Question

The coordinate that the chord $x\cos \alpha +y\sin \alpha -p=0$ 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

Answer: (A)

$a^2=2p^2$

According to given condition

$\begin{array}{c}{x}^2+y^2=a^2\\ centreis\left(0,0\right)\&a^2\end{array}$

equation of chord is

$\begin{array}{c}x\cos \alpha +y\sin \alpha =P\\ OA=\frac{\left|0\cos \alpha +0\sin \alpha -P\right|}{\sqrt{{\sin }^2\alpha +{\cos }^2\alpha }}=d\\ \left|P\right|=d\\ \angle AOC={45}^0\\ \cos \left(\angle AOC\right)=\frac{x}{r}\Rightarrow \frac{1}{\sqrt{2}}=\frac{x}{a}\\ \frac{1}{\sqrt{2}}=\frac{p}{a}\\ squarebothsides\\ \frac{1}{2}=\frac{p^2}{a^2}\Rightarrow p^2=\frac{a^2}{2}\end{array}$