Question

Let $S_1$ and $S_2$ be two fixed externally tangent circles with radius 2 and 3 respectively. Let $S_3$ be a variable circle internally tangent to both $S_1$ and $S_2$ at point $A$ and $B$ respectively. The tangents to $S_3$ at $A$ and $B$ meet at $T$ and given $TA=4$.
The square of the radius of circle $S_3$ is

Answer 1

Answer: 64

Let $O$ be the centre of circle $S_3\cdot P$ is the point of contact of $S_1$ and $S_2\tan \angle ATC=\frac{2}{4}=\frac{1}{2}$
$\tan \angle BTD=\frac{3}{4}$
$\angle ATC=\angle CTP$
$\angle PTD=\angle DTB$
$\angle OTA=\angle OTB$
$\Rightarrow \angle OAT=\angle ATC+\angle BTD$
$\tan \angle OAT=\frac{r}{4}$
$\frac{\frac{1}{2}+\frac{3}{4}}{1-\frac{1}{2}\cdot \frac{3}{4}}=\frac{r}{4}$