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The locus of the centre of the circle described on any focal chord of the parabola y 2 = 4 a x as the diameter is

Options

  1. Ay 2 = 2 a x + a
  2. By 2 = a x + a
  3. Cy 2 = 2 a x - a
  4. Dy 2 = 4 a x - a

Correct answer

C. y 2 = 2 a x - a

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 extremities of a focal chord P Q of the parabola y 2 = 4 a x . Then, t 1 t 2 = - 1 . Let, h , k be the coordinates of the centre of the circle described on P Q as diameter. Then, h = a 2 t 1 2 + t 2 2 and k = a t 1 + t 2 ⇒ 2 h a = t 1 2 + t 2 2 and k a 2 = t 1 + t 2 2 ⇒ 2 h a = t 1 2 + t 2 2 and k 2 a 2 = t 1 2 + t 2 2 + 2 t 1 t 2 ⇒ k 2 a 2 = 2 h a - 2 ∵ t 1 t 2 = - 1 ⇒ k 2 = 2 a h - a Hence, the locus of h , k is y 2 = 2 a x - a .

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