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

Options

  1. Ax 2 + y 2 - x 4 + 2 y - 24 = 0
  2. Bx 2 + y 2 - 4 x + 9 y + 18 = 0
  3. Cx 2 + y 2 - 4 x + 8 y + 12 = 0
  4. Dx 2 + y 2 - x + 4 y - 12 = 0

Correct answer

C. x 2 + y 2 - 4 x + 8 y + 12 = 0

Step-by-step solution

y 2 = 8 x is the equation of the given parabola. If P is a point at a minimum distance from ' 0 , - 6 ' , then it should be normal to the parabola at P . Normal to parabola y 2 = 8 x is y = m x - 2 · 2 · m - 2 · m 3 It passes through 0 , - 6 ⇒   m 3 + 2 m - 3 = 0 ⇒ m = 1   P a m 2 , - 2 a m = P ( 2 , - 4 ) Equation of circle with centre P and passes through ' C ' is x - 2 2 + y + 4 2 = 8 ⇒ x 2 + y 2 - 4 x + 8 y + 12 = 0

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