. Algebraic geometry; a new treatise on analytical conic sections . duce ? 11. If the tangent at (as, y) to the hyperbola -3-^=1 cuts the auxiliarycircle at points whose ordinates are yi, y^, show that i i=l 12. Prove that a; cos a+y sin o =p is a tangent to the hyperbola if p = tJt^ oos^a - 6* sin^a. 13. If a tangent to the hyperbola -2 —?2= 1 meets the transverse axis at T, and the conjugate axis at t, prove that a?- 6^ _CT2 0^~ ■ 14. Find, from first principles, the equation of the tangent to thehyperbola x^-y^=a? at the point (x-^, y-^. 15. If the normal at the point (asecS, Btanfl) meets
. Algebraic geometry; a new treatise on analytical conic sections . duce ? 11. If the tangent at (as, y) to the hyperbola -3-^=1 cuts the auxiliarycircle at points whose ordinates are yi, y^, show that i i=l 12. Prove that a; cos a+y sin o =p is a tangent to the hyperbola if p = tJt^ oos^a - 6* sin^a. 13. If a tangent to the hyperbola -2 —?2= 1 meets the transverse axis at T, and the conjugate axis at t, prove that a?- 6^ _CT2 0^~ ■ 14. Find, from first principles, the equation of the tangent to thehyperbola x^-y^=a? at the point (x-^, y-^. 15. If the normal at the point (asecS, Btanfl) meets the transverseaxis at G, prove that AG . AG=ai(e* sec^e - 1). 16. Tangents to the hyperbola 3a;^-2y=6 are drawn from the point(A, i) and make angles 9,, 0^ with the axis of x. If taneitane2=2, provethatF=2A2-7. __*_ —1 244 THE HYPERBOLA. [chap. XII. 260. Def. An asymptote to a hyperbola is a straight linewhich is itself not altogether at infinity, but which meets theconic at two points at inflhity. Tojmd the eguatims of the asymptotes to the hyperbolaa^~b^~. FlQ. 151. Where the straight line y = mx + c meets the curve, we have,by substitution, x^ (mx + c)^ _ 1 or a;2(62 - a^m^) - 2mahx - a^c^ - aW = 0. Now if y = mx + c is an asymptote, both roots of this equationmust be infinite; .■. J2-a%2 = o and ma^c = 0 (Art. 150); .. m= +- and c = 0: a • ., y= ± — are the equations of the asymptotes. ART. 268.]. ASYMPTOTES. 245 They may be written - + f = 0 and --1 = 0,a 0 ah or, in one equation, ~2 ~ fs ^ ^• The slopes of the asymptotes are ± - ; .. the angle between them = 2 tan~i-. a We see that the asymptotes are the diagonals of the parallelo-gram formed by drawing straight lines through A, A, B, Bparallel to the axes. 261. If 2a is the angle between the asymptotes of a hyperbolasec a = e, the eccentricity. If CE, CE are the asymptotes, when EAE is the tangent at A,CA = a, AE = 6;, CE2 a^ + ¥ „CA^ a^ .. sec a = e. 262. -We know that y^mx + Ja^m^-b^ is a tangent to
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