Question

Let the double ordinate $P{P}^{\prime }$ of the hyperbola $\frac{x^2}{4}-\frac{y^2}{3}=1$ is produced both sides to meet asymptotes of hyperbola in $Q$ and $Q^{\prime }.$ The product $\left(PQ\right)\left(P{Q}^{\prime }\right)$ is equal to

Answer 1

$\frac{x^2}{4}-\frac{y^2}{3}=1$


Let $P$ be $(2\sec \theta ,\sqrt{3}\tan \theta )$
$\Rightarrow P^{\prime }$ will be $(2\sec \theta ,-\sqrt{3}\tan \theta )$
Asymptote will be $y=\pm \frac{\sqrt{3}}{2}x$
​​​​​​​
$\therefore Q$ will be : $(2\sec \theta ,\sqrt{3}\sec \theta )$
and $Q^{\prime }$ will be : $(2\sec \theta ,-\sqrt{3}\sec \theta )$

So, $PQ=\sqrt{3}(\sec \theta -\tan \theta )$
$P{Q}^{\prime }=\sqrt{3}(\tan \theta +\sec \theta )$
$\therefore PQ\cdot P{Q}^{\prime }=3({\sec }^2\theta -{\tan }^2\theta )=3$