Question

If the cordinate of any point p on the hyperbola $9x^2-16y^2=144$ is produced to cut the asymptotes in the points Q and R. Then the product PQ.PR equals to:

• A. $9$
• B. $\frac{12}{5}$
• C. $\frac{144}{25}$
• D. $7$

Answer 1

Answer: (C)

$\frac{144}{25}$

$\frac{x^2}{16}-\frac{y^2}{9}=1$

Asymplote is $y=\pm \frac{3}{4}x$

Let us take $4y=3x,4y=-3x$

consider a parametric point $(4\sec \theta ,3\tan \theta )$ on the parabola

$Q$ is intersection with $4y=3x$

then $PQ=\left|\frac{12\sec \theta -12\tan \theta }{\sqrt{4^2+3^2}}\right|$

$PQ=\left|\frac{12}{5}\right(\sec \theta -\tan \theta \left)\right|$

$R$ is intersection with $4y=-3x$

then $PR=\left|\frac{12\sec \theta +12\tan \theta }{\sqrt{4^2+3^2}}\right|$

$PQ.PR=\frac{144}{25}(\sec \theta +\tan \theta )(\sec \theta -\tan \theta )$

$=\frac{144}{25}({\sec }^2\theta +{\tan }^2\theta )=\frac{144}{25}\left(1\right)$

`e`$=\frac{144}{25}$