Question

Let $P$ and $Q$ be $2$ circles externally touching each other at point $X$ . Line segment $AB$ is a direct common tangent to circles $P$ and $Q$ at points $A$ and $B$ respectively. Another common tangent to $P$ and $Q$ at $X$ intersects line $AB$ at a point $Y$ . If $BY=10$ units and the radius of $P$ is $9$ units, then the value of $9$ times of the radius of the circle $Q$ is equal to (in units)

Answer 1

$AY=BY \, \, $ ( $\because Y$ lies on the radical axis)

From the figure,
$\left(A B\right)^{2}=\left(r_{1} + r_{2}\right)^{2}-\left(r_{1} - r_{2}\right)^{2}=4r_{1}r_{2}$
$AB=2\sqrt{r_{1} r_{2}}$
Now, $BY=10$ (given)
$\Rightarrow AB=20=2\sqrt{9 r_{2}}$
$\Rightarrow r_{2}=\frac{100}{9}$ units.
$\Rightarrow 9r_{2}=100$ units.