The ellipse ( x^2 a^2 + y^2 b^2 =1(b>a) ) and the parabola (y^2=4 a x ) cut at right angles. If (e ) is the eccentricity of the ellipse, then (2 e^2= )

Step-by-step solution

Given equation of parabola is (y^2=4 a x ), then slope of tangent to be given parabola at point of intersection ( (a t^2, 2 a t ) ) is (m₁= 1 t^ ), and slope of tangent to the given ellipse ( x^2 a^2 + y^2 b^2 =1 ) at point of intersection ( (a t^2, 2 a t ) ) is ( aligned m₂ & =- b^2 a^2 ( t 2 ) m₁ m₂ & =-1 ( 1 t ) (- b^2 a^2 t 2 ) & =-1 b^2 a^2 =2 e^2 & =1- a^2 b^2 =1- 1 2 ( b > a) 2 e^2 & =1 aligned ) Hence, option (a) is correct.