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Let P be a point on the parabola, y 2 = 12 x and N be the foot of the perpendicular drawn from P , on the axis of the parabola. A line is now drawn through the mid-point M of P N , parallel to its axis which meets the parabola at Q . If the y - intercept of the line NQ is 4 3 , then :

Options

  1. AP N = 4
  2. BM Q = 1 3
  3. CM Q = 1 4
  4. DP N = 3

Correct answer

C. M Q = 1 4

Step-by-step solution

Comparing the equation of the parabola, with standard equation y 2 = 4 a x , we get a = 3 . Figure is drawn on the basis of the given information. Parametric form of parabola is x = a t 2 , y = 2 a t ∴ P = 3 t 2 , 6 t , N = 3 t 2 , 0 , M = 3 t 2 , 3 t ⇒ Q = 3 t 2 4 , 3 t Equation of N Q : y - 0 = 3 t - 0 3 t 2 4 - 3 t 2 x - 3 t 2 i.e., y = - 4 3 t · x - 3 t 2 ⇒ y = - 4 3 t · x + 4 t ⇒ 4 t = 4 3 ⇒ t = 1 3 P N = 6 t = 2 M Q = 3 t 2 - 3 t 2 4 = 9 t 2 4 = 1 4

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