Art. 122. The length of the perpendicular from the centre on the tangent, = 125. If CE is drawn parallel to the tangent, meeting HP in E, then PE AC 126. The equation to the normal Page 78 130. All the diameters of the ellipse pass through the centre; y = ax + c, a2a y + b2x = 0, are the chord and corresponding diameter 131. There is an infinite number of pairs of conjugate diameters; 133. Equation to the curve referred to any conjugate diameters, 136. The sq. on QV: the rectangle P V, VP' :: sq. on CD: sq. on CP 138. The ellipse being referred to its axes, the tangent is parallel to the conjugate diameter: the two equations are, a2 y y' + b2 x x = a b2, the tangent, a2 y y' + b2 xx′ = 0, the parallel conjugate 139. The square upon CD = the rectangle SP, HP 140. The perpendicular from the centre on the tangent, 141, 142. The product of the tangents of the angles which a pair of supplemental chords makes with the axis major is constant, 143. The tangent of the angle between two supplemental chords, 79 79 80 81 82 82 83 81 144. Supplemental chords are parallel to conjugate diameters 145. The equation to the ellipse, referred to its equal conjugate diameters, is X Art. 148. The focus, the pole, r = a (1-e2) -e cos. 2a (1-e2) cos. 149. The pole at the vertex, u = 155-7. Discussion of the equation: The sq. on MP: rectangle A M, M A':: sq. on BC: sq. on A C . 158. The equation to the equilateral hyperbola is y2 — x2 = — a2 159. The results obtained for the ellipse are applicable to the hyperbola, by changing b2 into - b 160. The Latus Rectum is defined to be a third proportional to the transverse and conjugate axes 161-3. The focus; the eccentricity: The rectangle AS, S A' = the square on B C ex-a, HP ex+a, HP – SPAA' 164. SP 91 165. To find the locus of a point the difference of whose distances from two fixed points is constant 168. The equation to the tangent, at the extremity of the Latus Rectum, is y = ex- a. The distances of any point from the focus and from the directrix are in a constant ratio 169. The length of the perpendicular from the focus on the tangent, 170. The locus of y is the circle on the transverse axis 171. The tangent makes equal angles with the focal distances, 94 95 96 96 97 Art. Page 174. If CE be drawn parallel to the tangent and meeting HP in E, then P EAC 98 175-7. The equation to the normal is The rectangle P G, PG' the rectangle SP, HP 178, 9. The diameters of the hyperbola pass through the centre, but do not all meet 180, 1. There is an infinite number of pairs of conjugate diameters, 182. The equation to the curve referred to conjugate axes is 185. The sq. on QV: the rectangle P V, V P' :: sq. on CD: sq. on CP 187. The conjugate diameter is parallel to the tangent. The equations are a2 y2 y — b2x2x′ = a2 62 the tangent. a2 yy - b2 x x′ = 0 188. The sq. on C D = the rectangle SP, HP 189. If P F be drawn perpendicular on CD, then the conjugate 190-2. If a and be the tangents of the angles which a pair of supplemental 102 103 plemental chords. Conjugate diameters are parallel to sup. chords 193. There are no equal conjugate diameters in general. In the equilateral hyperbola they are always equal to each other 194-6. The Asymptotes. The equation to the asymptote is the equation to the curve, with the exception of the terms involving inverse powers of x. Curvilinear asymptotes 197. The hyperbola is the only one of the lines of the second order that has a rectilineal asymptote 198. Method of reducing an equation into a series containing inverse powers of a variable. The asymptotes parallel to the axes 105 199. Discussion of the equation b x y +ƒ=0 200. Referring the curve to its centre and axes, the equations are a2 y2 — b2 x2 — — a2 b2, the curve, 201. In the equilateral or rectangular hyperbola (y2 - x2 = — ao) the angle between 202, 3. Asymptotes referred to the vertex of the curve; a line parallel to the asymptote cuts the curve in one point only Art. 204. Examples of tracing hyperbolas, and drawing the asymptotes 206. Reduction of the general equation of the second order to the form xy=k2 a (tan. 6)2 + b tan. 6 + c = 0 207. To find the value of b 209. Examples. If ca, the curve is rectangular 109 110 110 211. Given the equation xy=k, to find the equation referred to rectangular axes, and to obtain the lengths of the axes 212. From the equation a2 y2 - b2x2 — — a2 b referred to the centre and axes to obtain the equation referred to the asymptotes, 213. The parallelogram on the co-ordinates is equal to half the rectangle on the 214. The parts of the tangent between the point of contact and the asymptotes are equal to each other and to the semi-conjugate diameter 215. Given the conjugate diameters to find the asymptotes. If the asymptotes are given, the conjugate to a diameter is given 216. The equation to the tangent referred to the asymptotes 217, 8. The two parts of any secant comprised between the curve and asymptote are equal. The rectangle S Q, QS' sq. on CD 219. The general polar equation is 112 113 114 114 . 114 115 . 115 223, 4. The conjugate hyperbola. The locus of the extremity of the conjugate diameter is the conjugate hyperbola. The equation is 116 117 225, 6. The equation to the parabola referred to its axis and vertex is y2= px 229. The principal parameter, or Latus Rectum, is a third proportional to any abscissa and its ordinate. In the following articles 4m is assumed to be the value of the Latus Rectum Art. 230. To find the position of the focus 231. The distance of any point on the curve from the focus, S P = x+m 232. The equation to the tangent is 235. The equation to the tangent at the extremity of the Latus Rectum is 121 236. The Directrix. The distances of any point from the focus and directrix are equal 237. The length (Sy) of the perpendicular from the focus on the tangent =√m'r SP: Sy::Sy: SA 238, 9. The locus of y is the axis A Y. The perpendicular Sy cuts the directrix on the point where the perpendicular from P on the directrix meets that line 240. The tangent makes equal angles with the focal distance and with a parallel 242. The subnormal is equal to half the Latus Rectum : SGSP, and PG4mr 121 122 243. The parabola has an infinite number of diameters, all parallel to the axis 123 244, 5. Transformation of the equation to another of the same form referred to a 246. The new equation is y2=px; the new parameter p = 4 SP 247. Transformation of the equation when the position of the new origin and axes 125 248. The ordinate through the focus 4SP the parameter at the origin 249. The equation to the tangent 250. Tangents drawn from the extremities of a parameter meet at right angles in 251. The general polar equation is (yu sin. 4)2=p (x2+u cos. 0) |