The normal at the point $ P\left(a p^{2}, 2 a p\right) $ meets the parabola $ y^{2}=4 a x $ again at $ Q\left(a q^{2}, 2 a q\right) $ such that the lines joining the origin to $ P $ and $ Q $ are at right angle. Then

Question

The normal at the point $ P\left(a p^{2}, 2 a p\right) $ meets the parabola $ y^{2}=4 a x $ again at $ Q\left(a q^{2}, 2 a q\right) $ such that the lines joining the origin to $ P $ and $ Q $ are at right angle. Then, $ p^{2}=2 $, $ q^{2}=2 $, $ p=2 q $, $ q=2 p $

MARKS App · Accepted Answer

Since the normal at P a p 2 , 2 a p to y 2 = 4 a x meets the parabola at Q a q 2 , 2 a q , therefore P Q is the focal chord. Hence q = - p - 2 p . . . i Since O P ⊥ O Q , ∴ 2 a p - 0 a p 2 - 0 × 2 a q - 0 a q 2 - 0 = - 1 ⇒ p q = - 4 ⇒ p - p - 2 p = - 4 [Using ( i ) ] ⇒ p 2 = 2

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