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Tangents are drawn at the end points of a normal chord of the parabola y 2 = 4 a x . The locus of their point of intersection is

Options

  1. Ax - 2 a y 2 + 4 a 3 = 0
  2. Bx - 2 a y 2 - 4 a 3 = 0
  3. Cx + 2 a y 2 - 4 a 3 = 0
  4. Dx + 2 a y 2 + 4 a 3 = 0

Correct answer

D. x + 2 a y 2 + 4 a 3 = 0

Step-by-step solution

Let, P a t 1 2 , 2 a t 1 and Q a t 2 2 , 2 a t 2 be the end points of a normal chord of the parabola y 2 = 4 a x such that P Q is normal at P . Then, t 2 = - t 1 - 2 t 1 …(i) Let, h , k be the point of intersection of tangents to the parabola at P and Q . Then, h = a t 1 t 2 and k = a t 1 + t 2 ⇒ h = a t 1 - t 1 - 2 t 1 and k = - 2 a t 1 . [Using (i)] ⇒ h = - a t 1 2 + 2 and t 1 = - 2 a k ⇒ h = - a 4 a 2 k 2 + 2 ⇒ h + 2 a k 2 = - 4 a 3 Hence, the locus of h , k is x + 2 a y 2 = - 4 a

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