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JEE Main202427 Jan 2024Morning ShiftMathematicsParabolaActual

If the shortest distance of the parabola y 2 = 4 x from the centre of the circle x 2 + y 2 - 4 x - 16 y + 64 = 0 is d , then d 2 is equal to :

Options

  1. A16
  2. B24
  3. C20
  4. D36

Correct answer

C. 20

Step-by-step solution

Given: Equation of parabola is y 2 = 4 x and the equation of circle is x 2 + y 2 - 4 x - 16 y + 64 = 0 . We know that, equation of normal to parabola is given by, y = m x - 2 m - m 3 . Also, the centre of given circle is 2 , 8 . The normal of parabola is passing through center of circle. ⇒ 8 = 2 m - 2 m - m 3 ⇒ m = - 2 So point P on parabola is given by, am 2 , - 2 am = ( 4 , 4 ) And C = ( 2 , 8 ) ⇒ P C = 4 + 16 = 20 d 2 = 20 .

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