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NTA Abhyas JEE Main2020MathematicsParabolaPractice

Let the normals at points A 4 a , - 4 a and B 9 a , - 6 a on the parabola y 2 = 4 a x meet at the point P . The equation of the normal from P on y 2 = 4 a x (other than P A and P B ) is

Options

  1. A5 x + y - 135 a = 0
  2. B5 x - y + 115 a = 0
  3. C5 x + y + 115 = 0
  4. D5 x - y - 115 a = 0

Correct answer

A. 5 x + y - 135 a = 0

Step-by-step solution

Let, A = a t 1 2 , 2 a t 1 , B = a t 2 2 , 2 a t 2 and the foot of the required normal is a t 3 2 , 2 a t 3 , then 2 a t 1 = - 4 a , 2 a t 2 = - 6 a and t 1 + t 2 + t 3 = 0 ⇒ t 1 = - 2 , t 2 = - 3 and - 2 - 3 + t 3 = 0 ⇒ t 3 = 5 Hence, the equation of the required normal is y = - t 3 x + 2 a t 3 + a t 3 2 ⇒ y = - 5 x + 10 a + 125 a ⇒ y + 5 x - 135 a = 0

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