P and Q are two distinct points on the parabola, y 2 = 4 x , with parameters t and t 1 , respectively. If the normal at P passes through Q , then the minimum value of t 1 2 , is

Step-by-step solution

Given P t 2 , 2 t and Q t 1 2 , 2 t 1 , are parametric points on parabola y 2 = 4 x Slope of normal at P = - t Since normal passes through Q , ⇒ slope of normal at P = slope of P Q ⇒ - t = 2 t 1 - t t 1 2 - t 2 ⇒ - t = 2 t 1 + t ⇒ t 1 = - t - 2 t ⇒ t 1 2 = t 2 + 4 t 2 + 4 . . . 1 t 2 + 4 t 2 ≥ 2 t 2 . 4 t 2 ≥ 4 , Using AM ≥ GM ∴ t 1 2 ≥ 4 + 4 from 1 ⇒ t 1 2 ≥ 8 ∴ Minimum value = 8