The ordinates of the points P and Q on the parabola y 2 = 12 x are in the ratio 1 : 2 . The locus of the point of intersection of the normals to the parabola at P and Q is

Step-by-step solution

Here, a = 3 . So, let P 3 t 1 2 , 6 t 1 and Q 3 t 2 2 , 6 t 2 be two points on the parabola y 2 = 12 x . Then, 6 t 1 6 t 2 = 1 2 ⇒ t 2 = 2 t 1 . Let, h , k be the point of intersection of the normals at P and Q . Then, k = - t 1 h + 6 t 1 + 3 t 1 3 and k = - t 2 h + 6 t 2 + 3 t 2 3 ⇒ h = 6 + 3 t 1 2 + t 2 2 + t 1 t 2 and k = - 3 t 1 t 2 t 1 + t 2 ⇒ h = 6 + 3 t 1 2 + 4 t 1 2 + 2 t 1 2 and k = - 6 t 1 2 3 t 1 ⇒ h - 6 = 21 t 1 2 and k = - 18 t 1 3 ⇒ h - 6 21 3 = t 1 6 and - k 18 2 = t 1 6 ⇒ h - 6 21 3 = k 2 324 ⇒ k 2