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 O A of a circle C 1 . Let C 2 be the circle with O A 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 Q A :   A P is equal to

Question

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 O A of a circle C 1 . Let C 2 be the circle with O A 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 Q A : &#160; A P is equal to

Aicharya · Accepted Answer

The centre of the circle $C_1$ will be $(2, 0)$ 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 $OA = l$.

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

A circle C 1 passes through the origin O and has diameter 4 on the positive x -axis. The line y = 2 x gives…

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