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NTA Abhyas 2020

Exam: NTA Abhyas
Year: 2020
Subject: Mathematics
Chapter: Parabola

NTA Abhyas 2020 · Mathematics · Parabola

If the normals at two points P and Q of a parabola y ^ 2 = 4x...

If the normals at two points

P

and

Q

of a parabola

{y}^{2} = 4x

intersect at a third point

R

on the parabola

{y}^{2} = 4x

, then the product of the ordinates of

P

and

Q

is equal to

Correct answer

8

Step-by-step solution

Let, $P = \left({t}_{1}^{2}, 2{t}_{1}\right)$ , $Q = \left({t}_{2}^{2}, 2{t}_{2}\right)$ and $R = \left({t}^{2}, 2t\right)$

then ${t}_{1}, {t}_{2}$ are roots of the equation $t = - x - \dfrac{2}{x}$
$⇒ {t}_{1}, {t}_{2}$ are roots of the equation ${x}^{2} + tx + 2 = 0$
$⇒ {t}_{1}{t}_{2} = 2$
$⇒ \left(2{t}_{1}\right)\left(2{t}_{2}\right) = 8$
$⇒$ product of the ordinates of $P$ and $Q$ is equal to $8$

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