Let the hyperbola H : x 2 a 2 - y 2 b 2 = 1 pass through the point 2 2 , - 2 2 . A parabola is drawn whose focus is same as the focus of H with positive abscissa and the directrix of the parabola passes through the other focus of H . If the length of the latus rectum of the parabola is e times the length of the latus rectum of H , where e is the eccentricity of H , then which of the following points lies on the parab

Step-by-step solution

Given, H :   x 2 a 2 - y 2   b 2 = 1 So coordinates of foci will be : S a e , 0 , S ' − a e , 0 Now foot of directrix of parabola will be - a e , 0 Also focus of parabola is which is same as focus of H will be a e ,   0 Now, semi latus rectum of parabola = S S ' = 2 a e Given, 4 a e = e 2 b 2 a ⇒ b 2 = 2 a 2           . . . 1 Also given, 2 2 , - 2 2 lies on H :   x 2 a 2 - y 2   b 2 = 1 ⇒ 2 2 2 a 2 - 2 2 2   b 2 = 1 ⇒ 1 a 2 - 1 &