JEE Main 2022 — Circle Question with Solution

The centre of the circle $C_1$ will be $\left(2,0\right)$ and radius is $2$

So, the equation of the circle $C_1$ is $x^2+y^2-4x=0$

Let $\theta$ be the angle made by the line $y=2x$ with positive $x$-axis.

i.e. $\tan \theta =2$

Now $C_2$ is a circle with $OA$ as diameter.

So, tangent at $A$ on $C_2$ is perpendicular to $OA$

Let $OA=l$

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$\therefore \frac{QA}{AP}=\frac{l\cot \theta }{l\tan \theta }$

$\Rightarrow \frac{QA}{AP}=\frac{1}{{\tan }^2\theta }=\frac{1}{4}$