Question

The condition that the chord $x\cos \alpha +y\sin \alpha -p=0$ of the circle $x^2+y^2-a^2=0$ may subtend a right angle at the centre of the circle is given by?

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

Answer 1

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

$x^2+y^2=a^2,C=(0,0)r=a$

Let the chord

$x\cos \alpha +y\sin \alpha =p$ intersect the circle at A and B

$\angle AOB=90$

If C is the mid point

$OC\perp AB$ and $\angle COB=45$

OC is perpendicular distance from O to line

$=\frac{|0+0-p|}{{\cos }^2\alpha +{\sin }^2\alpha }=p$

In $△OCB$

$\cos 45=\frac{OC}{r}=\frac{p}{a}$

$⟹a=\sqrt{2}p⟹a^2=2p^2$