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

Answer: 100

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

From the figure,
${\left(AB\right)}^2={\left(r_1+r_2\right)}^2-{\left(r_1-r_2\right)}^2=4r_1{r}_2$
$AB=2\sqrt{r_1{r}_2}$
Now, $BY=10$ (given)
$\Rightarrow AB=20=2\sqrt{9r_2}$
$\Rightarrow r_2=\frac{100}{9}$ units.
$\Rightarrow 9r_2=100$ units.