Coordinate Geometry

Let P be the point on the parabola, y2 = 8x which is at a minimum distance from the centre C of the circle, x2 + (y + 6)2 = 1. Then the equation of the circle, passing through C and having its centre at P is:

  1. x2 + y2 – x/4 + 2y – 24 = 0
  2. x2 + y2 – 4x + 8y + 12 = 0
  3. x2 + y2 – 4x + 9y + 18 = 0
  4. x2 + y2 – x + 4y – 12 = 0

Answer

For minimum distance from the centre of circle to the parabola at point P, the line must be normal to the parabola at P.

Let P(at2, 2at) = (2t2, 4t)

y = –tx + 4t + 2t3

–6 = 4t + 2t3

t = –1

Equation of required circle:

(x – 2)2 + (y + 4)2 = 8

The correct option is B.