Question

The condition that the chord $xcos\alpha +ysin\alpha -p=0$ of $x^2+y^2-a^2=0$ may subtend a right angle at the centre of the circle is

• A. $a^2=2p^2$
• B. $p^2=2a^2$
• C. $a=2p$
• D. $p=2a$

Answer 1

The correct option is A $a^2=2p^2$

Homogenising the given circle gives
$x^2+y^2-a^2{\left(\frac{x\cos \alpha +y\sin \alpha }{p}\right)}^2=0$
as chord subtends right angle at the centre.
$a+b=0$
$\Rightarrow 1-a^2\frac{{\cos }^2\alpha }{p^2}+1-a^2\frac{{\sin }^2\alpha }{p^2}=0$
$\Rightarrow 2-\frac{a^2}{p^2}=0$
$\therefore a^2=2p^2$
Hence, option A.