Question

A chord $PQ$ of a conic subtends a right angle at the focus $S$ then ${\left(\frac{1}{SP}-\frac{1}{l}\right)}^2+{(\frac{1}{SQ}-\frac{1}{l})}^2=$

• A. $\frac{e}{l}$
• B. $\frac{l^2}{e^2}$
• C. $\frac{e^2}{l^2}$
• D. $\frac{l}{e}$

Answer 1

Answer: (C)

$\frac{e^2}{l^2}$

Let the equation of conic be $\frac{l}{r}=1+e\cos \theta$ ...(1)
It is given that the chord $PQ$ subtends a right angle at the focus $S.$
Therefore if the vertical angle of $P$ is $\alpha$, then $\angle Q=\frac{\pi }{2}+\alpha$
$\therefore \frac{1}{SP}=1+e\cos \alpha$ and $\frac{1}{SQ}=1+e\cos (\frac{1}{2}\pi +\alpha )=1-e\sin \alpha$
$\therefore {(\frac{1}{SP}-\frac{1}{l})}^2+{(\frac{1}{SQ}-\frac{1}{l})}^2$
$={(\frac{1+e\cos \alpha }{l}-\frac{1}{l})}^2+{(\frac{1-e\sin \alpha }{l}-\frac{1}{l})}^2$
$={\left(\frac{e\cos \alpha }{l}\right)}^2+\left(\frac{-e{\sin }^2\alpha }{l}\right)=\frac{e^2}{l^2}$