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10-1.Circle and System of Circles
hard
A circle $C_{1}$ passes through the origin $O$ and has diameter $4$ on the positive $x$-axis. The line $y =2 x$ gives a chord $OA$ of a circle $C _{1}$. Let $C _{2}$ be the circle with $OA$ as a diameter. If the tangent to $C _{2}$ at the point $A$ meets the $x$-axis at $P$ and $y$-axis at $Q$, then $QA : AP$ is equal to.
A
$1:4$
B
$1: 5$
C
$2: 5$
D
$1: 3$
(JEE MAIN-2022)
Solution
$C _{2}$ is a circle with $OA$ as diameter.
So, tangent at $A$ on $C _{2}$ is perpendicular to $OR$
Let $OA =\ell$
$\therefore \frac{ QA }{ AP }=\frac{\ell \cot \theta}{\ell \tan \theta}$
$=\frac{1}{\tan ^{2} \theta}=\frac{1}{4}$
Standard 11
Mathematics
