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

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} $
$\frac{\frac{1}{2}+\frac{3}{4}}{1-\frac{1}{2} \cdot \frac{3}{4}}=\frac{ r }{4}$