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BIBLIOGRAPHIC RECORD TARGET Graduate Library University of Michigan Preservation Office Storage Number: ABRO0I8 UL FMTBRTaBLmT/C DT07/18/88R/DT07/18/88CC STATmmE/L1 035/1: : |a (RLIN)MIUG86-B48728 035/2: : [a (CaOTULAS)160121345 040: : [a IC] |cMiU 100:1 : |a Loney, Sidney Luxton. 245:04: |a The elements of coordinate geometry, |c by S. L. Loney, M.A. 250: : la 2d edition, revised. 260: : |a London, |a New York, |b Macmillan and Co., |c 1896. 300/1: : fa ix, 416, xiii p. |b diagrs. |c19 cm. 650/1: 0: |a Geometry, Analytic 998: : |c WFA |s 9124 Scanned by Imagenes Digitales Nogales, AZ On behalf of Preservation Division The University of Michigan Libraries Date work Began: Camera Operator: THE ELEMENTS or COORDINATE GEOMETRY. First Edition (Globe Bvo.), 1895. Reprinted with corrections (Crown Bvo.), 1896, PREFACE. N the following work I have tried to present the elements of Coordinate Geometry in a manner suitable for Begimners and Junior Students. The present book (only y Áleald with Cartesian and Polar Coordinates. Within these limits 1 venture to hope that the book is fairly complete, and that no proposi- tions of very great importance have been omitted. The Straight Linc and Circle have been treated more fully than the other portions of the subject, since it is generally in the elementary conceptions that beginners find great difficulties. There are a large number of Examples, over 1100 in all, and they are, in general, of an elementary character. The examples are especially numerous in the earlier parts of the book, CHAP, II. NI, IV. VI. VIL CONTENTS. PAGE INIRODUCFION. ÁLGEBRAIC RESULTS , . 1 CooRDINATES. LENGTHS OF STRAIGHT LINES AND AREAS OF TRIANGLES . . . . . 8 Polar Coordinates . . . . . . 19 Locus. Equation tro A Locus . . . . 24 The Straight Line. RecrancuLAR COORDINATES, 31 Straight line through two points-. . . . 39 Angle between two given straight lines . . 42 Conditions that they may be parallel and per- pendicular . . . . . . . 44 Length of a perpendicular . . . . 51 Bisectors of angles . . . . . . 58 Tam Serargar Lise. PoLrar EQUATIONS AND OBLIQUE COORDINATES . . . . [o Equations involving an arbitrary constant . . B Examples of loci . . . . , . 80 EQUATIONS REPRESENTINS TWO OR MORE STRAIGHT Lints : . : . . . . .. 88 Angle between two lines given by one equation 90 General equation of the second degree . . 9 TRANSFORMATION OF COORDINATES . . . 109 Tnvaxiants . . . . . . . . 116 viil CHAP, VII IX. XI. NIE XUI. CONTENTS, The Circle . Equation to a tangent . Pole and polar . Equation to a circle in polar coordinates Equation referred to oblique axes Equations in terms of one variable SystEMS OF CIRCLES . Orthogonal circles . Radical axis Coaxal circles Conic Sections, Tam PARABOLA Equation to a tangent . Some properties of the parabola Pole and polar Diameters . Equations in terms of one variable Tur PARABOLA (continued) Loci connected with the parabola Three normals passing through a given point. Parabola referred to two tangents as axes Tue ELLIPSE Ausiliary cirele and eccentric angle Equation to a tangent . . Some properties of the ellipse Pole and polar . Conjugate diameters . Four normals through any point Examples of loci THE HyPERBOLA Asymptotes Equation referred to the asymptotes as axes . One variable. Examples PAGE 118 126 137 145 148 150 160 160 161 166 174 180 187 190 195 198 206 206 211 217 225 281 237 24% 249 asa 265 266 s71 284. 296 299 CHAPTER T INTRODUCTION. SOME ALGEBRAIC RESULTS. 1. Quadratic Equations. The roots of the quad- ratio equation aa? + ba + c=0 may easily be shewn to be —-b+ NiB— dao 2a and —-b- bi das , 2a É They are therefore real and unequal, equal, or imaginary, according as the quantity &º—- 4ac is positive, zero, or negative, i.e, according as bº E 4a. < 2. Relations between the roots of any algebrarc equation and the coeficients of the terms 0f the equation, If any equation be written so that the coeficient of the highest term is unity, it is shewn in any treatise on Algebra that (1) the sum of the roots is equal to the coeflicient of the second term with its sign changed, (2) the sum of the products of the roots, taken two at a time, is equal to the coefficient of the third term, (3) the sum of their products, taken three at a time, is equal to the coefficient of the fourth term with its sign changed, and so on. L. e 1 DETERMINANTS, 3 [Mi Cas Cal djs Das Ogleccescceremerererearea (1) Cis Cas Os is called a determinant of the third order and stands for the quantity 5. The quantity & ba ba] E Dol ais da + Os Cos Cy Cs Cs! [Em Ca %e. by Art. 4, for the quantity O (Dao; — bato) -— Eg (Dies = ba) + Oy (bit, — bat)» ERA Cy (Doc; — ba6a) + do (Bot — Des) + dt (Dito — Dos). 6. A determinant of the third order is therefore reduced to three determinants of the second order by the foliowing rule: Take in order the quantities which occur in the first row of the determinant; multiply each of these in turn by the determinant which is obtained by erasing the row and column to which it belongs; prefix the sign + and — al. ternately to the products thus obtained and add the results. Thus, if in (1) we omit the row and column to which «, ni bi and this is the Cu, Cai belongs, we have left the determinant coefficient of a in (2). Similarly, if in (1) we omit the row and column to which P ds] and this C1, Os] with the — sign prefixed is the coeficient of às in (2). «» belongs, we have left the determinant t 1,-2,-8 7. Ex. The determinant |-4, 5,-6 -7, 8-9 5 | 4, —6 =1x 8, o (-2 pa 7, e 3) x7 [0-9 8x ( 02x dt- -(- mM 6) IL 4)x8-(-7)x5) =[-45+48] +2/86-42)-34 82485) =8-12-9=-18, 1.2 4 COORDINATE GEOMETRY, Grs Coy Ugy Gy db ba, da, da Cs Coy Cs, Og da, dy da, dy is called a determinant of the fourth order and stands for the quantity 8. The quantity da, ds, d, Dis Das dba Cy X [Coy Css Caj— Go X | Cr Css Cy dos da, dy hs das da dos bs dy ss Day by] +OgX|0 Ca C|—CaX|C Ca Cy|y ds da, da [dh do, ds and its value may be obtained by finding the value of each of these four determinants by the rule of Art. 6. The rule for finding the value of a determinant of the fourth order in terms of determinants of the third order is clearly the same as that for one of the third order given in Art, 6. Similarly for determinants of higher orders. 9. A determinant of the second order has two terms. One of the third order has 3 x 2, 2.e. 6, terms. One of the fourth order has 4x3 x 2, %e. 24, terms, and so on. 10. Exs. Prove that 5, —3, T (1) h “ál=28, (2) [4 fes e) |-2, 4 gf=-s5 NA ) 9 3, -10) 19,8,7 -a bc (4) 16,5, 4/=0, (5) q -db, el=tade, 83,2,1 la db-c “hq (6) “hs d, f|=abe+2fgh — af? - dg? — chê, mnhe 6 COORDINATE GEOMETRY. By dividing each equation by 2 we have three equations between the two unknown quantities > and é. Two of these will be sufficient to determine these quantities. By substituting their values in the third equation we shall obtain a relation between the nine coefficients, Or we may proceed thus. From the equations (2) and (3) we have Doo — bata E Doc, — bycg E Bila — De, Substituting these values in (1), we have et (Docs — Duca) + ig (Duty — Dirty) + dog (Dyt, — Do0y) = 0... (4). This is the result of eliminating x, y, and z from the equations (1), (2), and (3). But, by Art. 5, equation (4) may be written in the form Ley, Oss Gy by, by, by = 0. Cy Cy 6] This eliminant may be written down as in the last article, viz. by taking the coefficients of x, 3, and z in the order in which they occur in the equations (1), (2), and (3), placing them to form a determinant, and equating it to zero. 13. Ex. What is the value of a so that the equations get + 32=0, 22-9y+42=0, and duty — 82=0 may be simultancoushy true ? Eliminating «x, y, and 2, we have a 2 3i 2, —8, 4/=0, 5 7-8 te al(-B)(-8)-4x7]-2[2x(-8)-4x5]+8[2x7-5x(-3)]=0, ie. a[-4]-2[-86]-+8[29]=0, 72+87 159 a= =>" so thai A Sa ELIMINATION. g 14, Tfagain we have the four equations Gê + Coy + gg + ago =0, d+ by+ bx+bu=0, CGx+ey+ce+cu=0, and dy + day + dg + du =0, it could be shewn that the result of eliminating the four quantities «, y, &, and w is the determinant Gr Gay Og; Mg bis Das bs Da. q Cy, Ca Cas Os dh, da, ds, da | A similar theorem could be shewn to be true for n equations of the first degree, such as the above, between m unknown quantities, Tt will be noted that the right-hand member of each of the above equations is zero. 
