The asymptotes of the hyperbola x 2 a 2 - y 2 b 2 = 1 , with any tangent to the hyperbola form a triangle whose area is a 2 tan α . Then its eccentricity equals

Step-by-step solution

Given hyperbola x 2 a 2 - y 2 b 2 = 1 Asymptotes of hyperbola are x a + y b = 0 . . . I   ,   x a - y b = 0 . . . I I Tangent at x 1 ,   y 1 is x x 1 a 2 - y y 1 b 2 = 1 . . . . I I I We know that area formed by a triangle with asymptotes and any tangent is equal to a b . ⇒ a b   =   a 2   tan   α ⇒ b   = a   tan   α Eccentricity of hyperbola are e   =   a 2 + b 2 a 2 ⇒ e   =   a 2 + a 2 tan 2 α a 2 =   1 + t