The locus of the mid points of the chords of the hyperbola x 2 - y 2 = 4 , which touch the parabola y 2 = 8 x , is :

Step-by-step solution

Equation of chord of hyperbola x 2 - y 2 = 4 with mid-point h ,   k is given by x h - y k = h 2 - k 2 ⇒ y k = x h - h 2 + k 2 ⇒ y = h x k + k 2 - h 2 k This will touch the parabola y 2 = 8 x , if ⇒ k 2 - h 2 k = 2 h / k ⇒ 2 k h = k 2 - h 2 k ⇒ h 3 + 2 k 2 - k 2   h = 0 ∴ Equation of locus is x 3 + 2 y 2 - y 2 x = 0   ⇒   x 3 = y 2 x - 2