similarly, it is found that the X- "■' points of contact from (.r'l, y/j) to the hyperbola -r, - 1 and yih to the parabola y' — lpx lie in the lines -—2^ — 72^ = 1 a,nd ?///, = j>(;« -f- x^ respectively. The coordinates of the points of contact of tangents through (.»■„ v/j) to a conic section are found prxOPEUTIES OF THE CONIC SECTIONS 115 by making the equations of the chord of contact and of the conic section simultaneous and solving for x and //. A theory of poles and polars with respect to any conic sec- tion might be constructed entirely analogous to the theory of poles and polars with respect to the circle. The equation y = mx + n, where m is a fixed constant, n a parameter, represents a system of parallel straight lines. For any value of n, the abscissas of the points of intersection of straight line and ellipse %,-\-% = l are found by solving the equation (b' + aha-) x- + 2 a-mnx + a" (a- - b-) = 0. These ab- scissas are equal, and the line ?/ = nix + n becomes a tangent to the ellipse -;, + -'^ = l when u' = b'- + a-ni-. Therefore u- b- y = mx ± (b- + a-m-y are the two tangents to the ellipse whose slope is m. In like manner it is found that the tangents to the hyperbola whose slope is m are y = nix ±(a-iii' — b'-)- ; the P tangent to tlu; parabola whose slope is vi is y — rax + — — The equations of the normals to ellipse, hyperbola, and parab- ola at the i)oint (.»•„, ?a,) of the curves are y — 7j,, = -^(x — x^^, y -?/„ = - "/"(.f - .f,), // - //„ = - -''^(x - .Vu) respectively. Problems. — 1. Find the eiiuatioiis of taiip:cnts to the ellipse whose axes arc S and at the points wliose distance from the T-axis is 1. 2. Find the eiiuatioiis of the focal tangents of ellipse, hyperbola, and parabola. 3. From the point (fi, 8) tangents are drawn to tlu' ellipse ^-|-^=1. F'ind the coordinates of the points of contact and the equations of the tangents. 4. At what point of the parabola ?/- = 10x is the slope of the tan- gent 1 h ? 5. On an elliptical track whose major axis is due east and west and 1 mile long, minur axis ! mile lonsr, in what direction is a man traveling 116 ANALYTIC GEOMETRY when walking from west to east and ] mile west of the north and south line ? 6. Write the equations of tangents to ^ + ^ = 1 making an angle 45° with the X-axis. 7. Write the equations of the tangents to — — ^ = 1 perpendicular to 2x-32/ = 4. ^ ^ 8. Write the equation of the tangent to y- = 8x parallel to ^ + -^ = ^• 9. Find the slopes of the tangents to — + ^- = 1 through the point 9 4 (4,5). ?/ = mx + (4 + 9m2)2 is tangent to— + ^= 1. Since (4, 5) is m i 9 4 the tangent, 5 = 4 wi + (4 + 9 m^) 2. Solve for m. 10. Find the slopes of tangents to ^ - ^ = 1 through (2, 3). 11. Find the slopes of tangents to y' = Gx through (-5, 4). 12. Find the points of contact of tangents to y" = {Jx through ( -5, 4). 13. Find the intercepts of normals to ellipse, hyperbola, and parabola on X-axis. 14. Find distances from focus to point of intersection of normal with axis for each of the conic sections. 15. Prove that tangents to ellipse, hyperbola, or parabola at the ex- tremities of chords through a fixed point intersect on a fixed straight line. 16. Prove that the chords of contact of tangents to a conic section from points in a straight line pass through a common point. 17. Show that the tangent to the ellipse at any point bisects the angle made by one focal radius to tlie point with the prolongation of the other focal radius to the point. rUOVERTIKS OF THE CONIC SECTIONS 117 The ratio of the focul nulii is "^^ = - — ^- Since PF' a + exo AF= AF' = ae and AT- Xo F'T~ a (« - ex^) — (« + ea;o) Hence EJL = I1L^ and Pr bisects FPS. F'T PF' 18. In the hyperbohx the tangent at any point bisects the angle in- cluded by the focal radii to the point. 19. In the parabola the tangent at any point bisects the angle included by the focal radius to and the diameter through the point.* \D' Fig. 104. On problems 17, 18, 19 is based a simple method of drawing tangents to the conic sections through a given point. With the given point as center and radius equal to distance from given point to one focus strike * Since it is true of rays of light, heat, and sound that the reflected ray and the incident ray lie on different sides of the normal and make equal angles with the normal, it follows that rays emitted from one focus of an elliptic reflector are concentrated at the other focus ; that rays emitted from one focus of an hyperbola reflector proceed after reflection as if emitted from the other focus ; that rays emitted from the focus of a parabolic reflector after reflection proceed in parallel lines. It is this property of conic sections that suggested the term focus or " burning point." 118 A NA L YTIC GEOMETR Y off an arc. In the parabola the parallels to the axis through the inter- sections of this circle with the directrix determine the points of tan- gency. For TF = TD, hence the triangles TPF, TPD are equal and PT is tangent to the parabola. In ellipse and hyperbola strike off another arc with the second focus as center and radius equal to transverse axis. Lines joining the second focus with the points of intersection of the two arcs determine the points of tangency. In the ellipse T'F' + T' F = 2 a, and by construction T'F' + TD' = 2 a, hence T'F - T'D'. The trian- gles T'PF, T'PD' are equal, and PT' is tangent to the ellipse. In the hyperbola TF' - TF = 2 a, TF' - TD = 2a; hence TD = TF, the triangles TPD, TPF are equal, and PT is tangent to the hyperbola. 20. Show that the locus of the foot of the perpendicular from the focus of the ellipse \--l-=zl to the tangent is the circle described on the a- b'^ major axis as diameter. The equation of the perpendicular from the focus {ae, 0) to the tan- gents y = mx ± (b- + n-m^) •2 is my + x = ae. Make these equations simultaneous and eliminate m by squaring both equations and adding. There results x'^ + y^ = a'^. 21. Show that the locus of the foot of the perpendicular from the focus of the hyperbola ^ - ^- = 1 to the tangent is the circle described on the a^ b'^ transverse axis as diameter. 22. Show that the locus of the foot of the perpendicular from the focus of the parabola y^ = 2pxto the tangent is the F-axis. PUOI-KliTIKS or THE CONIC SECTIONS 119 Problems 20, 21, 22 may be used to construct the cuuic sections as envelopes when the focus and the vertices are known. 23. Prove that for ellii)se and hyperbola the product of the perpen- diculars from foci to tangent is constant and eipial to h'-. 24. Prove that in the parabola the locus of the point of intersection of a line through the vertex perpendicular to a tangent with the ordinate through the point of tangency is a semi-cubic parabola. AkT. 62. (JONJUGATK DiAMETERS Let {xo, ?/„) be the point of intersection of the diameter w = tan^ • X with the ellipse — \-^^=l, and call the angle a' Ir made by the tangent to the ellipse at (.r,,, ij^^) with the X-axis $\ Then Y tan d = •^, tan^' = -^ tan 6 tan 6' = - Now let (a'l, ?/,) be the point of intersection of Fig. 107. y = tan 9' - x with the ellipse, and call the angle made by the tangent to the ellipse at (a;,, ?/i) with the X-axis 0. Then tan ?/i tan e=^- ^, tan tan 6' = - Ir Hence the condition tan 6 tan & = causes each of the diameters of the ellipse y = tan • x, y = tan 0' ■ x to be par- allel to the tangent at the extremity of the other. Such diameters are called conjugate diameters of the ellipse. 120 ANALYTIC GEOMETRY The equation of tlie ellipse —-{---^^^l referred to a pair of (r b'- conjugate diameters and in terms of the semi-conjugate diam- eters a' and b'is— + ^=l. (See Art. 35, Prob. 39.) This equation shows that each of a pair of conjugate diameters bisects all chords parallel to the other. The axes of the ellipse are a pair of perpendicular conjugate diameters. Let (xu, 2/o) be the point of intersection oi y — tan 6 • x with the hyperbola — — ^ = 1, and call the angle made by the tan- gent to the hyperbola at {xq, y^) with the X-axis 6'. Then .Vo 4-„„ flf _ ^^•^•o tan 6* tan 6>' = — . Since y = ~x and b y — X are a tan 0^'^, tan 0' x„ b%, tl le common asymptotes of the pair of con- jugate hyperbolas and — — - cr b'' l,it is evident that the condition tan d tan $' = — causes y = tan 0' • x to inter- sect --^=-1 if tan I intersects —-■£=!. Now suppose (x^, y^ to be the point of a- y- intersection of the line y = tan 6' • x with the conjugate hyper- bola 1, and call the angle made by the tangent to this hyperbola at (.Ti, ?/i) with the X-axis 6. Then tan & tan b-x, tan e tan 6' = Diameters of the hyperbola b^ satisfying the condition tan 6 tan 0' =-7, are called conjugate diameters of the hyi^erbola. PliOPEliTIEtS OF THE CONIC SECTIONS 121 The ecjuation of the hyperbola '- of conjugate diameters diameters a' and b', is -^. — r^=l. (See Art cr Ir and in terms of tl referred to a pair le semi-conjngate Prob. 38.) This equation shows that chords of an hyperbola parallel to any diameter are bisected by the conjugate diameter. The axes of the hyperbola are perpendicular conjugate diameters. The equation of the parabola referred to a diameter, and a tangent at the extremity of the diameter, is y^ — 2piX. (See Art. 35, Prob. 40.) This equation shows that any diam- eter of the parabola bisects all chords parallel to the tan- gent at the extremity of the diameter. The axis of the parabola is that diameter which bisects the system of parallel chords at right angles. It is now possible to deter- mine geometrically the axes, focus, and directrix of a conic section when the curve only is given. In the case of the ellipse draw any pair of parallel chords. Their bisector is a diameter of the ellipse. With the center of the ellipse as center strike off a circle intersecting the ellipse in four points. The bisectors of the two pairs of parallel chords joining the points of in- tersection are the axes of the ellipse. An arc struck off with extremity of minor axis as center, and radius equal to semi-major axis, inter- 122 ANALYTIC GEOMETRY sects the major axis in the foci. The directrix is perpendicular to the line of foci where the focal tangents cross this line. In the case of the hyperbola the directions of the axes are found as for the ellipse. The focus is determined by drawing a perpendicular to any tangent at the point of intersection of this tangent with the circumference on the transverse axis. Drawing the focal tangents determines the directrix. The conjugate axis is limited by the arc struck off with vertex as center and radius equal to distance from focus to center. In the case of the parabola, after determining a diameter by bisecting any pair of paral- lel chords, and the axis by bisecting a pair of chords per- pendicular to the diameter, the focus is determined by the property that the tangent bisects the angle included by diameter and focal radius to point of tangency. Art. 63. — Supplementary Chords Chords from any point of ellipse or hyperbola-to the extrem- ities of the transverse axis are called supplementary. Let (x', y') be any point of the The ellipse ^„ + j-^ equations of lines through (a;', ?/'), (a, 0) and (.-»', ?/'), ( — a, 0) are Fig. 112. •/ {X - a), x' — a , y' PliOPKRTlES OF THE CONIC SECTIONS 123 Calling the angles made l)y tlie supplementary eliords with the X-axis </> and <^', tan <^ tan 4>' — V From the equa- x'-). Hence tan </> tan (/>' = ;. d- — = 1 , tan <f) tan </> = — . a- Ir a' V are a pair of conjugate diam- 1 when tan ■ tan $' = — —. Hence d- tion of the ellipse, //'- = —(a' In like manner for the hyperbola y = tan 6 • x and ?/ = tan 6 eters of the ellipse '—4--^-- d- h- tan 6 • tan 6' = tan 4> • tan </>', from which it follows that if one of a pair of supplementary chords is parallel to a diameter the other chord is parallel to the conjugate diameter. This proposition is dem- onstrated for the hyper- ' bola in the same manner. On this proposition are based simple methods of drawing tangents to ellipse or hyperbola, either through a point of the curve or parallel to a given line. To draw a tangent to the ellipse at any point P, dra^v a diameter through P, a sup- plementary chord parallel to this diameter, and the line through /'parallel to the other supplementary chord is the tangent. To draw a tangent to the hyperbola parallel to a given straight line, draw one sup- plementary chord parallel to the given line, and the diameter parallel to the other supplementary chord determines the points of tangency. 124 ANALYTIC GEOMETEY To draw a pair of conjugate diameters. of an ellipse, includ- ing a given angle, construct on ^^^^^^ /^^^ \ ^^^ major axis of the ellipse a /^^ ^--V nh circular segment containing the — x ^""^^^ / "^ ~^ ^ -J ^ given angle. From the point of / V /' - -^^^^T"^-----/! intersection of the arc of the seg- 1 ^v^y^ 1^ ^^ ]\f) ment and the ellipse draw a pair I "^ / \ of supplementary chords. The PiQ 115 diameters parallel to these chords are the required diameters. Art. 64. — Parameters Since — + ^ = 1 is the equation of an ellipse referred to any pair of conjugate diameters, it is readily shown that the squares of ordinates to any diameter of the ellipse are in the ratio of the rectangles of the segments into which these ordi- nates divide the diameter. The same proposition is true of the hyperbola. Taking the pair of perpendicular conjugate diameters of tlie ellipse as reference axes and the points (cte, 'p), (0, IS), the proposition leads to the proportion ^ = — ^^ — v^^^y whence -^ — — - that is, the i:)arameter to the transverse axis of the 2& 2a' ' _ ellipse is a fourth proportional to the transverse and conjugate axes. Generalizing this result, the parameter to any diaineter of ellipse or hyperbola is the fourth proportional to that diame- ter and its conjxigate. In the common equation of the parabola, y''- = 2i')X, the parameter 2p is the fourth proportional to any abscissa and its corresponding ordinate. Generalizing this definition, the parameter to any diameter of the parabola is the fourth propor- tional to any abscissa and its corresponding ordinate with respect to this diameter. iniOPEIiTIES OF THE CONIC SECTIONS 125 ^Vhen (in, v) on the parabola y" = 2px- is taken as origin, the diameter througli {m, n) as X-axis, the tangent at (?//., n) as l''-axis, the e(i[uation of the pa- rabola takes the form Y yf = -J2h^i- (See Art. 35, I'rob. 40.) ^' sure Hence 22h = 4(?ji + J- jj); that is, the parameter to any di- ameter of a parabola is four times the focal radius of the vertex of that diameter. Calling the focal radius /, the equation of the parabola becomes r_ 1, find the equation of the Problems. — 1. In the ellipse rlianieter conjugate to y = x. 2. Find the angle between the supplementary chords of the ellipse '■!.' ^111=1 at the extremity of the minor axis. n- //- 3. Find the point of the ellipse ^ + ^'-1 at which supplementary chords include an angle of 45°. 4. Show that the maximum angle between a pair of supplementary x' , if , ..„ ..„_, 2 ah h chords of the ellipse 1 IS tan-' «•' o~ ¥ — a- 5. Show that a pair of conjugate diameters of an hyperbola cannot include an angle greater than 90°. 6. Construct the ellipse whose equation referred to a pair of conjugate -f^ Find focus and dircc- diameters including an angle of 45° trix of this ellipse. 7. Find the equation of the hyperbola whose axes arc 8 and 6 re- ferred to a pair of conjugate diameters, of which one makes an angle of 45° with the axis of the hyperbola. Find lengths of the semi-conjugate diameters. 126 ANALYTIC GEOMETRY 8. Find equation of parabola whose parameter is ter through (8, 8) and tangent at this point. Find the locus of the centers of chords of 2x referred to diame- ^ = 1 parallel to 4 is y^=- a parabola referred to 10. The equation of a pai-abola referred to oblique axes including an angle of 60° isy- = lOx. Sketch the parabola and construct its focus and directrix. 11. A body is projected from A in the direction AY with initial velocity of v feet per second. Gravity is the only disturbing force. Find the path of the body and its velocity at any instant. Taking the line of projection as F-axis and the vertical through A as X-axis, the coordinates of the body t seconds after projection are x = I gfi, y = vt; the equation of the path of the body, found by eliminating t, (J tangent and diameter through point of tangency. Comparing this e<iuation with 2/2 = 4 /x, the equation of parabola re- ferred to tangent and diameter, v'^=2 (jf ; that is, the initial velocity is the velocity acquired by a body falling freely from the directrix of the parabola to the start- ing point. If the body is projected from any point of the parabola along the tangent to the parabola at that point, and with a velocity equal to the velocity of the body projected from A wlien it reaches that point, the path of the body is the path of the body projected from A. Hence it follows that the velocity of the body at any point of the parabola is the velocity acquired by a body freely falling from the directrix of the parabola to that point. Art. 65. — The Elliptic Compass Let i^ -f -^ = 1 aud 3l + -^^ = 1 1)0, two ellipses ccmstructed on the same major diameter.- Let ?/, and ?/. be ordinates cor- respondintr to the same abscissa, then ■— = ^^; that is, if ellipses riiOPEIiTIES OF THE CONIC SECTIONS 127 are constructed on the same major diameter, corresponding ordinates are to each other as the minor diameters. The circle described on the major diameter of the ellipse is a variety of the ellipse, hence the ordinate of an ellipse is to the corre- sponding ordinate of the cir- cumscribed circle as the minor diameter of the ellipse is to the major diameter. On this principle is based a convenient instrument for drawing an ellipse whose axes are given. On a rigid bar take PH=a, PK^h. Fix pins at H and K which slide in grooves in the rulers X and "J perpendicular to each other Fo lY P' . ^.''- ' — ^< ^ ' 1 \ 1 \ / / / / 1 \^ x\ ^/ ^ D i^ 1 ■"^^^ / H ^y ^~~^ Y' PI) PII compass. P traces the ellipse ^ -f ^ = 1 . This instrument is called the elliptic Art. 66. — Area of the Ellipse Erect any number of perpendiculars to the major diameter of the ellipse, and beginniug at the right draw through the points of intersection of these perpendiculars with the ellipse and the circumscribed circle parallels to the minor diam- eter. There is thus inscribed in the ellipse and in the circle a series of rectangles. The corresponding rectangles in ellipse and circle have the same base, and their altitudes are in the ratio of h to a. 128 A NAL YTIC GEOMETR Y Hence the sum of the areas of the rectangles inscribed in the ellipse bears to the sum of the rectangles inscribed in the circle the ratio of h to a. By indefinitely increasing the number of rectangles, the sum of the areas of the rectangles inscribed in the ellipse approaches the area of the ellipse as its limit, and at the same time the sum of the areas of the rec- tangles inscribed in the circle approaches the area of the circle as its limit. At the limit therefore ^^-^ r-^^ = -, hence , area of circle a area of ellipse = - • 7ra^ = irah. a Art. 67. — Eccentric Angle op Ellipse At any point {x, y) of the ellipse ^, + 4, ■ 1 produce the ordinate to the transverse axis to meet the circumscribed circle and draw the radius of this circle to the point of meeting. The angle <^ made by this radius with the transverse axis of the ellipse is called the eccentric angle of the point (.r, ?/). Erom the figure x—.a • cos <)!>, y = -. 1\D=^ - ■ asinc^ (( a = h • sin <^. The coordinates of any point {x, ?/) of the ellipse are thus expressed in terms of the single variable ^. Let AP^ and AP^ be a pair of conjugate diameters of the 1, 6 and d' the angles these diameters make ^'. Let ellipse I + |: Avith the axis of the ellipse. Then tan 6 tan 6' I'llOPKUTIES OF THE CONIC SECTIONS 129 (.I'l, ?/i) be the coordinates, ^i the eccentric angle of J\; {.i:,,y.,) tlie coordinates, <f>.2 the eccentric angle of F2. Then tan e _?/, _ h sin •i'l a cos ^1 tai X., _ 6 sin <^o <<- cos </)j tan e tan 0' h- sin <^i sin <)!)2 a- cos </>! cos ^2 </>itan^2 - ^'l (r Hence tan c^, tan <^, = - 1 and c/,, and <^. differ l.y 00°; that is, the eccentric angles of the extremities of a pair of conjngate diameters of the ellipse differ by 90°. Call the lengths of the semi-conjugate diameters a, and />,. Then a{ = .c,- + v/f = or cos'' <^i + Ir sin- ^i, h{ = a- cos- (f)-. + b'' sin^ (f>2 = «^ sin^ ^1 + h- cos^ ^1, since cj^. = 90° + <^i. T.y addition ac + b^' = (r + b-; that is, the sum of the squares of any pair of conjugate diameters of the ellipse equals the sum of the squares of the axes. The conjugate diameters are of equal length when a- cos- <i> -\-b- sin- cji = a- sin' <f) + h- cos^ cf> ; that is, when tan- <^ = 1, tan </> = ± 1, <^ = 45° or 13")°. The cipiationsof the cc^ual conjugate diameters are y = ±~x, and their length Vw(u' + b'-). The area of the parallelogram circumscribed about the K 130 .1 NAL VTIC G EOMETll Y ellipse with its sides parallel to a pair of conjugate diameters is 4 6' • AN. The equation of the tangent to the el- lipse at (x', y') is xx' ?///' _ ^ The point (x', y') is the same as (a cos ^j, h sin <^i), and the tangent may be written X cos <^i y sin c^j 1. The length of the perpendicular from the origin to this tangent is AJS — / cos- </>! sin-</)| \ - ^1 1^ a? ^ U^ ) Hence 4 6' • AN= 4a6; that is, the area of the circumscribed parallelogram equals the area of the rectangle on the axes. Art. 68. — Eccentric Angle of the Hyperbola On the transverse axis of the hyperbola describe a circle. Through the foot of the ordinate of any point (x, y) of the hyperbola draw a tangent to this circle ; the angle made by the radius to the point of tangency and the axis of the hyperbola is called the eccentric angle of the point {x, y). From the hgure x = a • sec ^ and, since 7.2 y- — _^ (a- — X-), y = h • tan 4>. Let Al\ and AP-. be a pair of conjugate diameters of the I'llOl'EliTlES OF THE CONIC SECTIONS 131 hyperlxtla 1; (.i\,i/i) the coordinates, c/), the eeeeutric angle of the point I\ ; 6 and 0' the angles included by the conjugate diameters and the axes of the hyperbola. Then b tan <^, tan .Tj (( sec </>! Since tan 6 tan $' = , tane'= r a sm (pi Hence the equations of the conjugate diameters are y = ^ point of intersection of ?/ = x,y = a sm X- dtl Po, th -1, i a sin ^1 cr 0- (((tan^i, 6secc^i). Therefore APi — Ui' = a- sec"^ cl> + b- tmr 4>, AFi = bi^ = a^ tan^ <^i + b^ sec- <^i. By subtraction a^- — bc = a^ — b-; that is, the difference between the squares of any pair of conjugate diameters of the hyperbola equals the differ- ence of the squares of the axes. The area of the parallelogram whose sides are tangents to a pair of conjugate hyperbolas at the extremities of a pair of con- jugate diameters is Aby AN. The equation of the tangent to x^ y- i , , , 7 i. , \ • sec <f>. tan Aj , ^ mv,r> — — ^=1 at (a sec cb^, b tan <ii) is ^.r --^y = 1. ihe a^ b'^ a b perpendicular from tlie origin to this tangent is AN: + tan- </>, •a- b'- Hence the area of the parallelogram equals iab; that is, the area of the rectangle on the axes. sec<^..^. a tan (^, 6 '' - sec </,, ..+-^.. a tanc^,_^ a sec <^i _ b -^ — tan <^i a ■x + '-^^^-^.y 132 ANALYTIC GEOMETRY The equations of the sides of the parallelogram are 1, (1) 1, (2) 1, (3) - 1. (4) Making these equations simultaneous and combining by addi- tion or subtraction, it is found that the vertices of the parallelo- gram lie in the asymptotes y = ± -x. Problems. — 1. Find the area of the ellipse whose axes are 8 and 6. 2. What are the eccentric angles of the vertices of the ellipse ? of the ends of the focal ordinate to the transverse axis ? 3. The extremity of a diameter of the ellipse — -|-f-= 1 is (xi, yi), a-' 0^ the extremity of the conjugate diameter (x2, 2/2)- Find X2 and 2/2 in terms of Xi and yi. 4. Solve the same problem for the hyperbola. 5. In the hyperbola whose axes are 10 and the length of a diameter is 15. Find the length of the conjugate diameter. 6. Find the lengths of the equal conjugate diameters of the ellipse whose axes are 12 and 8. Also the equation of this ellipse referred to its equal conjugate diameters. CHAPTER X SECOND DEGKEE EQUATION AuT. 69. — Locus OF Second Deouee Equation Write the general second degree equation in two variables in the form (ur + 2 bx!f + qf + 2 dx + 2 ey + / = 0. (1) The problem is to determine the geometric iigure represented by this equation when interpreted with respect to the rectangu- lar axes X, Y. The equation of this geometric figure when referred to axes Xj, \\, parallel to X, Y, with origin at (a-o, y^, becomes aa;/ + 2 bx,y, + cy,- + 2 {ax, + hy^ + d)x, + 2 {bx, + ry, + e)y, + ax,' + 2 bx„yo + cy,' + 2 dx, + 2 cy, +f= 0. (2) The geometric figure is symmetrical with respect to the new origin (a^o, y^ if the coefficients of the terms in the first powers of the variables in equation (2) are zero. The coordinates of the center of symmetry of the figure are therefore determined by the equations ax, + by, + fZ = 0, bx, + c?/o + e = 0. Whence ^ eb - cd^ ^ db - ae rj.^^ center is a determinate finite ac — b- ac — ¥ point only when ac — b' ^ 0. Suppose ac — b^ =^ 0. The absolute term of ecpiation (2) be- comes ax,- + 2 6a-ov/o + c?/o' + 2 dx, -\-2eyo+f = Xo(axo + by, + (Z) + 2/o (c?/o + ^.I'o + <')+ dx, + ey, + f , , , ^ acf+2bde-ae--cd'-fb- = dx, + c!/o + / = -"^^^^ TV, ac — b' 133 134 ANALYTIC GEOMETRY Writing the last expression , equation (2) becomes ac — b- axf + 2 bx,y, + c?/f + —^^ = 0, (3) ac — c>- or axj- + 2 6a;i?/i + ci/f + A; = 0, (4) where k — dx,^ + e^y,, + ./". If A = 0, equation (3) becomes ax{' + 2 bx,ii, + ci/f = 0, (5) which determines two values real or imaginary for •— ; that is, the equation resolves into two linear equations, and hence represents two straight lines. An e(|uation which resolves into lower degree equations is called reducible, and the function of the coefficients, A, whose vanishing makes this resolution pos- sible, is called the discriminant of the equation. Turn the axes Xj, Yi about the origin (xq, ?/(,) through an angle 6. Equation (4) becomes (a cos^ e + c sin- ^ + 2 & sin ^ cos 6)x.f + (a sin- 6 -j-c cos^ 6 — 2b sin 6 cos 6)yi + 2 { (c - a) sin ^ cos ^ + 5(cos- 6 - sin- 0) I x.fli. + fc = 0. Determine 6 by equating to zero the coefficient of x^^^ whence tan 2 (9 = '^ ^ . Writing the res.ulting equation a — c 3Ixi + Ny^^ + k = Q, it follows that M+ N= a + c, il/- JV=(a - c)cos(2 ^)+ 2 6 sin(2^). From tan (2^)=-^, sin (2^) = — -, cos (2 6) = '^ — ^ -• \^b'+{a-c)X' Therefore, M-\-N=^a + c, 3/- iV^= ^6' +(« - c)-Ss and MN= ac — b-. Now the equation 3fx.f + Ny-r + A' = repre- sents an ellipse referred to its axes when M and N have like signs, an hyperbola referred to its axes when 31 and N have SECOND DEGREE EQUATION 135 unlike signs. Hence the second degree equation represents an ellipse when ac — b'^> 0, an hyperbola when ac — b^ < 0. tan (2^)= ^ determines two values for 2 $, and the radi- a — c cal \-ih- +(a — c)-J ^ has the double sign. To resolve the ambi- guity take 2 6 less than 180°, which makes sin (2 6) positive, and requires that the sign of the radical be the same as the sign of b. When a — c and the radical have the same sign, cos (2 9) is posi- tive and 2 6 is less than 90° ; when a — c and the radical have different signs, cos (2^) is negative and 2 6 is greater than 90°. The ambiguity may be resolved and the squares of the semi- axes calculated in this manner. The equation tan (2 6*) = -^^ — -, ■written "^ = '^ , determines two values for tan 0. 1 — tan- a — c Call these values tan 6i and tan Oo, and let 0^ locate the Xa-axis, $2 the Fg-axis. In the equation axi^ + 2 bxiy^ + cj/f +k = 0, substitute Xj = r cos 0, ?/i = r sin 0, and solve for ?-l There re- sults r' = -1c 'i- + t&^^'0 Calling the values of r a + 2 & tan 6^ -f c tan- corresponding to tan Oi and tan 6.^ respectively r^^ and ?•2^ the equation of the ellipse or hyperbola referred to the axes X2, Y^, IS ^,-|---^;=l. rr ?2- When oc — b- = 0, the general ecjuation becomes ax- -f 2 a)(^xy -f cf + 2dx + 2 e>j + / = 0, which may be written {a^x+chjf+2 dx-\-2 e//+f=(). Trans- form to rectangular axes with a-.c + c-y = for X-axis, the origin unchanged. Then tan 6=~— and sin 6 = ~ ^^' ^ , cos 6 = — ^— ^• c' (a + cY (a + c)-^ The transformation formulas become (a + c)'^ (a + c)^ 136 ANALYTIC GEOMETllY The transformed equation is 2,0 ft-^ + c^e c) ft-e - ckl _ f Vi + -^ T y^ — ' r ^1 I' {a+cy {a + cY {a + cy which may be written in the form 0/.-„y = 2 "'''~''y (.r,-m), the equation of a paraboLa whose parameter is 2 — ^, and (« + cy whose vertex referred to the axes Xj, Y^, is {m, n). The condition cfc — lr = causes the center (a'o, ?/o) of the conic section to go to infinity. Hence the parabola may be regarded as an ellipse or hyperbola with center at infinity. When the discriminant A also equals zero, the parabola be- comes two straight lines intersecting at infinity ; that is, two parallel straight lines. It is now seen that every second degree equation in two variables interpreted in rectangular coordinates represents some variety of conic section.* Problems. — Determine the variety, magnitude, and position of the conic sections represented by the following equations : 1. 14 x2 - 4 xy + 11 ?/2 - 44 X - 58 2/ + 71 = 0. ac — h'=-\- 150, therefore the equation represents an ellipse. The center is determined by the equations 14 xo - 2 2/0 - 22 = 0, - 2 xo + 11 yo - 29 = 0, * The three varieties of curves of the second order are plane sections of a right circular cone, which is for this reason called a cone of the second order. When the conic section becomes two parallel straight lines, the cone becomes a cylinder. Newton (1642-1727) discovered that the curves of the third order arc plane sections of five cones which have for bases the curves 21-25 on page 44. Pliicker (1801-18G8) showed that curves of the third order have 219 varieties. SECOND DEGUEE EQUATIOy 137 to be the point (2, 3). k = dxo + ei/o +/ mined in direction by tan (2 0) = — j, wlience 2 tan"- ^ - 3 tan 6-2 = 0, tan = 2 or - J. M+N= 25, 3IN = 150. If the X-axis corresponds to tan 6 = 2, M — Xniust have the same sign as b. Tiicrefore M _ ,Y = _ 5, .1/ = 10, .V = 15. The equation of tlie ellipse 10 a-2- -h 15 2/2- = 60, The axes are deter- 2/2- FiG. 125. 2. x:^ - 3 xij + 2/- + 10 X - 10 2/ + 21 = 0. ac — b" = ~:l, therefore the equation represents an hyperbola. The center, determined by the equations Xo — % ijo + 5 = 0, — ^ xo + 2/0 — 5=0, is (— 2, 2). A; = fZ.ro + eyo+f=+ \. The axes are determined in direc- tion by tan (2 0) =00, whence 0i=45°, 62 -IS^''- By substituting in r^- = -k "^tJ^^ , ,,. = 2, r^ = -l rt + 2 ?* tan + c tan- The equation of the hyperbola referred to its own axes is I x- — ly- = I. Fm. 127 3. 9 a:2 - 21 xij + 16 2/2 - 18 x - 101 y + 19 = 0. rtc — 62 — 0, therefore the equation represents a parabola. Write the equation in the form (3x-42/)2-18x-101 2/-M9=0. Take 3.x-42/ = as X-axis of a rectangular system of coordinates, the origin unchanged. Then tan = ?, sin0=i?, cos0 = v;, and the transformation formulas 138 ANALYTIC GEOMETRY become x = '^^^~^^S y = §_^lJiAll. The transformed equation is 5 5 25?/i2-75a-i-70?/o+19 = 0, wliicli may be written (?/i-|)2=3 (xi + g). Hence the parameter of the parabola is 3, the vertex referred to the new axes ( — I , I) . 4. 2/2 + 2 a;y + 3 x2 - 4 X = 0. 5. y^ + 2 xy - 3 oc^ - 4 x - 0. 6. ?/ -2xy + x^ + x = 0. 7. y^ -2 xij + 2 = 0. 8. ?/ + 4 x?/ + 4 x2 - 4 = 0. 9. 3 x2 + 2 xy + 3y'^= 8. 10. 4 x2 - 4 x?/ + 2/2 - 12 X + 6 ?/ + 9 = 0. 11. x^ — xy — 6 2/2 = 6. 12. x^ + xy + y-^ + x + y = 1. 13. 3 x2 + 4 xy + 2/2 - 3 X - 2 ?/ + 21 = 0. 14. 5 x2 + 4 X2/ + 2/"^ — 5 X — 3 2/ — 19 = 0. 15. 4 x2 + 4 X2/ + 2/- - 5 X - 2 2/ - 10 = 0. Art. 70. — Second Degree Equation in Oblique Coordinates To determine the locus represented by ax' + 2 hxy + c/ + 2 dx + 2 c?/ + /= 0, (1) when interpreted in oblique axes including an angle ft, let a'x" + 2 6 'x'y' + cY' + 2 rt'-^'' + 2e'y'+f' = (2) be the result obtained by transforming the given equation to rectangular axes, the origin unchanged. Since (x, y) repre- sents any point P referred to the oblique axes,- and {x\ y') the same point referred to rectangular axes, the expressions ^ + ?/ + 2 xy cos /3 and x'- + y'^ are each the square of the distance from P to the origin. Hence x^ + ?/- + 2 xy cos ^ = x'^ + 7j'\ (3) By hypothesis ax' + 2 &.^7/ + c//- = «'•'«'- + 2 6'.);'.v' + c^/'-. (4) Multiply the identity (3) by X and add the product to (4). There results the identity (a + X)x^ + 2(b + \ cos /3) xy + (c + A) ?/^ = (a' + X) x'2 + 2 b'x'y' + (c' + X) y". SECOND DEGREE EQUATION 130 Now any value of A wliicli makes the left-hand member of this identity a perfect square must also make tlie ri,L,^ht-han(l memlier a perfect square. Tlic left-hand mcialjor is a perfect , n> + X cos BV <■ + X. S(iuare when / — ^ = — — , \ a+X J a+X , , , • , . .> , a +c — 2 6 cos 8 ^ , ac — b'- ^ tliat IS, when X- -\ — — :-— ^^ X -\ ; = 0. sin^ /3 sin- /3 The riL,dit-hand nieniher is a perfect square when A- + 0-t' + b')X + a'c' - b'- = 0. Since these equations determine the same values for X, a'e'-b'-' = '-^^^^. sin- /5 Therefore ac — b^ is greater than zero when a'c' — b'- is greater than zero. When a'c' — b''^ > 0, equation (2) represents an ellipse when interpreted in rectangular coordinates. Conse- quently when ac — &- > equation (1) represents an ellipse when interpreted in oblique coordinates. In like manner it follows that equation (1) interpreted in oblique coordinates represents an hyperbola when ac — Z^- < 0, a parabola when ac - 62 = 0. Problems. — 1. Two vertices of a trianc;le move along two intersecting straight lines. Find the curve traced by the third vertex. From the figure are obtained the pro- portionsof y = ^'^Cg + °), /y b sin oj X _ sin (e + CO - p) a sin w whence 140 ANALYTIC GEOMETRY Substituting in sin^ d + cos- ^ = 1, there results af _ 2 sin (a - ;3 + co) _^ tf ^ sin'-^ (a + ^ - w) ^ a'^ " ah Ir sin- w tlie equation of an ellipse. 2. Find the envelope of a straight line which moves in such a manner that the sum of its intercepts on two intersecting straight lines is constant. Let - + - = 1 be the moving straight a b line, then must a -i- b = c, where c is a constant. The equation of the straight line becomes - -| — — 1, which may a c — a be written a^ -\-(y — x — c)a = ex. The equation determines for every point P(x, y) two values of a, to which cor- respond two lines of the system inter- secting at (x, ?/). When these two values of a become equal, the point (;*•, y) becomes the intersection of consecutive positions of the line; that is, a point of the envelope of the line. Hence the point (x, y) of the envelope must satisfy the condition that the equation in a has equal roots. The equation of the envelope is therefore (y — X — c)2 + 4 c.^ = 0, which reduces to ?/2 '- 2 xy + xr — 2 cy + 2 ex + C2 = and represents a parabola. This problem furnishes method frequently used to construct a parabola tan- gent to two given straight lines at points equidistant from their intersec- tion. Mark on the lines starting at their intersection the equidistant points 1, 2, 3, 4, 5, G, 7, 8, ••■, -1, -2, -3, -4, -5, - G, -7, -8, •••. SECOND DEGREE EQUATION 141 If the given points are + 5 on one line and + 5 on the other, the straight lines joining the points of the given lines the sum of whose marks is + 5 envelop the pai-abola required. 3. Through a fixed point a system of straight lines is drawn. Find the locus of the middle points of the segments of tliese lines includid by the axes of reference. 4. Find the envelope of a straight line of constant lungtli whose ex- tremities slide in two fixed intersecting straight lines. Art. 71. — Conic Section through Five Points Let (a-i, ?/i), (.«,,, ?/,), (a;,,, ?/..j), {x^, y^) be four points of which no three are in the same straight line. Let a = be the straight line through (x^, y{), (x.,, yS)] b = the line through (.i\,, y.^, (xs, 2/3) ; c = the line through (a%, y.), (x^, y^ ; d = the line through (a-4, 2/4), O^'i, Z/i)- The equation ac-\-'kbd=(), where k is an arbitrary constant, represents a conic section through the four points. For, since a, b, c, d are linear, the equa- tion ac + kbd = is of the second degree, and must therefore represent a conic section. The equation is satisfied by a = and b = 0, condi- tions which determine the point (ic^, 3/2) ; by a = and d — 0, determining the point (xi, y^); by c = 0, b = 0, determining (x.j, 1/3) ; by c = 0, d = 0, determining (a;^, y^). Since k is arbi- trary, ac + kbd = represents any one of an infinite number of conic sections through the four given points. If the conic section is required to pass through a fifth point (x'5, 2/5) not in the same straight line with any two of the four points {xi, yy), (x.,, y^, {x^, y^), (x^, y^), the substitution of the coordinates of (x^, 2/5) in ac + kbd = determines a single value 142 ANALYTIC GEOMETRY for Jc. Therefore five points of wliich no three lie in the same straight line completely determine a conic section. Problems. — Find the equations of conic sections tlirough the five points. 1. (1,2), (3,5), (-1,4), (-3, -1), (-4, 3). The equations of the sides of the quadrilateral whose vertices are the first four points are a = Sx — 2y + 1 =0, b = x — 4tj + 17 = 0, c = 5 X — 2 ?/ + 13 = 0, d = Sx — 4y + 5 = 0. The equation of a conic section through these four points is therefore (3x - 2?/ + l)(5x - 2?/ + 13)+ i-(a; - 4^ + 17)(3x - 4?/ + 5) = 0. Substituting the coordinates of the fifth point (—4, 3), k = W- The equation of the conic section through the five points is 79 x2 - 320 xy + 301 ?/2 + noi x - 1665 y + 1580 = 0. 2. (2, 3), (0, 4), (- 1, 5), (- 2, - 1), (1, - 2). 3. (1, 3), (4, - G), (0, 0), (9, - 9), (16, 12). 4. (- 4, - 2), (2, 1), (-6, 3), (0, 0), (2, - 1). 5. (- i, - i), (2, 1), (f, 2), (-J, - 3), (I,- I). 6. (3, V5), (-2, 0), (-4, - Vl2), (3, - V5) (2, 0). 7. (1,2), (2, 1), (3, -2), (0,4), (3,0). 8. (2,3), (-2,3), (4,1), (1,3), (0,0). Art. 72. — Conic Sections Tangent to Given Lines Let « = and b = represent two straight lines intersected hy the straight line c = 0. The equation ab — kc' = repre- sents a conic section tan- gent to the lines a = 0, 6 = at the points of inter- section of c — 0. For the equation ab — kc' = is of the second degree, and the points of intersection of the line a = with ab — kc^ = ^'"^ ''^^' coincide at the point of in- tersection of the lines a = 0, c = 0, which makes a = tangent to the conic section. For a like reason b = is tangent to SECOND DEGliEK EQUATION 143 ab — kc- — 0. Since k is arbitrary, an infinite number of conic sections can be drawn tangent to the given lines at the given points. The equation of a conic section tangent to the lines x = (), y = lit the i)oints (a, 0), (0, h) is a b Kxy = 0. (1) The points of intersection of this conic section and the line MK - + ^ = 1, lie in the m 11 locus of the equation i? + f_5_?'Y = A>,. (2) a b m nj This last equation is homo- geneous of the second degree, and hence represents two straight lines from the origin through the points of inter- r a" , 2/ 1 A Fig. 138. section of [--—1 = VI n and (- + - — 1) — Kxy = 0. The straight lines represented by equation (2) coincide, and - + - — 1 = is tangent to when is a perfect square ; that is, whei a b ' I b m - ' ^ \rt mj \b 71 J I \a mj \b nj 2 S 144 ANALYTIC GEOMETRY whence K= if^- lY/^i - ^Y (3) \(i iiij \h nj Similarly, — + - — 1 = is tangent to (1) when Equations (3) and (4) determine the values of - and - in terms of the arbitrary constant K, which shows that an infinite num- ber of conic sections can be drawn tangent to four straight lines no three of which pass through a common point. If [- — = 1 is also tangent to the conic section represented by equation (1), A-=4fl^i)fl-l> (5) Equations (3), (4), (5) determine -, -,aud A" uniquely, proving a h that only one conic section can be drawn tangent to five straight lines no three of which pass through a common point. This proposition is the reciprocal of the j)roposition of Art. 71 and might have been demonstrated by the method of reciprocal polars. Problems. — 1 . Find the equation of the parabola tangent to two straight lines including an angle of 60° at points whose distances from their point of intersection are 2 and 4. 2. Find the equation of the conic section tangent to two straight lines including an angle of 45° at (3, 0), (5, 0), and containing the point (7, 8), the given straight lines being the axes of reference. Art. 73, — Similar Coxic Sections The points P{x, y) and P^iinx, my) lie in the same straight line through the origin 0, and 0I\ = m • OP. The distance between any two positions of P^, (mx', my'), (mx", my") is m SECOND DEGREE EQUATION 145 times tlie distance between tlie correspondiut,^ positions of P,{x\y'),{x'\y"). For { (m.f' - mx")- + {my' - myy\ ^ = m \ (^x' - x'J + (//' - y"f\ i Representing the point P by {x, y), the point P^ ])y (X, Y), when {x, y) traces a geometric figure, the point (A", Y) traces a figure to scale m times as large. The effect of tlie substitution X Y . X——, ?/ = — is therefore simijly to change the scale of tlu; drawing. Figures thus related are said to be similar. When the two equations /(;r, y) = 0, fi~-, — j = are interpreted in the same axes, their loci are similar and similarly placed ; when interpreted in differ- ^ ent axes but including the same angle, the loci are similar. Ellipses similar to — + -i- = 1 are repre- sented by Al + ^ 1. All ellipses of a similar system have the same ec- centricity, for «-^ r - 7;* m-a^ ^ _ a- - -b\ Y2 In like manner, all hyperbolas of a similar system -^ • ??i^a^ m'b- have the same eccentricity. The parabolas similar to y^ = 2x)x are represented by the equation Y^ = 22)mX. Taking as corresponding points J'(^, y), Pi(mx, my), !'.,(- mx, 'Z/), 146 ANALYTIC GEOMETRY the figure traced by Pi is similar to that traced by P, the figure traced by P2 is symmetrical to that traced by P^. The change of scale of a drawing may be effected mechani- cally by means of an instrument called the pantograph, which consists of four rods jointed together in such a manner as to form a parallelogram ABOC with sides of constant length, but whose angles may be changed with perfect freedom. On the rods AB and AC fix two points P and Pj in a straight line with 0. If the point is fixed in the plane, and the point P is made to take any new position P', and the cor- responding position of Pj is P/, the points P', 0, P/ in Fig. 134 are always in a straight line, the triangles Pi CO and Pi'A'P' are similar and ' P hence qpi OP' P,'C CA' m CA a constant which may be denoted by m. Taking as origin of a system of rectangular coordinates, if Pis (a-, 2/), Pi is (— mx, — my). If the point P is fixed in the plane and taken as origin of a system of rectangular coordi- nates, if the point is (x, y), the point Pj is {mx, my). There- fore, if the point is made to trace any locus, the point P, traces a similar figure to a scale m times as large. The equation Art. 74. — Confocal Conic Sections 1, If (1) a^ -f- A 6' -f A where a? > W represents an ellipse when A > — 6-, an hyperbola when — a^ < A < — &^, an imaginary locus when A < — al The SECOND DEGREE EQUATION 147 distance from focus to center of the ellii)se,s and liyperbolas represented by eqiiation (1) is \(i' + \ — Ir — \\'- = {tC- — U')-. Hence equation (1) when interpreted for different values of A. in the same rectangular axes represents ellipses and hyperbolas having common foci ; that is, a system of confocal conic sections. Through every point {x', y') of the plane there passes one ellipse and one hyperbola of the confocal system a^ + A h' + X For the conic sections passing through {x\ y') corresptmd to the values of X satisfying the equation .. 1. (2) a- + X b- + X This function of A, .^1^ + ^ 1, a- + A h- + X is negative when A = + co, positive just before A be- comes — 6-, negative when A is just less than — li' and again positive when A is just greater than — cr. Hence equation (2) deter- mines for A two values, one between + x and — h'-, the other between — Jr and — a'. The ellipse and hyperl)o]a of the confocal system (t- -H A //- -f A til rough the point (.c', //') intersect at right angles. Let Ai and A. be the values of A satisfying the equation X' + a- + A h- + X = 1. Then + a- + Ai &' -f Ai a- + X., U- + A. 148 ANALYTIC GEOMETRY represent ellipse and hyperbola tlirougli (x-', y'). The tan- gents to this ellipse and hyperbola at (x', y') are iiy a" + Ai h- + Ai xx' yy' a^ + A2 b^ + A2 (1) (2) From the equations + r7^^=l. -.-^^ + a^ + Ai b- + \i a- + \2 b'' + X., is obtained by subtraction (cr + Ai) (a' + \,) {b' + Aj) (6^ + A.) which is the condition of perpendicularity of tangents (1) and (2). Since through every point in the plane there passes one ellipse and one hyperbola of the confocal system, the point of the plane is determined by specifying the ellipse and hyperbola in which the point lies. This leads to a system of elliptic coordi- nates. If heat flows into an infinite plane disc along a finite straight line at a uniform rate, when the heat conditions have become permanent, the isothermal lines are the ellipses, the lines of flow of heat the hyperbolas of the confocal system. The same is true if instead of heat any fluid flows over the disc, or if an electric or magnetic disturbance enters along the straight line. CHAPTER XI LINE OOOKDINATES Art. 75. — Coordinates of a Straight Line If the equation of a straight line is written in the form ux -\-vy + 1 = 0, u and v are the negative reciprocals of the intercepts of the line on the axes. To every pair of values of H and V there corresponds one straight line, and conversely; that is, there is a " one-to-one correspondence " between the symbol («, v) and the straight lines of the plane, u and v are called line coordinates.* If {u, V) is fixed, the equation ux -{- vy -\- 1 = expresses the condition that the point {x, y) lies in the straight line («, v). The system of points on a straight line is called a range of points. Hence a first degree point equation represents a range of points and determines a straight line. If (x, y) is fixed, ux + vy + 1 = expresses the condition that the line (», v) passes through the point (x, y). The system of lines through a point is called a x^encil of rays. Hence a first degree line equation represents a pencil of rays and determines a point. The equations ?/.ri + vy^ -|- 1 = 0, nx^ + vy., + 1=0 determine the points {x^, y,), {x^, y^ respectively. \ \\^' ^ '\+^) represents for each value of X one point of the line through (a^i, ?/i), (.i\,, 7/2). X is the ratio of the segments into Avhich the point corresponding to X divides the finite line from (.r„ y,) to * riiickcr in Germany and Cliasles in France developed the use of line coordinates at about the same time (1829). 149 150 ANALYTIC GEOMETUr {x2, 2/2)- There is a " one-to-one correspondence " between A and tlie points of the line throngh (a-j, y^, (.i-^, y^. 1-f-A 1 + A which reduces to {ux^ + vy^ + 1) -|- A {ax^ + vy2 + 1) = 0, is the line equation of the point A. Denoting m.Ti + vy^ -f 1 by L, UX2 + v?/2 + 1 by 3f, L + \3I— represents the range of points determined hy L = 0, 3£— 0. The rays of the pencil determined by the lines UiX + Viy + 1 = 0, ti.^x -\- v^y + 1 = are represented by the equation (u,x -f v,y + 1) + A (n.x + v-^y + 1) = 0, which may be written '^!l±^x + !!l±J^ + 1 = 0. •^ 1+A 1 + A Hence (n, + Xv v,±M:: V 1+A 1+A are the lines of the pencil determined by (11^, Vi), (xi2, v^). There is a " one-to-one correspondence " between A and the rays of the pencil. Denoting UiX -[■ Viy -\-l by P, u^x + Vjy + 1 by Q, P + AQ = represents the rays of the pencil determined by P=0, Q = 0. ' Problems. — 1. Construct the lines (4, 1); (- 2, 5); (- i, - J). 2. Construct the pencil represented by 3 ?i - 2 y ^- 1 = 0. 3. Construct the range represented by 2 .x — 3 ?/ -f 1 = 0. 4. Locate the point determined by 4 ?( -^ 5 v -|- 1 = 0. 5. Draw the line determined by 3x — 5 ?/ + 1 = 0. 6. Write the equation of the range of points determined by 2ti-Sv + l = 0, i?t-f^w-fl=0. 7. Write the equation of the pencil of rays determined by 2x-Sy + l=0, J a;-f i?/ + l = 0. LINE COORDINATES 151 Akt. 76. — Line Equatioxs of the Conic Sections The equation of the tangent to the ellipse ^—■^•-L^—l at a- b- (.T,,, y/o) is ^' + "''•'^ = 1. Comparing the equation of the tangent a- U- with nx + (•>) + 1 = it is seen that the line coordinates of the tangent are a = — ' ", i' = — ^", whence a-,, = — u-u, ?/„ = — Irv. If the point of tangency (.Tn, ?/„) generates the ellipse "^ + ii — •'-> the tangent {ii, v) envelopes the ellipse. Hence the line equa- tion of the ellipse, when the reference axes are the axes of the ellipse, is a-ir + b'-v^ — 1. Problems. — 1. Show that the Hue equation of the circle x"^ + y- — r" is iC- + y- = — r- 2. Show that the Hue eiiuatiou of the hyperbola 1 is d-xi- — b-v- = 1. a- h- 3. Show that the Hue equatiou of the parabola y"^ = 2px is pv'^ = 2 u. Construct the euvelopes of the equations 4. - + - = - 5. G. ifi-\- v"^ = ' 8. 9 1*2 - 4 u2 = i. U V 5. uv-\. 7. 9!t2 + 4tj2-i, 9. 8i;2-u = 0. Art. 77. — Cross-ratio of Four Points The double ratio -^--. — ^ is called the cross-ratio of the four CB I)B points A, B, C, D, and is denoted by the symbol (ABCB). If the point A is denoted by i = 0, the point B by J/= 0, > ^ »< ^ the points C and D respec- ^'"- '"■ tively by L + XiM=Q and L + XoM^O, it follows that ^ = Ai, ^ = X,,, and (ABCD) = ^- Take any four points of CB JJB Ao 152 ANALYTIC GEOMETRY tlie range L + X3/= corresponding to Aj, X.,, A3, A4, and repre- sent L + Aji)/ by Li, L + LM by i»/i, whence /v + A,,.1/ is represented by L,-^^^M„ L + A, J/ by L, - ^i^il/-i. Ao — A3 Ao — A4 The four points corresponding to Aj, A., A3, A4 are represented by the equations L, = 0, 3/1 = 0, A - ^^^^^^3/, = 0, A - ^i-^^1A = 0, A2 — A3 A2 — A4 and their cross-ratio is ^^ ~ ^ -~ ^ Since the four points ^2 — X^Xi — A4 Ai, Ao, Ag, A4 can be arranged in 24 different ways, the cross- ratio of four points takes 24 different forms, but these 24 different forms are seen to have only six different vaUies. i-f A3/=0, L' + XM' = represent two ranges of points. By making the point of one range determined by a value of A correspond to the point of the other range determined by the same value of A, a " one-to-one correspondence " is established between the points of the two ranges, and the cross-ratio of any four points of one range equals the cross-ratio of the corre- sponding four points of the second range. Such ranges are called projective. Art. 78. — Second Degkke Line Equations Remembering that each of the equations L + \M=0, i'-fAJ/' = for any value of A represents the entire pencil of rays through the point of the range corresponding to A, it is evident that the equation LM' — L'M= 0, obtained by eliminating A between L + \M=^ 0, L' + AIT' = 0, represents the system of lines join- ing the corresponding points of the two projective point ranges. This equation is a second degree line equation, and it becomes necessary to determine the locus enveloped by the lines repre- sented by the equation. LINE COOliDINATES 153 Let nx + vy + 1 = represent any i^oint (x, y) of the plane. Writing the values of L, M, L', 31' in full, the elimination of u and V from the equations i(x + t'^ + 1 = 0, iixi + viji + 1 + X(ux2 + vyo + 1) = 0? ux' + vy' + 1 + X(ux" + vy" + 1) = 0, determines a quadratic equation in X with coefficients of the first degree in (x, ?/), GX^ + HX + K= 0. To the two values of X which satisfy this equation there correspond the tangents from (x, y) to the envelope of LM' — VM — 0. When these tangents coincide, the point (.r, ?/) lies on the envelope. 4 IP - GK^ causes the coincidence of the tangents, and is therefore the point equation of the envelope. The point equation being of the second degree, the envelope is a conic section. The degree of a line equation denotes the number of tan- gents that can be drawn from any point in the plane to the curve represented by the equation, and is called the class of the curve. Art. 79. — Cross-ratio of a Pencil of Four Eays Let a pencil of four rays, P=0, Q = 0, P-|-X,Q = 0, P-fA,Q = 0, be cut by any transversal in the four points A, B, C, D. j^ is the common altitude of the triangles whose common vertex is 0, and whose bases lie in the transversal. Then i> • CA = OA - OC • sin COA, p • DA = OA ■ OD ■ sin DO A, p -03=00 -OB- sin COB, p - DB = OD • OB ■ sin DOB, and (ABCD) = ^"^ ^^^^ - ^"^^^^^ . This double sine ratio is ^ ^ sin COB sin DOB 154 ANALYTIC GEOMETRY called the cross-ratio of the pencil of four rays. It is evident that central projection does not alter the cross-ratio of four points in a straight 'PA \ "^^ line. Writing the equation P-|-XQ = in the com- Q=0 plete form u^x + i\y + 1 FiG' 138. and this in the form the factor ^ll^ll+i^iX is seen to be the negative ratio of the distances from any point of the line P + \Q = to the lines P = 0, Q = 0. Hence (7a_ sinC0^1 __. Da' ^ ^mPOA ^ ^ Cb sin COB " Db' sin DOB " ^ sinCO^^sin^DOA^Xi^^j^g cross-ratio of the four rays sin COB sin X)0i5 Xg P^O, Q = 0, P4-AiQ = 0, P + X,Q = 0. Representing P+X,Q by Pi, P + XoQ by Q„ P-f-XgQ is represented by P^ _ k^A^ Q„ p + X.Q by P. - ^^ Qi. Xa — X3 Ao — A4 Hence the cross-ratio of the four points of the pencil P+XQ—0 , . , , - , 1 • Xi — Xi Xo — X) corresponding to Xj, Xo, X;,, A4 is -^ — • X2 — A.3 Aj^ • — A4 LINE COORDINATES 155 By making the ray of F+kQ = determined by a value of \ correspond to the ray of F' +XQ' = determined by the same value of X, a " one-to-one correspondence " is established between the rays of the two pencils, and the cross-ratio of any four rays of one pencil equals the cross-ratio of the correspond- ing four rays of the other pencil. Such pencils are called projective pencils. The equation of the locus of the points of intersection of the corresponding rays of the two projective pencils F + XQ = 0, F' -t- XQ' = is FQ' — F'Q = 0. This is a second degree point equation and represents a conic section.* Art. 80. — Construction" of Projective Ranges and Pencils If there exists a " one-to-one correspondence " between the points of two ranges, between the rays of two pencils, or be- tween the points of a range and the rays of a pencil, the ranges and pencils are projective. Let F=0, Q = 0, determining the range or pencil F-}-XQ=0, correspond to F^ = 0, Qi = 0, determining the range or pencil Pi+ A,Qi = 0, and let a "one-to-one correspondence" exist between the elements X of the first system and the elements Ai of the second system. This "one-to-one correspondence" interpreted algebraically means that Xi is a linear function of X; that is, Xj = ^Jhj±A. By hypothesis, Xi = when X = 0, ' ' cX + d ^ ^^ ^ ' and X, = CO when X = cc, hence b = 0, c = 0, and X, = - X. Let X = / and X, = /, be a third ])air of corresponding ele- * A complete projective treatment of conic sections is developed in Steiner's Theorie der Kcgelschnitte, 1800, and in Chasles' G^om^trie Sup^rieure, 1852, and in Cremona's Elements of Projective Geometry, translated from the Italian. 156 ANALYTIC GEOMETRY ments;tlien - = -, Ai = -X, and the equations of tlie systems P+XQ = 0, Pi + Ai(3i = become P + XQ = 0, IP^ + XI,Q,^0. Now the elements of ^ IF, + XIQ, = are the elements of P, + XQ, = 0, hence the systems between whose elements there exists a '^ one-to-one correspondence " are the projective systems Fm. m P-{-XQ = 0, Pi + AQ, = 0. This analysis also shows that the correspondence of three ele- ments of one system to three elements of another makes the systems projective. Projective systems are constructed geometrically, as follows : Let the points 1, 2, 3 on one straight line mm correspond to the points 1, 2, 3, respec- tively, on another straight line nn. Place the two lines with one pair of corresponding points 2, 2 in coincidence. Join the point of intersection of the lines through 1, 1 and 3, 3 with 2. Take the 3 points of intersection of lines through with mm and 7in as corresponding points, and a " one-to-one correspondence " is established between the points of the ranges mm, nn, which are therefore projective. In like manner, if three rays 1, 2, 3 of pencil m correspond to the rays 1, 2, 3, respectively, of pencil w, by placing the cor- responding rays 1, 1 in coincidence, and drawing the line 00 LINE COORDINATES 157 through the points of intersection of the corresponding rays 2, 2 and 3, 3, and taking rays from m and 7i to any point of 00 as corresponding rays, a " one-to-one correspondence " is estalv lished between the rays of the two pencils, and the pencils are projective. Art. 81. — Coxic Section through Five Points It is now possible by the aid of the ruler only to construct a conic section through five points or tangent to five lines. Take two of the given points 1, 2 as the vertices of pencils, the pairs of lines from 1 and 2 to the remaining three points 3, 4, 5, respectively, as corresponding rays of projective pencils. The 2 /^ 7 1(5)/ / ). y (7) (3^ 3 line 11 is a transversal of pencil 1, 22 of pencil 2. 0, the intersection of 51 and 32, is the vertex of a pencil of which 11 and 22 are transversals. Hence the pencils 1 and 2 arc projective, and corresponding rays are rays to the points of intersection of the rays of pencil with 11 and 22. The 158 ANALYTIC GEOMETRY intersections of these corresponding rays are points of the required conic section. Take tAvo of the five given lines 11 and 22 as bearers of point ranges on which the points of intersection of the other lines 33, 44, 55 respectively are corresponding points. The line 00 is a common transversal of the pencils (11), (22). Hence corresponding points of the projective ranges 11 and 22 are located by the intersection with 11 and 22 of lines connecting (11) and (22) respectively with any point of 00. The straight lines connecting corresponding points are tangents to the required conic section. Notice that the construction of the conic section tangent to five straight lines is the exact reciprocal of the construction of the conic section through five points. The figure formed by joining by straight lines six arbitrary points on a conic section in any order whatever is called a six- LINE COORDINATES ir)9 7-f-\!(2)\4 .''5 point Taking 1 and 5 as vertices of pencils whose correspond- ing rays are determined by the points 2, 3, 4, the points of intersection of 16 with 11 and of 5iy with 55 must lie in the same ray of the auxiliary pencil 0; that is, in any six-point of a conic section the inter- ^i section of the three pairs \ 1\ of opposite sides are in a straight line. This is Pascal's theorem.* Reciprocating Pascal's theorem, Brianchon's theo- rem is obtained. — In the figure formed by drawing tangents to a conic section at six arbitrary points in fig. u3. any order whatever (a six-side of a conic section), the straight lines joining the three pairs of opposite vertices pass through a common point.f By Pascal's theorem any number of points on a conic section through five points may be located by the aid of the ruler ; by Brianchon's theorem any number of tangents to a conic section tangent to five straight lines may be drawn by the aid of the ruler. * Discovered by Pascal, 1040. t Discovered by Brianclion, 180G. CHAPTER XII ANALYTIC GEOMETRY OP THE COMPLEX VAEIABLE Art. 82. — Graphic Represextation op the Complex Variable The expression x + iij, where x and y are real variables and i stands for V— 1, is called the complex variable, and is fre- quently represented by z. Vx' + if is called the absolute value of z and is denoted by 1 2; | or | a; + iy |. If a; + iy is represented by the point (x, y), a " one-to-one correspondence" is established be- ^x-^iy tween the complex variable x + iy j and the points of the XF-plane. [ The X-axis is called the axis of X \ reals, the F-axis the axis of imagi- ^ naries. Denoting the polar coordi- FiG. 144. nates of (x, y) by r and 6, x= r cos 6, y = r sine, and 2=x-+ «/=?• (cos ^+r sin ^), where r=Vo^+f, 6 = tan-^^. r is the absolute value, and 6 is called the anipli- X tude of the complex variable x + iy. Hence to the complex variable x + iy there corresponds a straight line determinate in length and direction. A straight line determinate in length and direction is called a vector. Hence there is a " one-to-one correspondence" between the complex variable and plane vectors. As geometric representative of the complex variable may be taken either the point (x, y) or the vector which deter- mines the position of that point with respect to the origin.* * Argand (1806) was the first to represent the complex variable by points in a plane. Gauss (1831) developed the same idea and secured for it a permanent i>lace in mathematics. 160 COMPLEX VAIIIABLE 161 Calling a liiu^ etiual in Icnii^'tli to the linear unit and laid olF from the origin along the positive direction oi' the axis of reals the uuit vector, the complex variable z — X -\- i>i — r (cos 6 + I sin 6) represents a vector obtained by multiplying the unit vector by the absolute value, then turning the resulting line about its ex- tremity at the origin through an angle equal to the amplitude of the complex variable. When the complex variable is written in the form r(cos ^ + « sin(9), r is the length of the vector, cos 6 -}- i sin 9 the turning factor. In analytic trigonometry it is proved that cos 6 + i sin 6 = c'".* Hence the complex variable r{vo9,e + i sin^)= r-c'^ whore the stretching factor (tensor) and turning factor (versor) are neatly sc[)a,ratcd. Problems. — 1. Locate the points rcpresmtud by 2 + t5; 3-i2; - 1 + i2 ; i5 ; - i4 ; - 3 - i ; - t 7 ; + i7. 2. Draw the vectors represented by 3 + i2; 1— i3; — 2+i3; - 1 -i4; - i5 ; 3 -/; 1 + /. 3. Show that e-'"^' = 1, when n is any integer. 4. Show that r • e'(9+-"'^' represents the same point for all integral values of n. 2nni 5. Locate the different points represented by e"^ for integral values of n. I(g+2n7r) 6. Locate the different points represented by 5 • e •« for integral values of n. *This relation was discovered by Eulcr (1707-1783). 162 ANALYTIC GEOMETRY Akt. 83. — Arithmetic Operations applied to Vectors The sum of two complex variables Xj + ii/i and X2 + iy2 is (.I'l + X,) + / (^1 + //,) . H euce |(a-i + iyi) + (•'^•2 + ^'^2)1 = y/{xi + x^y+ivi + 2/2)', and. the amplitude of the sum is tan"' -'' A The graphic Xj ~|~ ^2 representation shows that the vector corresponding to the sum ^-rC^ + ii/) is found by constructing the vec- ' "" / tor corresponding to Xi + i?/i ^iid. / using the extremity of this vector as origin of a set of new axes w^ ^. parallel to the first axes to con- struct the vector corresponding ^ to X2 + ^2/2• The vector from the F'G- ^■^6. origin to the end of the last vec- tor is the vector sum. The vector sum is independent of the order in which the component vectors are constructed. From the figure it is evident that I {^1 + Wi) + {^2 + iVi) I > l-^'i + m\ + \^2 + il/2\- The difference between two vectors x^ + iyi and Xo + iy^ is (x^ — x.^+ i{yi — y^. The graphic representation shows that the vector corresponding to the difference is found by construct- ing the vector corresponding to x-^ + iyi and adding to it the vector corresponding to — x.2 — iy-,- It is seen that I (x, + iy,) - {x. + iy^ | = V(a-i - x.y- + (?/i - y^, the amplitude of the difference is tan-^ -^^' ~^^- , and that the a-, - x. equality of two complex variables requires the equality of the coefficients of the real terms and the imaginary terms separately. CO^fPLEX V Alii MILE 1G: The product of two complex variables is most readily found by writing these variables in the form r^ • e''^', r.^ • e'^^. The product is i\r.2 • e''(«»+*2', showing that the absolute value of the product is the product of the absolute values of the factors and the amplitude of the product is the sum of the amplitudes of the factors. Hence, writing the complex variables in the form Ti (cos 6i + i sin ^i), n (cos 60 + i sin O.j), the product is 'V*2[cos (^1 + ^2) + ^ sill (^1 + ^2)]? which of course can be shown directly. Construct tlie vector corresponding to the multiplier r, • e'^* and join its extremity Pj to the extremity of the unit vector 01. Construct the vector corre- sponding to the midtiplicand 7-2 • e'^% and on this vector OP., as a side homologous to 01 construct a triangle OP^P similar to OPjl ; then OP is the product vec- tor. For, from the similar tri- angles 0P= Ti • r^, and the angle XOP=6i+62- The product vec- tor is therefore formed from the vector which is the multiplicand in the same manner as the vector which is the multiplier is formed from the unit vector. The product vector is indepen- dent of the order of the vector factors and can be zero only when one of the factors is zero. The quotient of two complex variables i\ • e''^', ?*2' e'^- is '"1 . e'(9i 62) . that is, the absolute value of the quotient is the quotient of the absolute values, and the amplitude of the quotient is the ampli- tude of the dividend minus the amplitude of the divisor. Construct the vectors OP, and OPo corresponding to dividend and divisor respectively, and let 01 be the unit vector. On 164 ANALYTIC GEOMETRY OPi as a side homologous to 01\ construct the triangle OI\P similar to OP^l, then OP is the quotient vector, for OP = -j and the angle XOP is ^i — $2- The quotient vector is obtained from the vector which is the dividend in the same manner as the unit vector is obtained from the vector which is the divisor. Extracting the ??i root of z = r • e'^ = r • e'f^+^'w) there results z"^ = 9-™ . e ^ "» Since n and m are integers, Avhere q is an integer and r can have any value from to m — l. <'^ + ^ Hence 2;"' = r™-e~'" " ; that is, the m root of z has m values which have the same absolute value and amplitudes differing by — beginning with — .* Problems. — 1. Add (2 + 1 5) , ( - 3 + 1 2) , (5 - i 3). 2. Find the value of (3 - i2) + (7 + i4) - (0 - i3). 3. Find absolute value and amplitude of (4-j.3) + (2 + i5)-(-3 + i4). 4. Construct (2 - i 3) x (5 + i 2) h- (4 - i 5) . 5. Find absolute value and amplitude of (10 - i 7) x (4 - iS ). 6. Find absolute value and amplitude of (15 + f8) x (5 - i2). 7. Construct (2 + 1 3)3. 9. Construct (7 + i 4) ^ 8. Construct (8 - 1 5)^. 10. Construct (9 - i 7) t. * In mechanics coplanar forces, translations, velocities, accelerations, and the moments of couples are vector quantities ; that is, quantities which are completely determined by direction and magnitude. Hence the laws of vector combination are the foundation of a complete graphic treatment of mechanics, COMPL EX VA R I A BLE 165 11. Construct the five fifth roots of unity. 12. Construct the roots of 2- — 3 2 + f) = 0. Put 2 = a; + ill. There results (x^ _ 2/2 - 3 x + 5) + t (2 xy - 3 ?/) = 0. Plot ofl - y- - 3 X + 5 = and 2xy - Sy = 0. The values of z deter- nihied by the intersections of these curves are the roots of z'^— 3 2 + 5=0. AiiT. 84. — Algkp.raic Functions of thk Complex Variable The geonietric representative of the real variable is the point system of the X-axis and the geometric representation of a function of a real variable 7/ = f(x) is the line into which this function transforms the X-axis. The geometric representative of the complex variable is the point system of the XF-plane, and the geometric representation of a function of a complex variable u + iv=f(x-\- i;/) is the system of lines into which this function transforms systems of lines in the XF-plane. ^ : 1 ' Y 1 I fi 5 ^a A rf d' ' - ' ' 2 V 1 : I 5 '—Ct \' A' d' Fio. 149. When the complex variable is written in the form x -f- ??/, it is convenient to use the systems of parallels to the X-axis and to the F-axis. Take the function iv = z-\- c, where iv stands for u 4- iv, z for x -\- vj, c for a + ih, then ?t + iv = (x -f a) + / (// + '>) and ii. = x-\- a, v = .y + h. 166 ANALYTIC GEOMETRY If in the XF-plane a point moves in a parallel to the F-axis, X is constant, and consequently u is constant. Hence the func- tion 10 — z + c transforms parallels to the F-axis into parallels to the F-axis in the f/F-plane. In like manner it is shown that lo — z + c transforms parallels to the X-axis into parallels to the {/-axis. If the variables to and z are interpreted in the same axes, the function io = z + c gives to every point of the XF-plane a motion of translation equal to the translation which carries A to c. When the complex variable is written in the form r • e'^, it is convenient to use a system of concentric circles and the system of straisfht lines throudi their common center. Take the func- tion lu — c-z, when to stands for R • e'®, z for r • e'^ and c for r' • e'« ; then R • e'® = rr' ■ e'(«+e ^ and R = rr', © = 6 + 6'. If a point in the XF-plane describes the circumference of a circle center at origin, r is constant, and consequently R is constant, and the corresponding point describes a circumference in the C/F-plane, center at origin, and radius r' times the radius of the corresponding circle in the XY-plane. If the point in the XF-plane moves in a straight line through the origin, 6 is con- stant, and consequently © is constant, and the corresponding coMrLEx V. 1 in A hle 107 p(iiiit in the CF-plane moves in a straight line through the origin. If the variables w and z are interpreted in the same axes X and Y, the function xo = c • z either stretches the XY- plane outward from the origin, or shrinks it toward the origin, according as r' is greater or less than unity, and then turns the whole plane about the origin through the angle 0'. R = The function lo 1 ^ may be written R • e'® : ■'^', whence z r A circle in the XF-planc with center at the origin is transformed into a circle in the (/F-plane with center at the origin, the radius of one circle being the reciprocal of the radius of the other. A straight line through the origin in V the XF-plane making an angle 6 with the X-axis, is trans- formed into a straight line through the origin in the t/F-plane making an angle — Avith the {7-axis. If iv and z are inter- preted in the same axes, the function lo =- is equivalent to a z transformation by reciprocal radii vectors with respect to the unit circle, and a transformation by symmetry with respect to the axis of reals. 168 ANALYTIC GEOMETRY In the equation xv = z^, or R • e'® = ?-^ • e'''^, iv is a single valued function of z, but 2 is a three-valued function of w. Since r = i2% ^ = ® + ^^^, the absolute values of the three 3 3 values of z are the same, but their amplitudes differ by 120°. The positive half of the (7-axis, ^ = 0, corresponds to the posi- tive half of the X-axis, and the lines through the origin making angles of 120° and 240° with the X-axis. The entire C/F-plane is pictured by the function iv = z^ on each of the three parts into which these lines divide the XF-plane. Art. 85. — Generalized Transcendental Functions Since z = x^ iy = r • 6'^^+""-', log z = log r + i{e + 2 mr). The equation w = log 2 may be written u + iv = log ?• -f- i (^ -f- 2 mr). Y V 277 77r i 37r 55 TT ¥ y2 A U 1 f ■'s Hence u = log r, v = ^ + 2 htt. To the circle r = constant in the XF-plane there corresponds in the C7F-plane a straight line parallel to the F-axis ; to the straight line 6 = constant in the XF-plane there corresponds in the f/F-plane a system of parallels to the i7-axis at distances of 2 tt from one anotlier ; w - is an infinite valued function of z, but 2 is a single valued func- COMPLEX VARIABLE 1G9 tion of IV. Tlie entire Xl'-plane is i)ictui'ed between any two successive parallels to the C-axis at distances of 2 tt. Writing the function ?o = sin (x + itj) in the form u -\- IV = sin X cos iij + cos x sin iij, and remembering that cosh ?/ = 1 (e" + e *) = cos «//, sinh ?/ = i(e^' - 6-")= - i sin iy, there results M -j- iv = cosh ?/ sin .r — / sinh ?/ cos x, whence ?t = cosh ?/ sin x, v = — sinh ?/ cos ^'j and sin x = cosh u cos a;= ^, cosh ?/=-r^, sinh^ = sinh >j sm x cos x sma; Substituting in sin-.v+cos-a;=l and cosh-.y-sinli-// = l, there results w V 1, — V = i- cosh-?/ ' sinh'-^y sui^x cos-iK Y 3 2 1 I/, TT H^ ,,,. T X \V4^ \ V _3___ 2 //X 1 Uax V^' / ffA\ irn 1 " Iv/ ] / yyi — W- -'^/jU These equations when x and ?/ are respectively constant rep- resent a system of confocal conic sections with the foci at (4- 1, 0), (- 1, 0). The entire system of ellipses filling up the C7F-plane is obtained by assigning to y values from + co to — oc ; the entire system of hyperbolas filling up the C/F-plane 170 ANALYTIC GEOMETRY is obtained by assigning to x values from to 2 7r. Hence v: = sin (x + iy) pictures that part of the XF-plane between two parallels to the F-axis at a distance of 2 tt from each other on the entire C/F-plane.* Problems. — 1. Show that ■?« = - transforms the system of straight z lines through a + ih, and the system of circles concentric at this point into systems of orthogonal circles. 2. Find what part of the AT-plane is transformed into the entire {/ ^''-plane by the function w = z"^. 3. Into what systems of lines does tp = cos 2; transform the parallels to the X-axis and to the F-axis ? *The geometric treatment of functions of the complex variable has been extensively developed by Riemann (1826-G6) and his school. ANALYTIC GEOMETRY OF THREE DIMENSIONS CHAPTER XIII POINT, LINE, AND PLANE IN SPACE Art. — Rectilinear Space Coordinates Through a point in space draw any three straight lines not in the same plane. The point is called the origin of coordi- nates, the lines the axes of coordinates, the planes determined by the lines taken two and two, the coordinate planes. The distance of any point P from a coordinate plane is meas- ured on a parallel to that axis which does not lie in the plane, and the direction of the point from the plane is denoted by the algebraic sign prefixed to ^^^ the number expressing the dis- ___' tance. The interpretation of these signs is indicated in the figure. If the distance and direction of the point from the yZ-plane is given, x — a, the ^'"- ''"* point must lie in a determinate plane parallel to the J'Z-plane. If the distance and direction of the point from the XZ-plane is given, y = h, the point must lie in a determinate plane parallel to the XZ-plane. If it is known that x = a and y = b, the point must lie in each of two planes parallel, the one to the 171 172 ANALYTIC GEOMETRY FZ-plane, the other to the XZ-plane, and therefore the point must lie iu a determinate straight line parallel to the Z-axis. If the distance and direction of the point from the XF-plane 2; = c is also given, the point must lie in a determinate plane parallel to the XF-plaue and in a determinate line parallel to the Z-axis ; that is, the point is completely determined. Conversely, to every point in space there corresponds one, and only one, set of values of the distances and directions of the point from the coordinate planes. For through the given point only one plane can be passed parallel to a coordinate plane, a fact which determines a single value for the distance and direction of the point from that coordinate plane. The point whose distances and directions from the coordi- nate planes are represented by x, y, z is denoted by the symbol {x, y, z), and x, y, z are called the rectilinear coordinates of the point. There is seen to be a "one-to-one correspondence" between the symbol {x, y, z) and the points of space. Observe that x = a interpreted in the ZX-plane represents a straight line parallel to the Z-axis ; interpreted in the XF-plane a straight line parallel to the F-axis ; but when interpreted in space it represents the plane parallel to the FZ-plane containing these two lines. The equations x = a, y = b interpreted in the XF-plane represents a point ; inter- preted in space they represent a straight line through this point parallel to the Z-axis. If the axes are perpendicular to each other, the coordinates are called rectangular, in all other cases oblique. Problems. — 1. Write the equation of tlie plane parallel to the rZ-plane cutting the X-axis 5 to the right of the origin. 2. What is the equation of the FZ-plane ? 3. What is the locus of the points at a distance 7 below the XF-plane ? Write equation of locus. 4. Write the equations of the line parallel to the X-axis at a distance -f 5 from the XF-plane and at a distance - 5 from the XZ-plane. POINT, LINE, AND PLANE IN SPACE 173 5. Write the equations of the origin. 6. What are the coordinates of the point on the Z-axis 10 below the Xr-plane ? 7. What arc the equations of the Z-axis ? 8. What are the equations of a line parallel to the Z-axis ? 9. Explain the limitations of the position of a point imposed by placing X = -I- 5, then y = — 5, then z — - 3. 10. Locate the points (2, - 3, 5); (- 2, 3, - 5). 11. Locate (0, 4, 5); (2, 0, - 3). 12. Locate (0, 0, - 5); (0, - 5, 0). 13. Show that (o, b, c), {-a, b, c) are symmetrical with respect to the rZ-plane. 14. Show that («, b, c), (— «, — b, c) are symmetrical with respect to the Z-axis. 15. Show that (a, b, c), (- a, - b, - c) are symmetrical with respect to the origin. Art. 87. — Polar Space Coordinates Let (x, y, z) be the rectangular coordinates of any point F in space. Call the distance from the origin to the point r, the angle made by OP with its projection OP' on the XF-plane 0, the angle made by the projec- tion OP' with the X-axis <^. r, (fi, 6 are the polar coordinates of the point P. From the figure OP' = r ■ cos e, X = OP' ■ cos cf) = r cos 6 cos 4>, y = OP' • sin ^ = r cos 6 sin (^, z = r sin 6, formulas which express the rec- /Y tangular coordinates of any point ^"'' ^^' in space in terms of the polar coordinates of the same point 174 ANAL YTIC GEOMETll Y From the figure are also obtained r ={x- + 9/ + z^)'^, sin^ =-, tan 4> = '-, formulas which express the polar coordinates of any point in space in terms of the rectangular coordinates of the same point. Problems. — 1. Locate the points whose polar coordinates are 5, 15°, 60" ; 8, 90°, 45^ 2. Find the polar coordinates of the point (3, 4, 5). 3. Find the rectangular coordinates of the point (10, 30°, 60°). 4. Find the distance from the origin to the point (4, 5, 7). Art. Distance between Two Points Let the rectangular coordinates of the points be {x', y\ z'), (x", y", z"). From the figure If- = D" + (z' - z"y, D" = (x' - x"f + (?/' - y")\ hence (1 ) D' = {xJ - x"f + (ij' - ?/")- + (2' - z"f. x' — x" is the projection of D on the X-axis; y' — y" the pro- Z ^-v., POINT, LINE, AND PLANE IN SPACE 175 jection of D on the I'-axis; z' — z" the projection of D on the ^axis.* Calling the angles which B makes with the coordi- nate axes respectively X, Y, Z, x' — x" = D cos X, y' — y" = D cos Y, z' — z" = D cos Z. Substituting in (1), there results D- cos- X + D- cos- Y+ D- cos- Z = D% whence cos- X + cos' Y+ cos- Z=l; that is, the sum of the squares of the cosines of the three angles which a straight line in space makes with the rectangu- lar coordinate axes is unity. The distance from (x', y', z') to the origin is Va;'" + y'^ + z'^. If the point {x, y, z) moves so that its distance from {x', y', z') is always R, the locus of the point is the surface of a sphere and (x - x'Y + {y- y'-) + {z-z'f = Er, which expresses the geometric law governing the motion of the point, is the equa- tion of the sphere whose center is (x', y\ z'), radius R. Problems. —1. Find distance of (2, - 3, 5) from origin. 2. Find the angles which the line from (3, 4, 5) to the origin makes with the coordinate axes. 3. Find distance between points ( - 2, 4, - 5), (3, - 4, 5), 4. Write equation of locus of points whose distance from (4, - 1, 3) is 5. 5. Write equation of sphere center at origin, (2, 1,-3) on surface. 6. The locus of points equidistant from (x', y', z'), (x", y", z") is the plane bisecting at right angles the line joining these points. Find the equation of the plane. 7. Find the equation of the plane bisecting at right angles the line joining (2, 1,3), (4,3, -2). * The projection of one straight line in space on another is the part of the second line included between planes through the extremities of the first line peri^endicular to the second. The projection is given in direc- tion and magnitude by tlie product of the line to be projected into tlie cosine of the included angle. 176 A NA L YTIC GEOMETR Y 8. Show that y' + ir + z< is the point midway be- tween (x', j/', z')^ {%", y", z"). 9. Find the point midway between (4, 5, 7), (2, — 1, 3). 10. Find the equation of the sphere wliich has the points (4, 5, 8), (2, — 3, 4) at the extremities of a diameter. 11. Write the equation of the spliere with the origin on the surface, center (5,0, 0). 12. Find angles which the line through (2, 3, - 5), (4, - 2, 3) makes with the coordinate axes. 13. The length of the line from the origin to (x, ?/, z) is ?•, the line makes with the axes the angles a, /8, 7. Show that x = rcosa, ?/ = rcos 3, z = r cos 7. Art. Equations of Lines in Space Suppose any line in space to be given. From every point of the line draw a straiglit line perpendicular to the XZ-plane. There is formed the surface which projects the line in space on the XZ-plane. The values of x and z are the same for all points in the straight line which projects a point of the line in space on the XZ-plane. Hence the equation of the projection of the line in space on the XZ-plane when interpreted in space repre- sents the projecting surface. The projection of the line in space on the XZ-plane deter- mines one surface on which /Y the line in space must lie. The projection of the line in space on the I''Z-plane determines a second surface on which the line in space must lie. The equations of the pro- jections of the line in space on the coordinate planes XZ and POLXT, LINE, AND PLANE IN SPACE 177 YZ therefore determine the line in space and are called the equations of the line in space. By eliminating z from tlie equations of the projections of the line on the planes XZ and YZ, the equation of the projection of the line on the XF- plane is found. Art. 90. — E(juATroNs of the STUAKiiix Link iglit line on the h «, y = bz + (3. The equations of the projections of the sti coordinate planes XZ and YZ are x = az The geometric meaning of a, b, a, fi is indicated in the figure. The elimina- tion of z gives y-(3 = l{x-a), the equation of the pro- jection of the line in the XF-plane. Two points, (x', y', z'), {x", y", z"), determine a straight line m space. The projection of the line through the points {x\ y,' z'), (x", y", z") on the ZX-plane is determined by the projections (a;', z'), (x", z") of the points on the ZX-plane, likewise the projection of the line on the ZF-plane is deter- mined by the points {z', y'), (z", y"). Hence the equations of the straight lines through {x', y', z'), (x", y", z") are Fm. 159. ^-(z-z'), y-y' = -(z-z'). A straight line is also determined by one point and the direc- tion of the line. Let {x', y', z') be one point of the line, «, /3, y N 178 ANALYTIC GEOMETRY the angles which the line makes with the axes X, Y, Z respec- tively. Let {x, y, z) be any point of the line, d its distance from {x', y', z'). Then z-x' ^y-y' ^z-z' ^_^ cos a cos 13 cos y is the equation of the line. This equation is equivalent to the equa- tions x=x' + d cos a, y=y'+d cos (3, z = z' + d cos y, which express the coordinates of any point of the line in terms of the single variable d. If the straight line (1) contains the point (x", y", z"), x"-x' ^ y"-y' ^ z" -z' , cos a cos /8 cos y Eliminate cos a, cos (3, cos y from (1) and (2) by division, and the equation of the straight line through two points is obtained x — x' _ y — y' _ z — z' x" — x' y" — y' z" — z' as found before, a, (3, y are called the direction angles of the straight line. Problems. — 1. The projections of a straight line on the planes XZ y z Find the projection on the XY - 5, ?/ = 2 2 — 3 with the coordinate and YZ are 2 x + 3 plane. 2. Find the intersections of x planes. 3. Write the equations of the straight line through (2, 3, 1), ( - 1, 3, 5) . 4. Write the equations of the straight line through the origin and the point (4, - 1, 2). 5. Write the equations of the straight line through (3, 1, 2) whose direction angles are (60°, 45°, G0°). 6. The direction angles of a straight line are (45°, 60°, 60°) ; (4, 5, 6) is a point of the line. Find the coordinates of the point 10 from (4, 5, 6). POINT, LINE, AND PLANE IN SPACE ITU Akt. 91. Let X — a cos « Angle hetwkex Two Stuaigux Lines X y-b ^z~c cos (i cos y if cos /8' cos a' cos li' cos y be the straight lines. The angle between the lines is by definition the angle between parallels to the lines through the origin. Let OM' and OM" be these parallels through the origin. From any point P'{x\ y\ z') of OM' draw a perpendicular P'P" to OM". Then OP" is the projection of OP' on OM", and OP" is also the projection of the broken line {x' + y' + z') on OM".* Hence / r' cos 6 = x' cos a' + y' cos ^' + Z' cos y, = '-cos a -|-— cos r' y' + -C0Sy 7 that cos = cos a COS a' + cos /3 cos (3' + cos y cos y'. (1) * The sum of the projections of the parts of a broken line on any straight Hne is the part of the line included between the projections of tlie extremities of the broken line. a!) is the projection of AB ; be is the projection of BC; ac is the projec- tion of AB + BC. Fio. 162. 180 ANALYTIC GEOMETRY If the equations of the lines are written in the form X = az + a, y — bz + (3] x= a'z + «',?/ = b'z + (3', the equations of parallel lines through the origin are X = az, y = bz ; x = a'z, y — b'z. Let (x', y', z') be any point of the first line, its distance from the origin r'. Then x' — az', y' — bz', r'? = x'- + y'^ + z'-, whence cos a x' ~ r' r' = a VI cos/8 cosy b + b' vr + a- 1 T 1?' Likewise if {x", y", z") is any point of the second line, r" its distance from the origin. Substituting in (1) cos ii' x" r" a' V'l -f- «'- ■■ + b'^ cosfi' r" b' VI + a"^ '■ + b" cosy' _z" r" 1 VI + a" + b" l+aa' + bb' Vl + a^ + bWl + a" + b" When the lines are perpendicular, cos ^ = 0, whence 1 + aa' + bb' = 0. When the lines are parallel, cos^ = 1, whence ^ _ 1 + aa,' + bb' Vl + a- + bWl + a" + b" which reduces to (a' - ay + (b' - by + (ab' - a'by = 0. POINT, LINE, AND PLANE IN SPACE 181 This eciuation requires that a = a', b = b' ; that is, if two lines are parallel, their projections on the coordinate planes are parallel. The equations of the straight line through (x', ?/', z') parallel to x=az + a, y==bz-{- ^ are a; — x'= a {z — z'), y ~ y' = b{z — z'). The straight line (1) x — x' = a'(z — z'), y — y' = b' {z — z') through the point (x', y', z') is perpendicular to the straight line (2) X = az + a, y = bz + ft when a' and b' satisfy the equation 1 + cm' + bb' = 0. This equation is satisfied by an infinite number of pairs of values of a' and b'. This is as it ought to be, for through the given point a plane can be passed perpen- dicular to the given line, and every line in this plane is perpen- dicular to the given line, and conversely. Hence if the straight line (1) is governed in its motion by the equation l + aa'-j-6^' = 0, it generates the plane through (x', y', z') perpendicular to the straight line (2). 1 -f aa' -\- bb' = is the line equation of the plane. To find the relation between the constants in the equa- tions of two straight lines x = az + a, y = bz + ^, x = a'z + a', y — b'z + /3', which causes the lines to intersect, make these equations simultaneous and solve the equations of the projec- tions on the XZ-plane, also the equations of the projections on the I'Z-plane, for z. The two values of z, and ^^ ^ a — a' b — b' must be equal if the lines intersect. Hence for intersection the equation (a — a') (y8' — ft) — (b — b') {a' ~ a) = must be satisfied, and the coordinates of the point of intersection are aa' — a'a bB' — b'B «' — a -.^r, , -, x = , y = -^ -, z = When a = a and a — a' b — b' a — a' I) = //, the point of intersection is at infinity, and the lines are parallel, as found before. Problems. — 1. Find the angle between the lines x = Sz + \, y = ~ 22 + 5; x = z + 2, y = - z + i. 2. Find the angle between the lines through (1, 1, 2), (-3, - 2, 4) and (2, 1, - 2), (3, 2, 1). 182 ANALYTIC GEOMETRY 3. Find equations of line through (4, - 2, 3) parallel to a: = 4 2: + 1, y = 2 z — b. 4. Find line through (1, - 2, 3) intersecting x = -2z-\-o, y=z + 5 at right angles. 5. Find distance from (2, 2, 2) to line x = 2 z -\- l, y = - 2 z + S. 6. Find equations of line intersecting each of the lines x = 3^ + 4, y = -z + 2 and y = 2 z - 5, x - - z + 2 at right angles. 7. For what value of a do the lines x = Sz + a, y = 2z + 5 and X = - 2 z - o, y = i z - d intersect ? 8. Find the equations of the straight line through the origin intersect- ing at right angles the line through (4, 2, - 1), (1, 2, - 3). 9. Find distance of point of intersection of lines x = 2z-\-l, y = 2z + 2 and x = z + 5, y-iz-6 from origin. 10. Find distance from origin to line x-iz-H, y = — 2z + 3. Art. The Plane A plane is determined when the length and direction of the perpendicular from the origin to the plane are given. Call the length of the perpendicular p, the direction angles of the per- pendicular a, (3, y. Let P(.i-, ?/, z) be any point in the plane. The projection of the broken line (;f + y + z) on the perpendicular OP' equals p for all points in the plane and for no others. Hence xcosa + y cos /8 + 2: cos y = p is the equation of the plane. This is called the normal equation of the plane. Every first degree equation in three variables when inter- preted in rectangular coordinates represents a plane. The POINT, LINE, AND PLANE LW SPACE 183 locus reprostMiiod by Ax + B>i + Cz + D = is the same as the locus represeuted by u;cos a + ij (ios fi + z cos y - 2' = ^ i^' Con cos a A cos^ B _ cos y c D ibiniiu ; witli c + cos- 13 + cos r-y=\ , cos a A V^i- -\-B'+G' cos 3 B cos y = C "^ V^- +B' + C V^l' + B' + C P = - D V^l- + B' + C ice the factor ] _ Va:' + B-+ c- transfonns Ax + Bi/ + Cz + D = into an e(iuatiou of the form .1- cos a + y cos (3 + z cos y = p, which is the equation of a plane. •^ ^- •'/-)-?=: 1 is the equation of the plane whose intercepts a b c on the coordinate axes are a, b, c. This is the intercept equation of the plane. The plane represented by the equation Ax + By + Cz-{- D — depends on the relative values of the coefficients. Hence the equation of the plane has three parameters. To find the equa- tion of the plane through three points {x\ y', z'), (_x", y", z"), {x'", y'", z'"), substitute these coordinates for x, y, z in (1) A'x + By + C'a; + 1 = 0, solve the resulting e(iuations for A', B', C, and substitute in (1). The intersections of a plane with the coordinate planes are called the traces of the plane on the coordinate planes. The equation of the trace of Ax + By +Cz + D = on the X^-plane is found by making ?/ = in the equation of the plane. The trace is therefore Ax + Cz + D = 0. The trace on YZ is By + Cz + D = 0, on XY is Ax + By + D = 0. 184 ANALYTIC GEOMETRY For points in the intersection of the planes Ax + By+Cz + D = and A'x + B'y + C'z-\-D' = these equations are simultaneons. Eliminating >j, {AB' - A'B)x + (CB' - C'B)z + {DB - D' D) = 0, the equation of the projection of the intersection on the co- ordinate plane XZ. In like manner the equations of the pro- jections of the intersection on the planes YZ and XF are obtained. Problems. —1. Write the equation of the plane whose intercepts on the axes are 2, — 4, — 3. 2. Find the equation of the plane through (2, - 3, 4) perpendicular to the line joining this point to the origin. 3. Find the equation of the plane through (2, 5, 1), (3, 2, -5), (1, -3,7). 4. Find the equations of the traces of3a;-?/ + 5z— 15 = 0. 5. Find the equations of the intersection of Sx + by ~ 7 z + 10 = 0, bx -Utj + Sz - lb = 0. 6. Find the equation of the plane through (3, - 2, 5) perpendicular to a: -1 _ ?/ + 2 _ g - 3 cos GO'^ cos 45"^ cos G0° n. Find the direction angles of a perpendicular to the plane 2x-3?/+52 = 6. 8. Find the length of the perpendicular from the origin to 2x-3y + bz=^G. Art. 93. — Distance from a Point to a Plane Let (x', y\ z') be a given point, cc cos « + 2/ cos ^ + 2 cos y =i>, a given plane. Through {x\ y', z') pass a plane parallel to the given plane. The equation of this parallel plane is a; cos « -h 2/ cos ^ + » cos y = OF". POINT, LINE, AND PLANE IN SPACE m The point (x', y', z') lies in this plane, therefore x' cos a + y' cos ^ + 2' cos y = OP". Subtracting OP from both sides of this equation, x' cos a + y' cos fi -f- z' cos y — i^ = P-P" ; that is, the perpendicular distance from (x', y', z') to xcosa + y cos (3 + 2 cos y — p = is the left-hand member of this equation evaluated for (x', y', z'). The sign of the perpendicular is plus when the origin and the point (x', y\ z') are on different sides of the plane, minus when the origin and the , /,,,■. jl 1' 1(1. K«. point (a; , y', z) are on the same side of the plane. The distance from {x', y', z') to the plane Ax-\-By+Cz-\-D=0 is found by transforming the equation of the plane into the form X cos a -{- y cos 13 + z cos y — j? = to be Ax' + By' + Cz' + D Let .Tcos a-\-y cos /? + 2: cos y — p = and X cos «' -f ?/ cos )8' -f 2 cos y' — ;y = bo the faces of a diedral angle, (.« cos «+?/ cos ^+2;cos y— ^))±(.); cos «'+?/ cos ^' + z cosy'— p')=0 is the equation of the locus of points equidistant from the faces: that is, the eipiation oC the bisectors of the diedral angle. 186 ANALYTIC GEOMETRY Problems. — 1. Find distance from origin to plane Ux-13y+nz + 22 = 0. 2. Find distance from (3, - 2, 7) to 3 a; + 7 ?/ - 10 s + 5 = 0. 3. Write the equations of the bisectors of the diedral angles whose faces are 2 X + 5 y — 7 z = 10, and ix-y + Gz — l!i = 0. 4. Find distance from (0, 5, 7) to - + | + ? = 1. 5. Find distance from origin to | ce — | ?/ -| | ^ = 1- Art 94. — Angle between Two Planes Let a; cos a -\- y cos ^+z cos y=2^, a; cos «'+?/cos^'+2:cosy'=jy be two given planes, their included angle. The angle be- tween the planes is the angle between the perpendiculars to the planes from the origin. Hence cos 6 = cos a cos a' + cos (3 cos ^' -\- cos y cos y'. If the equations of the planes are in the form Ax + B>j + Cz + D = 0, A'x + B'y + C'z + Z)' = 0, cos a = ■ — > V.4^ + B'+ C COS «■ = ? cosB — ^ > cos IS' - ^' , V^- + B'+ C ^AJ' + B'-'+C" pi cos y' - V^» + B' + C- V^'- + B" + C" r.no. .„...- AA' + BB'+CC . VA' + B'+ CWA'^ + B'' + C"2 The planes are perpendicular when AA' + BB' + CC = ; ,, , , . AA' + BB' + CC 1 . -, parallel when 1 = — -'— — — ' r — =r — , which V2- + B' + C-VA" + B'- + C- reduces to {AB' - yl'i^)^ + {AC - A'Cf + (7JC" - B'Cf = 0, 1 A B C POINT, LINE, AND PLANE IN SPACE 1ST The angle between the plane xcos a -\->/ con f3 + zcosy =p y ~ •' =z ~ '^ is the conipUnnent of the cos «' cos /8' cos y' angle between the line and the perpendiculai- to the plane. Hence sin 6 = cos a cos a' + cos ft cos /3' + cos y cos y'. If the equations of line and plane are in the form x = az + «, y = hz + p, and Ax + By -\- Cz-{- D = 0, cos a = Vyl- + B'-^ C cos/8 = B V^l- + B' + C" COSv = C a coS|8' COS y' Vl + a2 + 62 Vl + a- 4- b^' 1 V^- + i3- + C - ' V 1 + a- + 62 Hence sin^ = ^a + m +C _. VJ^M^- + C- Vl + a- + b^ The line is parallel to the plane when ^hi -\- Bb -j- C=0; perpendicular when 1= Aa + B b + C ^^^^^^^^ VA' + B'+ (J- Vl + a' + b^ reduces to {Ab - Baf -\- {A - Ccif + (B - Cbf = 0, whence .1 , B a = —, b = — (f C To find the intersection of the line x = az -{- a, y — bz -j- (S, and the plane Ax + By + Cz + J9 = 0, make these equations simultaneous, and solve for x, y, z. There results ^ _ Au + B^ + D Aa + Bb + C If Aa -j- Bb + C—0, the point of intersection goes to infinity, and the line and plane are parallel, as found before. If Aa -I- B(3 + C also vanishes, z lieconies indeterminate, likewise x and //, and the line lies wholly in the plane. 188 ANALYTIC GEOMETRY If the plane Ax-\- By + Cz-\- D = contains the point (x', y', z') and the line x = az + a, y = bz-{-^, Ax' + By' + Cz'+D = 0, Aa -\- Bb + C= 0, Aa + B(3-\-D = 0. These equations determine the relative values of A, B, C, D, hence the plane is determined. The plane Ax + By -{- Cz + D = contains the two lines X = az + a, y = bz + (^ and x = a'z + a', y = b'z + (3' when Aa + Bb-^C^O, Aa + Bf3 + D^0, Aa' + Bb' + C^O, Aa' + B/3' + D =0. These four equations are consistent only when — — '^^'^ that is, when the lines intersect, and b—b' /? — /3 then the relative values of A, B, C, D, which determine the plane, are found by solving any three of the four equations. Problems. — 1. Find angle between planes 10 x — 3 y + 4 £• + 12 = 0, 15 X + 11?/ -7 2 + 20 = 0. 2. Find angle between line x = 5 z + 7, y = S z — 2, and plane 2x- 15?/ + 200 + 18=0. 3. Find equation of plane through (4, — 2, 3) parallel to 3x-2y + z- 5 = 0. 4. Find equation of plane through (1, 2, — 1) containing the line x = 2z — S, y = z + a. 5. Find equation of line through (4, 2, — 3) perpendicular to x + Sy -2z + 4 = 0. 6. Find equation of plane containing the lines x = 2z + \, y = 2z + 2, and x = z + 5, y = 4:Z — G. 7. Find angles which Ax + By + Cz + D = makes with the coordi- nate axes. 8. Find angles which Ax + By + Cz + D ■= makes with the coordi- nate planes. 9. Show that if two planes are parallel, their traces are parallel. 10. Show that if a line is perpendicular to a plane, the projections of the line are perpendicular to the traces of the plane. POINT, LINE, AND PLANE IN SPACE 189 11. Show that ^ ~ ^' = ^''s:Jl!. = ?-IiA is perpendicular to A B C Ax + ny + a:: + D = 0. 12. Show that (x>-x"){x-x") + (7j' -y")(>/~y") + (z' -z")(:-z")=0 is a plane through (x", y", z") perpendicular to the line through (x', y\ z') and (x", y", z"). 13. Find the equation of the plane tangent to the sphere x^ + y'^+z'^ — R- at the point (x", y", z") of the surface. 14. Find the equation of the plane tangent to the sphere (X - X')- + (2/ - y'y + {s- z'Y = R- at the point (x", y" , s") of the surface. CHAPTER XIV OUKVED SURPAOES trix of a cylindrical surface Z Art. 95. — Cylindrical Surface.s Let the straight line x = az + a, y = bz -\- (3 move in such a manner that it always intersects the XF-plane in the curve F(x, y) = 0, and remains parallel to its first position. The straight line is the generatrix, the curve F{x, y)=0 the direc- The generatrix pierces the XF-plane in the point («, (3), and therefore F(a, (3)^0. This is the line equation of the cylindrical surface, for since a and b are constant, to every pair of values of a and 13 there corresponds one posi- tion of the generatrix, and to all pairs of values of a and 13 satisfying the equation F(a, 13) =0 there corresponds the generatrix in all positions *''*"■ '^^^- while generating the cylindri- cal surface. To obtain the equation of the cylindrical sur- face in terras of the coordinates of any point (.r, y, z) of the surface, substitute in F{(z, /8)=0 the values of a and ^ ob- tained from the equations of the generatrix. There results F(:x-az, y-bz)=0, the equation of the cylindrical surface whose directrix is F(x, y) = 0, generatrix x=az-\-a, y=bz + (3. 190 CURVED SURFACES 11)1 + ^ = 1. What does this equation become when elements are parallel 6- ele- Problems. — 1. Find the equation of the right circular cylinder whose directrix is x- + y~ — f-, and axis the Z-axis. 2. TIic directrix of a cylinder is a circle in the A'l'-plane, center at origin. The element of tlie cylinder in the ZA'-plane makes an angle of 45° with the A'-axis. Find equation of surface of cylinder. 3. Find general equation of surface of cylinder whose directrix is X- , y^ to Z-axis ? 4. Find equation of cylindrical surface directrix y- — \(ix ments parallel to x = 22 + 5, ?/= — 3^ + 5. 5. Determine locus represented by a; = a sin </>, y = a cos 0, z = r(p. Since x- + y- = a-, the locus must lie on the cylindrical surface whose axis is the Z-axis, radius of base a. Points corre- sponding to values of (p differing by 2 tt lie in tlie same element of the cylindrical sur- face. The distance between the successive points of intersection of an element of the cylindrical surface with the locus is 2 wc. The locus is tlierefore the thread of a cylin- drical screw with distance between threads 2 TTC. The curve is called the helix. Art. 96. — Conical Surfaces Let the straii^^ht lino x = az -f «, y = hz + (i move in such a manner that it always intersects the XT-plane in the curve F(x, ?/) = 0, and passes through the point {x\ ?/', z'). The straight line generates a conical surface whose vertex is (:r', ?/', z'), directrix F{x, y) = 0. The equations of the generatrix are x — x' = a(z — z'), y — ?/' = h{z — z'), which may be written x=az-{-{x'—az'), y=bz-{-(y'—hz'). This line pierces the XY- 192 Analytic geometry plane in {x' — az\ y' — bz'), and therefore F(x'—az', y' — bz')=0. This is the line equation of the conical surface, for to every pair of values of a and b there corre- sponds one position of the gen- eratrix, and to all pairs of values of a and b satisfying the equa- tion F(x' — az', y'—bz) = there corresponds the generatrix in all positions while generating the conical surface. To obtain the equation of the conical surface in terms of the coordinates of any point (x, y, z) of the surface, substitute in F(x' — az', y' — bz')= for a and b their values obtained from the equations of the generatrix. There results j^r x'z - xz' y'z-yz' \^ ^^^ equation of the conical surface \ z — z' z~z' J whose vertex is (x', y', z'), directrix F(x, y) = 0* Problems. — 1. Find the equation of the surface of the right circular cone whose axis coincides with the Z-axis, vertex at a distance c from tlie origin. 2. Find the equation of the conical surface directrix ^ + l^= 1, ver- tex (5, 2, 1). 3. Find the equation of the conical surface vertex (0, 0, 10), directrix 2/2 = 10 X - x~. 4. Find the equation of the conical surface vertex (0, 0, c), directrix ^ + ?^'=1. 5. Find the equation of the conical surface vertex (0, 0, 10), directrix a;'^ + 2/2^0. * Surfaces which may be generated by a straight line are called ruled surfaces. CURVED sun FACES 19a Art. 97. — Sukkacks ok Kkvolution Let 3fN be any line in the ZX-plane. When MN revolves about the Z-axis, every point /* of JfiV deseribes the circumfer- ence of a (drcle witli its center on the Z-axis and whicli is pro- jected on the XF-plane in an equal circle. The equation of the circle referred to a pair of (' axes through its center parallel to the axes X and Y is or -j-y- = t^. This is also the equation of the })rojectiou of the circle on the Xl''-})lane. The radius r is a function of z whicli is given by the equation of the generatrix r = F(z). Hence the equation of the surface of revolution is obtained by eliminating /• from the equations af' + y- — r' and r = F(z). Problems. — 1. Find equation of surface of sphere, center at origin, radius li. This sphere is generated by the revolution about the Z-axis of a circle whoso e(iualion is r~ + z- = R-. Eliminate r from this equation and x- -|- y- — ?•-, and the ecpiation of the sphere is found to be X-' -f if -V Z'^:= R\ 2. Find equation of right circular cylinder vyliose axis is the Z-axis. 3. Find equation of right circular cone whose axis is Z-axis, vertex (0, 0, c). 4. Find equation of right circular cone whose axis is Z-axis, vertex (0, 0, 0). 5. Find equation of surface generated by revolution of ellipse about its niajiir axis. This is the prolate spheroid. 6. Find ((juation of surface generated by revolution of ellipse about its minor axis. This is the oblate spheroid. o 194 A NA L YTIC GEOMETK Y 1. Find equation of surface generated by revolution of hyperbola about its conjugate axis. This is the hyperboloid of revolution of one sheet. 8. Find equation of surface generated by revolution of hyperbola about its transverse axis. This is the hyperboloid of revolution of tv?o sheets. 9. Let PP' be perpendicular to the JT-axis, but not in the ZX-plane. Suppose PP' to re- volve about the Z-axis. The equation of the surface gener- ated is to be found. The equations of the projec- tions of PP' on the planes ZX and ZY are x = a.,y = bz. The point P describes the circum- ference of a circle whose equa- tion is x2 + ^2 _ ,.2. The value i of r depends on z, and from the ^^«- 1'^"- figure r''= a?+ Vh'K Hence the equation of the surface generated is x- + y'^ = &%- -f a'-. The surface is thei'efore an hyperboloid of revolution of one sheet. Akt. The Ellipsoid In the XF-plane there is the fixed ellipse ^-f--^ = 1, in the .2 2 «' ^ ZX-plane the fixed ellipse - + -" = 1. The figure generated a- c- by the ellipse which moves with its center on the X-axis, the plane of the ellipse perpendicular to the X-axis, the axes of the ellipse in any position the intersections of the plane of the ellipse Avith the fixed ellipses, is called the ellip- soid. The equation of the ellipse From the equations of the fixed ellipses C Uli \ 'ED S URFA CES 195 — +-- = 1, — + --=1, whence rs =lrll- \,rt—c-(l ]• a- b- a- c- \ a-j \ a-j Hence tlie equation of the generating ellipse in any position, that is, the e(uiation of the ellipsoid, is — + 4, + — , = 1- When (r b- & a, b, c arc unequal, the figure is an ellipsoid with unequal axes; when two of the axes are equal, the figure is an ellipsoid of revolution, or spheroid; Avhen the three axes are equal, the elli[)soid becomes the sphere. Art. The Hyperboloids In the ZX-plane there is the fixed hyperbola — — r, = 1, in the Zl'plane the fixed hyperbola ^ — ^, cr 1. The figure gen- Z crated by the ellipse which moves with its center on the Z-axis, the plane of the ellipse perpendicular to the Z-axis, the axes of the el- lipse in any position the intersections of the plane of the ellipse with the fixed hyperbolas, is called the hyperboloid of one sheet. The equation of the ellipse Fig. i7t. 1. From the equations of the fixed hyperbolas rs' rt ri _?'=:!, ^-^=1, whence ^s'=a-(l + '^, 7r=b'(l+''\ rr c- b' c- ' \ c-J \ c-J Hence the equation of the generating ellipse in any posi- tion, that is, the equation of the hyperboloid of one sheet, is a- b- c- yS ^2 In the ZX-plane there is the fixed hyperbola — — = ^^ 196 ANALYTIC GEOMETRY in the XT-plane the fixed hyperbola — — ^-=^1. The figure a^ h- generated by the ellipse which moves with its center on the X-axis, the plane of the el- ^ '' lipse perpendicular to the X-axis, the axes of the el- lipse in any position the in- tersections of the plane of the ellipse with the fixed hyperbolas, is called the hyperboloid of two sheets. The equation of the ellipse is ^ + |r,= l- Fi'oi^ tlie Fig. 172. '>'^ ^'^ 9 — 2 o -p equations of the fixed hyperbolas — — -—-=1, — -^—^1 whence ri^h'{-^-\\, '^V^ (?(^^-\\. Hence the equation of the generating ellipse in any position, that is, the equation of the hyperboloid of two sheets, is — ■ Art. 100. — The Paraboloids In the XF-plane there is the fixed parabola -if' — 2 hx, in the ZX-plane the fixed parabola z^=2 ex. The figure generated by an ellipse which moves with its center on the X-axis, its plane per- pendicular to the X-axis, the axes of the ellipse in any position the intersections of the plane of the X ellipse with the fixed parabola, is called the elliptical paraboloid. The equation of the ellipse is rs rt CUR VED S UIIFA CES 11)7 From the equations of the fixed paraboUis rs' = 2 bx, rt = 2 ex. Hence the equation of the generating ellipse in any position, that is, the e(iuation of the elliptical paraboloid, is •—-{ — = 2x. b c In the ZX-plane there is the fixed parabola 2- = 2 ex, in the Xl'-plane the fixed parabola y^ — — 2bx. The figure gener- ated by an hyperbola which moves with its center on the X-axis, the plane of the hy- ])erbola perpendicular to the X-axis, the axes of the hy- l)erbola the intersections of the plane of the hyperbola with the fixed parabolas, is called the hyperbolic para- boloid. The equation of the hyperbola is = 1. z' _ y rs rt' From the equations of the fixed parabolas rs = 2 ex, ri'=—2bx. Hence the equa- FiG. 174. tion of the generating hyper bola in any position, that is, the equation of the hyperbolic paraboloid, is —=2x. c b Art. 101, —The Conoid The center of an ellipse moves in a straight line perpendicu- lar to the plane of the ellipse. The major axis is constant for all positions of the ellipse, the minor axis diminishes directly as the distance the ellipse has moved, becoming zero when the 108 ANALYTIC GEOMETRY ellipse has moved the distance c. The figure generated is called the conoid with elliptical base. The equation of the ellipse is y- where 4-^^ — 1 -:: ^ — 2 — -^' s rt a, and, from similar z triangles, — = -, whence rt b c -(c — z). The efjuation of the c generating ellipse in any posi- tion, that is, the equation of the conoid, is lP{c Art. 102. Surfaces represented by Equations in Three Variables An equation ffi{x, ?/, z)=0, when intei-preted in rectangular space coordinates, represents some surface. For when z is a continuous function of x and y, if (x, y) takes consecutive posi- tions in the XF-plane, the point {x, y, z) takes consecutive positions in space. Hence the geometric representation of the function </> (x, y, «) = is the surface into which this function transforms the XF-plane. To determine the form and dimen- sions of the surface represented by a given equation, the inter- sections of this surface by planes parallel to the coordinate planes are studied. Problems. — Determine the form and dimensions of the surfaces rep- resented by the following equations. 1. E--|-^-)- ^2— \ Tiie equation of the projection on the Xr-plane 9 4 of the intersection of the surface represented by this equation and a plane g = c parallel to the AT-plane is ^ + ^ = 1 - c-. This equation repre- CURVED SURFACES 199 sents an ellipse whose dimensions are greatest when c = 0, diminish as the" numerical value of c increases to 1, and are zero when c = ± 1. The ellipse is imaginary when c is numerically greater than 1. The equation of the projection on the ZA'-plane of the intersection of the surface by a plane tj = b parallel to the ZX-plane is — \- z'^= I — — 9 4 which represents an ellipse whose dimensions are greatest when ft = 0, diminish as b increases numerically to 2, are zero when ft = ± 2, and become imaginary when ft is numerically greater than 2. The equation of the projection on the TZ-plane of the intersection of the surface by a plane x a parallel to the FZ-plane is ^ + 2^ 4 a2 which represents an ellipse whose dimensions are greatest when a = 0, diminish as a increases numerically to 3, are zero when a = ± 3, and be- come imaginary when a is numeri- cally greater than 3. The sections of the surface made by planes parallel to the coordinate planes are all ellipses, the surface is closed and limited by the faces of the rectangular parallelopiped whose faces are x — ±S, y = ±2, z = ±1. From the equation it is seen that the origin is a center of symmetry, the coordinate axes are axes of symmetry, the coordinate planes are planes of symmetry of the surface. The figure is the ellipsoid with axes 3, 2, 1. 0. 10x = 0. Fig. 176. X- + 2/-^ - z^ x"^ + >j- + z"^ r/ + z^ ■ X2 + J/2 . 10x = 0. ^2:^1. z' - 2 X + o !i j2 + 2 X + 4 i' : 12. Show that the conoid is a ruled surface. CHAPTER XV SECOND DEGEEE EQUATION IN THKEE VAEIABLES Art. 103. — Transformation of Coordinates Take the point (a, b, c) referred to the axes X, Y, Z as the origin of a set of axes X', Y', Z' parallel to X, Y, Z respec- tively. Let (.r, y, z), {x', y', z') represent the same point referred to the two sets of axes. From the figure x = x' + a, y = y'-\-b, z = z' -\- c. z' Let X, F, Z be a set of rectangular axes, X\ Y\ Z any set of rectilinear axes with the same origin. Denote the angles made by A'' with A, F, Z by a, (3, y respectively, the angles made by Y' with A, Y, Zhj a', y8', y', the angles made by Z' with A, Y, Z by a", j8", y". If (a-, y, z) and (x', y', z') represent the same point F, x is the projection of the broken line 200 SECOND DEGREE EQUATION 201 (x' + If' + z') on tlio X-axis, y the projection of this broken line on the i'-axis, z the projection of this broken line on the Z-axis. Hence X = x' cos a -\- ?/' cos a' + z' cos a", y = x' cos f^ + ?/' cos /3' + z' cos /3", Z = x' COS y -f- )/' COS y' + ;<;' cos y". Since X, F, Z are rectangular axes, cos^ a + cos^ /? + cos^ y = 1 , COS-«'+ C0S-/5'+ COS-y' = 1, cos^«" + cos-;8"-f cos^y" = 1. If X', I"', Z' are also rectangular, cos a cos a' -f cos ft cos /?' + cos y cos y' — 0, cos « cos a" + cos /3 cos (3" + cos y cos y" = 0, cos a' cos «" -f cos ft' cos /3" + cos y' cos y" = 0. Problems. — 1. Transform x- + y^ + z- = 2G to parallel axes, origin (-5,0,0). 2. Transform x- + 7j- + z- = 25 to parallel axes, origin ( — 5, - 5, — 5). 3. Transform ^ + ^ -f IT = 1 to parallel axes, origin ( - a, 0, 0). a'^ b- c^ 4. Show that the first degi-ee equation in three variables interpreted in oblique coordinates represents a plane. 5. Show that the equation of an elliptic cone, vertex at origin, and 3.2 y2 5-2 axis the Z-axis, is ~ + ^ — — = 0. ' a^ 62 c-i 6. Derive the formulas for transformation from one rectangular sys- tem to another rectangular system, the Z'-axis coinciding with the Z-axis, the X'-axis making an angle d with the A'-axis. Art. 104. — Plane Section of Quadric Surfaces represented by the second degree equation in throe variables Ax"^ + By- + Cz- + 2 Dxy + 2Exz + 2 Fyz * ■^2Gx + 2Hy + 2Kz + L = (1) are known by the general name of quadrics. 202 ANALYTIC GEOMETRY To find the intersection of the surface represented by this equation by any plane transform to a set of axes parallel to the original set, having some point (a, b, c) in the cutting plane as origin. The transformation formulas are x = x' -\-a, y = y' -\- b, z = z' -\-c, and the transformed equation is Ax'' + By'' + Cz" + 2 D'x'tj' + 2 E'x'z' + 2 F'y'z' + 2 G'x' + 2 H'y' + 2 K'z' + L' = 0, (2) where G' = .la + Db + Ec + G, H' = Bb + Da + Fc + H, K' = Cc + Ea + Fb + K, L' = Aa' + Bb- + Cc- + 2 Dab + 2 Eac + 2 F6c + 2 (^a + 2 /f & + 2 /ic + L. Now turn the axes X', Y', Z' about the origin until the X' F'-plane coincides with the cutting plane. This is accom- plished by the transformation formulas x' = Xi cos a + ?/i cos «' + Zi cos a", y' = Xi cos y8 + v/i cos /5' + ^1 cos |8", 2' = .Tj cos y + ?/i cos y' + 2:1 cos y". These formulas are linear, hence the equation of the quadric in terms of {x^, y^, Zj) is of the form A,x,' + B,y,' + C,z,' + 2 D,x,y, + 2 E,x,z, + 2 jPj^^i^i + 2 (^^.Ti + 2 if,y, + 2 /r.^i + Xi = 0. (3) Since the plane of the section is the Xj^Vplane, the equa- tion of the intersection referred to rectangular axes in its own plane is A.x^' -f JB,2/i' + 2 D.x.y^ + 2 G,x, + 2 H,y, + L,= 0, which represents a conic section. Hence every plane section of a quadric is a conic section. For this reason quadrics are also called conicoids. SECOND DEGIIEE EQUATION 20'.] Art. 105. — Ckxtku of Quadric The surface represented by eq\iation (2) is symmetrical with respect to the origin (a, b, c) wlien the coefficients of x', y', z' are zero, for then if {x\ y', z') is a point of the surface, (-•< -y', -2') is also a point of the surface. Hence the center of the quadrie (1) is found by solving the equations Aa+ Db + Ec + G = 0, Bb + Da + Fc + 11 = 0, and Cc + Ea + Fb + K= 0. Problems. — 1. Find the center of the quadrie represented by a:2 + ?/2 ^ 4 ^2 _ 8 a;5: + 2/ = 0. 2. Find the center of the quadrie represented by .r2 - 2/2 4- 2;2 - 10 X + 8 2 + 15 = 0. Art, 106. — T.vxgent Plane to Quadric Let (.To, ?yo> 2:0) be any point of the quadric (1). The equa- tions cc = .To + d cos (it.,y = ?/o + d cos p,z = Z(i-\- d cos y represent all straight lines through (.t,„ ?yo> ^^- By substituting in (1) = + ^aV + 2 oleosa a'o d+^lcos-« + By,' + 2Bcos^ 2/0 + 5cos^/8 + Cz,? + 2 Coos y 2^0 + C'cos-y + 2Z>.r,,Vo + 2I>cos« •Vo + 2 Z> cos a cos /? + 2Ex^, + 2 Z) cos ^ •Ty + 2 £" cos a cos y + 2FyoZo + 2£oosy .To + 2i^C0S)8C0Sy + 2Gx, + 2£cosa 2^0 + 2 By, + 2Fcosy 2/0 + 2Kz, + 2FcoS|8 2^0 + L + 2 (7 cos a + 2//cos^ + 2 A" cosy 204 ANALYTIC GEOMETRY an equation is found which determines the two values of d corresponding to the points of intersection of straight line and quadric. Since the point (a^o, yo, z^) lies in the quadric, the term of this equation independent of d vanishes. If the co- efficient of the first power of d also vanishes, the equation has two roots equal to zero ; that is, every straight line through the point (;Xq, ?/„, ^o), and whose direction cosines satisfy the equation A cos a • x^+B cos (3 • ?/o+ C'cos y • z^,-\-D cos a • ?/(, + Z) cos /5 • .Tq -f^cosy • Xo+Ecosa ■ Z(t-\-Fcosy ■ y^+Fcos/B • Zq -f GrCos« + HcosfS -f A'cos y = is tangent to the quadric. To determine the surface repre- sented by this equation multiply by d and substitute x — Xq for d cos a, y — ?/o for dcos (3, z — z^ for d cos y. There results the equation AxX(, + Byyo -f Czz^ + D (;xy^, + x^y) + E (xz^ + x^) + F(yZo + y,z) + G(x + x^) + H{y + y,) + K{z -f ^o) + ^ = 0, which, since it is of the first degree in {x, y, z) represents a plane. This plane, containing all the straight lines tangent to the quadric at {xq, y^, Zq) is tangent to the quadric at (.Tu, ?/„, z^^. Notice that the equation of the plane tangent to the quadric at (xq, yo, Zq) is obtained by substituting in the equation of the quadric xxq for x^, yy^ for y^, zz,, for z^, xy^ + x^y for 2 xy, xZq + X(^ for 2 xz, yzo -f- y^z for 2yz, x -\- x^ for 2x, y -\- y^ for 2 y, z + Zq for 2 z. Let (x', y', z') be any point in space, (x^, yo, Zq) the point of contact with the quadric (1) of any plane through (x', y', z') tangent to the quadric. Then (Xq, y^, z^, (x', y', z') must satisfy the equation .4.r'.T„ + By'yo + Cz'z^ -f D (x'y, + ?/'a'„) -f ^(^'.^o -f .x-'^o) -f F(z'y, + y'z,) + G (;«' + x^) + U{y' -{- y^) +K{z' + z,) +L = 0. SECOND DKCREK EQUATION 205 Hence the points of contact (.t,„ ?/„, z^■) must lie in a ]>lane, and the locus of the points of contact is a conic section. Problems. — 1. Write the equation of the phiiie tangent to .T- + y- + z^ = B^ at (xo, 2/0, Zu). 2. Write the ciiuation of the plane tangent to a:2 + y2 4. ^2 _ 10 a; + 25 = at (5, 0, 0). 3. Write the equation of the plane tangent to ^ + ?^' + -'=lat (Xo, 2/0, 20). a^ 62 c- 4. Write the equation of the plane tangent to t^?l = 2xat (Xo, 2/0, 2o). b c 5. Find equations of projections on planes ZX and ZY of locus of points of contact of planes tangent to x- + y"^ + z"^ — 25 through (7, - 10, 6). 6. Find equation of normal to— + ^ + — =1 at (x',ij',z'). The a^ b'^ c^ normal to a surface at any point is the line through that point perpen- dicular to the tangent plane at that point. 7. Find the angle between the normal to — + ^^-{-^=1 at (x', y', z') cfi h^ c'^ and the line joining (x', y', z') and the center of the ellipsoid. Art. 107. — Reduction of General Equation of Quadric To determine the form and dimensions of the surfaces repre- sented by the general second degree equation in three variables when interpreted in rectangular space coordinates it is desirable first to simplify the equation. This simplification is effected by changing the position of the origin and the direction of the axes. The change of direction of rectangular axes is effected by the formulas X = x' cos a + y' cos «' + ^' cos a", y = x' cos (3 ■{- y' cos /3' + 2' cos /3", z = a;' cos y 4- y' cos y' -f- z' cos y", 206 ANALYTIC GEOMETRY where the nine cosines are subject to the six conditions cos^ a + cos^ /3 + cos- y = 1 , cos'^w' + cos^)8' + cos-y' = 1, COS^«" + C0S-/3" + COS-y" = 1, cos a cos «' + COS (3 cos (3' + cos y cos y' = 0, cos a cos «" + cos (3 cos (3" + cos y cos y" = 0, cos a' COS a" + cos yS' cos /3" + cos y' cos y" = 0. Three arbitrary conditions may therefore be imposed on the nine cosines. Substituting for x, y, z in Ax" + By- -^Cz--^2 Dxy + 2 Exz + 2Fyz + 2Gx + 2Hy-^2 Kz + L = 0, there results Ax" + By" + Cz'- + 2 D'x'y' + 2 ^'a-'^' + 2 F'?/'^' + 2 G'x' + 2H'y' + 2K'z' + L' = 0, when D', E', F' are functions of the nine cosines. Equate B', E', F' to zero and determine the directions of the rectangu- har coordinates in space in accordance with these equations. This transformation is always possible, hence Ax' + Bf- + Cz' + 2 G'x + 2 IFy + 2 K'z + L' = interpreted in rectangular coordinates represents all quadric surfaces. Now transform to parallel axes with the origin at (a, b, c). The transformation formulas are X = a -It x', y = h + y', z = c + z' and the transformed equation Ax"+By" + Cz" + 2(Aa + G')x' + 2(Bb +H')y'+2(Cc + K')z' + (Aa' + BW + Cc- + 2 G\i + 2 //7> + 2 K'c + L') = 0. SECOND DEGREE EQUATION 207 Take advantage of the three arbitrary constants a, b, c to cause the vanishing of the coefficients of x', y', z'. This gives = -^ b = -— ' = -=^ " A' B' '' C' values whicli are admissible when .1 ^^ 0, B :^ 0, C ^ 0. The resulting equation is of the form Lx' + 3Ii/ + iVV = P. When A ^ 0, B ^ 0, C = 0, the transformation x = a + x',y = h-\-y',z = c + z' gives Ax" + By" + 2(Aa+ G)x> + 2(Bb + H')y' + K'z' + (Aa' + Bb' + 2 G'a + 2H'b + 2 K'c + L') = 0. Equating to zero the coefficients of x', y' and the absolute term, the values found for a, b, c are finite when A^O, -B =5^ 0, K' =^ 0. The resulting equation is of the form Lx? + My- + N'z = 0. When A^O, B^O, C = 0, K' = 0, the equation takes the form LiT + My- + X'x + M'y + P = 0. When url ^ 0, iJ = 0, C = 0, the equation takes the form Mx"" + M'x + N'y + P = 0. When A = 0, B — 0, C — 0, the equation is no longer of the second degree. Since x, y, z are similarly involved in Ax" + By- + 0x^ + 2 G'x + 2 IVy + 2 K'z + L' = 0, the vanishing of A and G' or of B and H' would lead to equa- tions of the same form as the vanishing of C and 7i '. Collecting results it is seen that the following equations interpreted in rectangular coordinates represent all quadric surfaces — ^1:^0, B^Q, C4-(), Lx'+My- + Nz-=P I A^^O, B^O, C=0, K'^0 Lx-+My''+N'z=0 II A^O, B^O, C=0, K' = Lx^-^My^+M'y-{-L'x+P=0\ A^O, B=0, C=0 Lx^+L'x-{-M'y+N'z + P=0\ These equations are known as equations of the first, second, and third class. 208 ANALYTIC GEOMETRY Art. 108. — Sukfaces of the First Class The equation of the first class may take the forms (a) ix- + 31 f + Nz^ = P, (b) Lx" + 3Iy- - Nz- = P, (c) Lx^ + 3bf-Nz'^ = -P, or similar forms with the coefficients of ar and z- or of 'if and z' positive. (a) The intersections of planes parallel to the coordinate planes with Lx^ + 3Iy/ + Nz' = P are for X = x', 3Iif + Nz~ = P- Lx'\ an ellipse whose dimensions are greatest when x' = 0, diminish as x' increases numerically, are zero for x' = ± \-jr, imaginary — ' X/ — ; for y = ?/', Lx- + Nz- = P — My'-, an ellipse whose dimensions are greatest when _?/' = 0, diminish as y' increases numerically, are zero for ?/' = ±a/— , imaginary when y' is numerically greater than -v/ — ; for z = z', Lx- + 3ry- = P — Nz'-, an ellipse whose dimensions are greatest for z' = 0, diminish as z' increases numerically, are zero when z' = ±\^, imaginary when z' is numerically greater than a / - • Calling the semi-diameter on the X-axis a, on the I''-axis b, on the .^-axis c, the equation becomes — -f^ + — = 1, the ellipsoid. " ^ ^ The figure represented by Lx^ -\- 3Df + Nz^ = — P is imagi- nary. The equation Lx- + 3[y- + Nz- = represents the origin. (b) Lx^ -\- 3fy^ — Nz^ = P. The intersections are for x = x', SECOND DKGRKE EQUATION 201) 3/)/- — Xz- = r — Lx'-, ;iu hyperbola whose real axis is parallel to the I'-axis when — \/. <-<^'< +\-,> parallel to the Z-axis when x' is numerically greater than -il -, and which becomes two straight lines Avhen x' = ±-1/—; ' L for !/ = y', Lx"" - .V^- = P- My", an liy}ierbola whose real axis is parallel to the X-axis when parallel to the Z-axis when y' is numerically greater than (/> jp \/— , and which becomes two straight lines when ^' = ±\/— : V M ^ M for z-z', Lx' + My- = P + Nz'-, an ellipse, always real, whose dimensions are least when z' = 0, and increase indefinitely when z' increases indefinitely in nu- merical value. Calling the intercepts of this surface on the X-axis a, on the F-axis b, on the Z-axis cV— 1, the equation becomes — -f -'^ — - = 1, the hyperboloid of one sheet. a^ b' & (c) Lx' -f My' -Nz' = - P. The intersections are for X = x', My' -Nz' = -P- Lx", an hyperbola with its real axis parallel to the Z-axis, dimen- sions least when a;' = 0, increasing indefinitely with the numeri- cal value of x' ; for y = y', Lx" -Nz' = - P- My", an hyperbola with its real axis parallel to the Z-axis, dimen- sions least when y' — 0, increasing indefinitely with the numeri- cal value of y' ; for z = z', Lx' + My- = Lz" - P, 210 ANALYTIC GEOMETRY an ellipse, imaginary when —a/— < 2;' < -\-\y' dimensions zero for z' — ±\j—, increasing indefinitely with the numerical value of z'. Calling the intercepts of this surface on the axes X, Y, Z respectively, aV— 1, 6V— 1. c, the equation becomes a? Ir c- the hyperboloid of two sheets. The surfaces of the first class are ellipsoids and hyperboloids. Art. 109. — Surfaces of the Second Class The equation of the second class may take the forms (a) Lx' + Ml/ ± N'z = 0, (&) Lx" - Mxf ± N'z = 0. (ct) Lx- + 3fy- = N'z. The intersections are for x = x', Mf- = N'z - Lx'\ a parabola Avhose parameter is constant, axis parallel to Z-axis, and whose vertex continually recedes from the origin ; for y = y', Lx^ = N'z - My", a parabola whose parameter is constant, axis parallel to Z-axis, and whose vertex continually recedes from the origin ; for z = z', Lx^ + 3fy- = N'z', an ellipse whose dimensions are zero for z' = and increase indefinitely as z' increases from to + co, but are imaginary for2;'<0. This surface is the elliptic paraboloid. The equation Lx^ + My^ = — N'z represents an elliptic paraboloid real for negative values of z. (b) Lx^ — My- = N'z. The intersections are for x = x', My- = Lx'- - N'z, a parabola of constant parameter whose axis is parallel to SECOyi) DEGREE EQUATION 211 the Z-axis and whose vertex recedes from the origin as x' increases numerically ; for y = ij', Lx^ = N'z + My'-, a parabola of constant parameter whose axis is parallel to the Z-axis and whose vertex recedes from the origin as y' increases numerically ; for z = z', Lxr — My- = N'z', an hyperbola whose real axis is parallel to the X-axis when z > 0, paralh'l to the I'-axis when z' < 0, and which becomes two straight lines when z' = 0. The surface is the hyperbolic paraboloid. The surfaces of the second class are paraboloids. Art. 110. — Surfaces of the Third Class The equation Lx^ + 3fy^ + L'x + 31 'y + P = does not con- tain z and therefore represents a cylindrical surface whose elements are parallel to the Z-axis. The directrix in the XF-plane is an ellipse Avhen L and 3f have like signs, an hyperbola when L and 31 have unlike signs. The surface represented by the e(piation Lx- + L'x + 31' y is intersected by the A'l"- })lane in the parabola Lx- + L'x + 3['y + P - 0, by the ZX-plane in the parabola Lx- + L'x -\-N'z + P = 0, 212 ANALYTIC GEOMETRY by planes x = x' parallel to the I'Z-plane in parallel straight lines N'y + L'z + Mx'- + 3I'x' + P = 0. Hence the surface is a parabolic cylinder with elements parallel to the ZF-plane. The surfaces of the third class are cylindrical surfaces with elliptic, hyperbolic, or parabolic bases. It is now seen that the second degree equation in three variables represents ellipsoids, hyperboloids, paraboloids, and cylindrical surfaces with conic sections as bases. Conical sur- faces are varieties of hyperboloids. Art. 111. — QuADRics as Ruled Surfaces The equation of the hyperboloid of one sheet '—^ ^ = 1 — --^ is satisfied by all values of x, y, z, which satisfy simultaneously the pair of equations l-l=t.(l-^, ^ + ?=.lfl+f\ (1) or the pair a c \ bj a c fx - - ■^+A - + ^=lfl-?A (3) a c \ oj a c fjL c a'V b when fx and fx' are parameters. For all values of /x equations (1) represent two planes whose intersection must lie on the hyperboloid. Likewise equations (2) for all values of fx' repre- sent two planes whose intersection must lie on the hyperboloid. There are therefore two systems of straight lines generating the hyperboloid of one sheet. Each straight line of one system is cut by every straight line of the other system. For the four equations (1) and (2) made simultaneous are equivalent to the three equations SECOND DEGREE EQUATION 21:J from ^vlli{•]l // M — /a' X 1 + llfx' Z _\ — fjifx.' b /u. + /a' « /A + /a' c /a + /a' Ko two straight lines of the same system intersect. "Write the equations of lines of the first system corresponding to //j and fx.0. INfaking the equations simultaneous (/jLi—fi^)! 1 — -- Wo, /"l 1 \ / \ \ ^/ and (- .Yl+?^]=0. Hence either n-i—fx-., or y = h and II = — h. Since _?/ cannot be at once + h and — 6, //.i = yu,o ; that is. two lines of the same system can intersect only if they coincide. Observing that the equation of the hyperbolic paraboloid '^ = 2 .T is satisfied by the values of x, y, z, which satisfy either of the pairsj, of equations z y _ I X Vc Vb /* ' Vc Vb 0) z y 2x Vc Vb /' Vc V6 (-0 it can be shown that this surface niaj^ be generated by two systems of straight lines ; that each line of one system is in- tersected by every line of the other, and that no two lines of the same system intersect. The equations of ellipsoid, hy})erl)()l()id of two sheets and of elliptical paraboloid cannot be resolved into real factors of the first degree, consequently these surfaces cannot be gener- ated by systems of real straiglit lines. Akt. 112. — AsvMi'i'oTic Sri;FA(;?:s From the equaticm of the hypcrlxiloid of one sheet b- 214 ANALYTIC GEOMETRY it is found that ^x^y the powers of a^'if + h-x^ in the denominators increasing in the expansion by the binomial formula. Hence the z of the hyperboloid -„ + ^ — - = 1, and the z of the cone o} b^ c- •ii + ^_ _ ^ = ce h"- & approach equality as x and y are indefinitely increased ; that is, the conical surface is tangent to the hyperboloid at infinity. ■^ — ^ = is shown to be asymp- a- h^ <T totic to the hyperboloid of two sheets '-, — ^ — ;, = 1- a^ IP- & Art. 113. — Orthogonal Systems of Quadrics The equation (1 ) — ^ h tt^^ + -^r^— = 1' where a>h>c and A is a parameter, represents an ellipsoid when co > A > — c^, an hyperboloid of one sheet when — c^ > A > — Ir, an hyper- boloid of two sheets when — 6^>A> — a', an imaginary sur- face when A < — al Through every point of space {x\ y\ z') there passes one ellipsoid, one hyperboloid of one sheet, and one hyperboloid of two sheets of the system of quadrics represented by equa- tion (1). Por, if A is supposed to vary continuously from + oo to — (X) through 0, the function of A, a^ -I- A b- + X c- + A. 1, SECOND DEGREE EQUATION 215 is — when A = -f vd and + when A is just greater than — r, — when A is just less than — c" and + when A is just greater than — //-, — wlien A is just less than — b^ and again + when A is just greater than — cr. Hence must determine three real values for A; one between + oo and — r, another between — c- and — b^, a third between — b' and — ((-. Let A„ Ao, A3 be the roots of equation (2) ; that is, let g-'-' y'- z'- _ . -, ^.'2 -,,»2 ^i-' - 1, (4) a- + A., b'^ + A, c- + A; a- + A3 6- + A3 C- + A3 ■ ^'^ The equations of tangent planes to the quadrics of system (1) corresponding to Ai, As, A3 at the point of intersection {x\ y\ z') are XX + b' + A, + - 22 r + A, = 1, xx' «■ + A. + b' + \, + zz' = 1, xx' + + zz' = 1. ((•- + A3 b' + A3 t- + A3 The condition of perpendicularity of the iii'st two [tlanes ^ +__.'/"' .+ ^ (a' + A,)(«- + A.) {b-' + X,){b-' + Ao) (c- + AO(r + A.) is a conse(pience of (.'i) and (4). In like manner it is shown that the three tangent planes are mutually perpendicular. 216 ANALYTIC GEOMETRY Hence equation (1) represents an orthogonal system of quad- rics. Since through every point of space there passes one ellipsoid, one hyperboloid of one sheet, and one hyperboloid of two sheets of the orthogonal system of quadrics, the point in space is determined by specifying the quadrics of the orthogo- nal system on which the point lies. This leads to elliptic coordinates in space, developed by Jacobi and Lame in 1839, by Jacobi for use in geometry, by Lame for use in the theory of heat. If a bar kept at a constant temperature is placed in a homo- geneous medium, when the heat conditions of the medium have become permanent the isothermal surfaces are the ellip- soids, the surfaces along which the heat flows the hyperboloid s, of the orthogonal system of quadrics. NEW AMERICAN EDITION OF HALL AND KNIGHT'S ALGEBRA, FOR COLLEGES AND SCHOOLS. Revised and Enlarged for the Use of American Schools and Colleges. By FRANK L. SEVENOAK, A.M., Assistant Principal of the Acadi-mic Di-partmcnt, Stevens Institute of TiJinolosy. Half leather. 12mo. $ 1 . lO. JAMES LEE LOVE, Instructor of Mathematics, Harvard University, Cambridge, Mass. : — I'mfessor Sevcnoak's rcvisimi of tlie F.lriiieiitnry AlgLbra is WW L\i client l>i»ik. I wish I could persuade all the teachers titling boys for the Laurence Sucntihc Sch.iol to use it. VICTOR C. ALDERSON, Professor of Mathematics, Armour Institute, Chicago, 111.: — We have Msea the lOn-lish Eililiou for the past two years in our Scienlilic Academy The new edition is superior to the ohl, ami we shall certainly use it. In my opiiiiou it is the best of all the elementary algebras. AMERICAN EDITION OF ALGEBRA FOR BEGINNERS. By H. S. HALL, M.A., and S. R. KNIGHT. NEVISED BY FRANK L. SEVENOAK, A.M., Assistant Principal of the Academic Department, Stevens Institute of Technology. 16mo. Cloth. 60 cents. An edition of this book containing additional chapters on Radicals and the Binomial Theorem will be ready shortly. JAMES S. LEWIS, Principal University School, Tacoma, Wash.: — 1 have examined Hall and Knight's "Algebra for Beginners " as revised by Professor .Sev- enoak, and consider it altogether the best book for the purpose intended that I know of MARY McCLUN, Principal Clay School, Fort Wayne, Indiana: — I have examined the .Mgebra (piite carefully, and I fuul it the best 1 have ever .seen. Its greatest value is found in the simple and clear language in which all its definitions are expressed, and in the fact that each new step is so carefully explained. The ex- amples in each chapter are well selected. I wish all teachers who leach Algebra might be able to use the "Algebra for Beginners." THE MACMILLAN COMPANY, 66 FIFTH AVENUE, NEW YORK. AMERICAN EDITION OF LOCK'S TRIGONOMETRY FOR BEGINNERS, WITH TABLES. Revised for the Use of Schools and Colleges By JOHN ANTHONY MILLER, A.M., Professor of Mechanics and Astronomy at the Indiana University. 8vo. Cloth. $1.10 net. IN PREPARATION. AMERICAN EDITION OF HALL and KNIGHT'S ELEMENTARY TRIGONOMETRY, WITH TABLES. By H. S. HALL, M.A., and S. R. KNIGHT, B.A. Revised and Enlarged for the Use of American Schools and Colleges By FRANK L. SEVENOAK, A.M., Assistant Principal of the Academic Department, Stevens Institute of Technology. THE MACMILLAN COMPANY, 66 FIFTH AVENUE. NEW YORK. ELEMENTARY SOLID GEOMETRY. HENRY DALLAS THOMPSON, D.Sc, Ph.D. Professor of Mathematics in Princeton University. i2mo. Cloth. $i.io, net. This is an elementary work on Geometry, brief and interesting, well and well written. — School of Mines Quarterly. THE ELEMENTS OF GEOMETRY. By GEORGE CUNNINGHAM EDWARDS, Associate Professor of Mathematics in the University of California. i6mo. Cloth. $i.io, net. PROF. JOHN F. DOWNEY, University of Minnesota : — There is a gain in its being loss formal th;in many of the works on this siiljict The arrangement and treatment ;ire such as lo develop in the student ability to do geometrical work. The book would furnish the preparation necessary for admission to this University. PRIN. F. 0. MOWER, Oak Normal School, Napa, Cal.:-Of the fifty or more English and American editions of Geometry which I have on my shelves, I consider this one of the best, if not the best, of them all. I shall give it a trial in my next class beginning that subject. THE MACMILLAN COMPANY, 66 FIFTH AVENUE, NEW YORK. MATHEMATICAL TEXT-BOOKS SUITABLE FOR USE IN PREPARATORY SCHOOLS. SELECTED FROM THE LISTS OF THE MACMILLAN COMPANY, Publishers. ARITHMETIC FOR SCHOOLS. By J. B. LOCK, Author qf " Trigonometry for Beginners," "Elementary Trigonometry," etc. Edited and Arranged for American Scliools By CHARLOTTE ANGAS SCOTT, D.SC, Head of Math. Dcpt., Bryn Maiur College, Pa. 1 6mo. Cloth. 75 cents. " Evidently the work of a thoroughly good teacher. The elementary truth, that arithmetic is common sense, is the principle which pervades the whole book, and no process, however simple, is deemed unworthy of clear explanation. Where it seems advantageous, a rule is given after the explanation. . . . Mr. Lock's admirable ' Trigonometry ' and the present work are, to our mind, models of what mathematical school books should be." — The Literary World. FOR MORE ADVANCED CLASSES. ARITHMETIC. By CHARLES SMITH, M.A., Author of " Elementary Algebra," "A Treatise on Algebra," AND CHARLES L. HARRINGTON, M.A., Head Master of Dr. J. Sach's School for Boys, New York. 1 6mo. Cloth. 90 cents. A thorough and comprehensive High School Arithmetic, containing many good examples and clear, well-arranged explanations. There are chapters on .Stocks and Bonds, and on Exchange, which are of more than ordinary value, and there is also a useful collection of miscellaneous examples. THE MACMILLAN COMPANY, 66 FIFTH AVENUE, NE^AT YORK. UNIVERSITY OF CALIFORNIA LIBRARY BERKELEY Return to desk from which borrowed. This book is DUE on the last date stamped below. OCT 7 1955 Lin - I6Dec'55GB RECTD Ltli" I'' ■V?^ 3 135? LD 21-95»ir-ll,'50(2877sl6)476 REC'D LD NOV 29 ,:: #v^'' ilAV^'ii '"^ APR 3 C ';4 '' 9 I ■ U C RhRKFLFY LIBRARIES C0t.l3SlbDb H0()554 IS 4-« UNIVERSITY OF CALIFORNIA LIBRARY