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JEE Main202330 Jan 2023Morning ShiftMathematicsParabolaActual

If P ( h , k ) be point on the parabola x = 4 y 2 , which is nearest to the point Q ( 0 , 33 ) , then the distance of P from the directrix of the parabola y 2 = 4 ( x + y ) is equal to:

Options

  1. A2
  2. B4
  3. C8
  4. D6

Correct answer

D. 6

Step-by-step solution

Given parabola is x = 4 y 2 ⇒ y 2 = 1 4 x ⇒ y 2 = 4 × 1 16 × x Equation of normal is y = - t x + 2 a t + a t 3 ⇒ y = - t x + 2 16 t + 1 16 t 3 It passes through Q 0 , 33 , so 33 = t 8 + t 3 16 ⇒ t 3 + 2 t - 528 = 0 ⇒ t - 8 t 2 + 8 t + 66 = 0 ⇒ t = 8 So, P ≡ h , k ≡ a t 2 , 2 a t ≡ ( 4 , 1 ) Now, we have parabola y 2 = 4 x + y ⇒ y 2 - 4 y = 4 x ⇒ y - 2 2 = 4 x + 1 Equation of directrix is x + 1 = - 1 ⇒ x = - 2 So, distance of point 4 ,

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