Mathematics - Circle Question with Solution | TestHub

We have, a circle $C$ which touches the $y-$ axis at $\left(0,6\right)$ and cuts off an intercept $6\sqrt{5}$ on the $x-$ axis.

We know that, length of the chord of the circle using perpendicular distance from the center is $2\sqrt{r^2-d^2}$ where $r$ is the radius of the circle and $d$ is the perpendicular distance from the center to the chord.

From the given data, we have the following diagram

Hence, length of the chord of the circle $\Rightarrow 2\sqrt{r^2-d^2}=6\sqrt{5}$

$\Rightarrow \sqrt{r^2-d^2}=3\sqrt{5}$

On squaring on both sides

$\Rightarrow r^2-d^2=45$

$\Rightarrow d^2=r^2-45$

Given that circle touches the $y-$axis at $\left(0, 6\right)$. Therefore, $d=6$.

$\Rightarrow r^2=45-36$

$r=\sqrt{36+45}$

$r=9$.