Let $a, r, s$ and $t$ be non-zero real numbers. Let $P (a t^{2}, 2 a t), Q, R (a r^{2}, 2 a r)$ and $S (a s^{2}, 2 a s)$ be distinct points on the parabola $y^{2} = 4 a x$ . Suppose that $P Q$ is the focal chord and lines $Q R$ and $P K$ are parallel, where $K$ is point $(2 a, 0)$ .
If $s t = 1$ , then the tangent at $P$ and the normal at $S$ to the parabola meet at a point whose ordinate is

$\frac{(t^{2} + 1)^{2}}{2 t^{3}}$

No worries! We‘ve got your back. Try BYJU‘S free classes today!

$\frac{a (t^{2} + 1)^{2}}{2 t^{3}}$

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

$\frac{a (t^{2} + 1)^{2}}{t^{3}}$

No worries! We‘ve got your back. Try BYJU‘S free classes today!

$\frac{a (t^{2} + 2)^{2}}{t^{3}}$

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is B
$\frac{a (t^{2} + 1)^{2}}{2 t^{3}}$

Equation of tangent and normal at $(a t^{2}, 2 a t)$ are given by $t y = x + a t^{2}$ and $y + t x = 2 a t + a t^{3},$ respectively.
Tangent at $P : t y = x + a t^{2}$ or $y = \frac{x}{t} + a t$
Normal at $S : y + \frac{x}{t} = \frac{2 a}{t} + \frac{a}{t^{3}}$
Solving $2 y = a t + \frac{2 a}{t} + \frac{a}{t^{3}}$
$\Rightarrow y = \frac{a (t^{2} + 1)^{2}}{2 t^{3}}$

Suggest Corrections

Similar questions

Let $a, r, s$ and $t$ be non-zero real numbers. Let $P (a t^{2}, 2 a t), Q, R (a r^{2}, 2 a r)$ and $S (a s^{2}, 2 a s)$ be distinct points on the parabola $y^{2} = 4 a x$ . Suppose that $P Q$ is the focal chord and lines $Q R$ and $P K$ are parallel, where $K$ is point $(2 a, 0)$ .
If $s t = 1$ , then the tangent at $P$ and the normal at $S$ to the parabola meet at a point whose ordinate is