Let P Q be a focal chord of the parabola y 2 = 4 x such that it subtends an angle of π 2 at the point 3 , 0 . Let the line segment P Q be also a focal chord of the ellipse E : x 2 a 2 + y 2 b 2 = 1 , a 2 > b 2 . If e is the eccentricity of the ellipse E , then the value of 1 e 2 is equal to

Step-by-step solution

Since P Q is focal chord of the parabola y 2 = 4 a x so let P t 2 , 2 t , Q 1 t 2 , - 2 t and R 3 , 0 Given m P R · m P Q = - 1 2 t t 2 - 3 × - 2 t 1 t 2 - 3 = - 1 t 2 - 1 2 = 0 ⇒ t = 1 i.e. P Q must be the latus rectum We get P 1 , 2   &   Q 1 , - 2 For ellipse 2 b 2 a = 4   &   a e = 1 Also b 2 = a 2 1 - e 2 ∴   a = 1 + 2 and   e 2 = 3 - 2 2 Hence 1 e 2 = 1 3 - 2 2 = 3 + 2 2