Let a tangent to the curve y 2 = 24 x meet the curve x y = 2 at the points A and B . Then the mid-points of such line segments A B lie on a parabola with the

Step-by-step solution

Given: y 2 = 24 x Comparing with y 2 = 4 a x , we get 4 a = 24 ⇒ a = 6 Also given x y = 2 Let any point Q ≡ a t 2 , 2 a t ≡ 6 t 2 , 12 t on the parabola. Equation of tangent at Q is 12 y t = 12 x + 6 t 2 ⇒ y t = x + 6 t 2 ⇒ x - y t + 6 t 2 = 0       . . . . i Let mid-point of A B be P h , k . Equation of chord of hyperbola is x k + y h 2 = h k ⇒ x k + h y - 2 h k = 0       . . . ii Since, i   &   ii are same, so on comparing, we get k 1