Let the eccentricity of an ellipse x 2 a 2 + y 2 b 2 = 1 is reciprocal to that of the hyperbola 2 x 2 - 2 y 2 = 1 . If the ellipse intersects the hyperbola at right angles, then square of length of the latus-rectum of the ellipse is _____.

Step-by-step solution

Given, The eccentricity of an ellipse x 2 a 2 + y 2 b 2 = 1 is reciprocal to that of the hyperbola 2 x 2 - 2 y 2 = 1 , Now eccentricity of rectangular hyperbola 2 x 2 - 2 y 2 = 1 is e H = 2 So, e e = 1 2 Now, focus of hyperbola = ± 1 ,   0 Now given that both curve intersect orthogonally, so ellipse and hyperbola are confocal So, for ellipse a e e = 1 ⇒ a = 2 Now length of latusrectum L . R . = 2 b 2 a = 2 a 1 - e e 2 = 2 2 . 1 2 = 2