. Algebraic geometry; a new treatise on analytical conic sections . Pio. 166. Let {asec6, htaxiO) be the co-ordinates of P, and draw CKperpendicular to the tangent at P. 276 PROPERTIES OF THE HYPERBOLA, [chap. xn. As in the ellipse (Art. 242), PG2, Also PF = CK = ab whence PF. PG = i that COKOLLARY. In the same way, as for the ellipse, we can = CA2. 297. If the diameter POP meets the hyperbola at P and P, and theconjugate diameter DCD meets the conjugate hyperbola at D and D,OP^ - CD2 = a^ - W:See Art. 287. With the assumption of the above proposition, SP. SP = CD^.SP. SP = {aesee6-
. Algebraic geometry; a new treatise on analytical conic sections . Pio. 166. Let {asec6, htaxiO) be the co-ordinates of P, and draw CKperpendicular to the tangent at P. 276 PROPERTIES OF THE HYPERBOLA, [chap. xn. As in the ellipse (Art. 242), PG2, Also PF = CK = ab whence PF. PG = i that COKOLLARY. In the same way, as for the ellipse, we can = CA2. 297. If the diameter POP meets the hyperbola at P and P, and theconjugate diameter DCD meets the conjugate hyperbola at D and D,OP^ - CD2 = a^ - W:See Art. 287. With the assumption of the above proposition, SP. SP = CD^.SP. SP = {aesee6-a)(aesec6-{-a) = aVsec^^-a^= (a2 + 62) see2^ -a^ = a^ tan^^ + b^ sec^S= Fia. 167. If CP, CD, conjugate semi^iarheters, meet the hyperbola and theconjugate hyperbola at P and D respectively, and tangents at P and Dare drawn to meet at T, the area of the parallelogram PODT = AC. BC- Use the method of Art. 241, remembering that if (asec^, 5tan^)are the co-ordinates of P, the co-ordinates of D are (a tan 6, JsecS). ] PROPERTIES OF TH^ HVPERBOLA. 277 Tangents at the ends of a chord intersect on the diameter whichbisects that chord. Use the method of Art. 230. If from T, a point in the diameter POP, tangents TQ, TQ aredrawn to the hyperbola, and 0,0! meets CT inW, = CP^. (SeeFig. 167, p. 276.) Use the method of Art. 243 with the necessary changes in theequations. Writing - ¥ for b% Arts. 244-247 hold for the hyperbola. Given a curve, which is known to be a hyperbola, find its centre, thepositions and lengths of its principal axes, and its foci. If both branches of the curve are given, the method of holds, If only one branch is given, t
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