If e is the eccentricity of the hyperbola x^2 a^2 - y^2 b^2 =1 and is the angle between the asymptotes, then ( 2 ) is equal to

Step-by-step solution

The equations of the asymptotes of the hyperbola x^2 a^2 - y^2 b^2 =1 are y= b a i e , y= b a x and y= -b a x If is the angle between the two asymptotes, then, = b a + b a 1- b a b a = 2 b a a^2 a^2-b^2 = 2 a b a^2-b^2 From this, we get = a^2-b^2 a^2+b^2 2 ^2 2 -1= a^2-b^2 a^2+b^2 2 ^2 2 =1+ a^2-b^2 a^2+b^2 2 ^2 2 = 2 a^2 a^2+b^2 ^2 2 = a^2 a^2+b^2 But b^2=a^2 (e^2-1 )=a^2 e^2-a^2 a^2+b^2=a^2 e^2 ^2 ( 2 )= a^2 a^2 e^2 = 1 e^2 ( 2 )= 1 e