Let the focal chord of the parabola P : y 2 = 4 x along the line L : y = m x + c , m > 0 meet the parabola at the points M and N . Let the line L be a tangent to the hyperbola H : x 2 - y 2 = 4 . If O is the vertex of P and F is the focus of H on the positive x -axis, then the area of the quadrilateral O M F N is

Step-by-step solution

Given, parabola y 2 = 4 x and line L :   y = m x + c , Now plotting the diagram we get, Now focus of hyperbola, H : x 2 4 - y 2 4 = 1 will be a e , 0 ⇒ F 2 2 , 0 Now given line L : y = m x + c pass focus of parabola 1 , 0 ⇒ 0 = m + c           ⋯ 1 Now also given line L is tangent to Hyperbola. x 2 4 - y 2 4 = 1 So by using condition of tangent to hyperbola in slope form we get, c = ± a 2   m 2 - l 2 ⇒ c = ± 4 m 2 - 4 Now from 1 we get, - m = &#177