Circle problem in which i wanted to find value of $PQ$ Find a value of $PQ$:  
Let the radius of bigger circle be $R$, and that of the smaller circles be $r_1$ and $r_2$.
Hence we can find value of $R = 10$.
I don't know how to proceed further to find the Value of $r_1$ and $r_2$.


Using the power of the point $O$ 
with respect to circle $O_1$ 
we have
\begin{align} 
|OT_2|\cdot|OT_1|&=|OP|^2
,\\
\text{or }\quad
R(R-2r_1)&=(R-(|AD|-r_1))^2
,\\
r_1&=
\sqrt{2R(2R-|AD|)}-(2R-|AD|)
\tag{1}\label{1}
.
\end{align} 
Similarly, 
\begin{align} 
r_2&=\sqrt{2R\cdot|AD|}-|AD|
\tag{2}\label{2}
.
\end{align}
Note that given 
$a=16=4\cdot4$,
$b=12=4\cdot3$ 
means that 
$\triangle ABC$ 
is a scaled version of the famous $3-4-5$ triangle,
so $R$ and $|AD|$ can be easily calculated,
\begin{align} 
R&=4\cdot\frac 52=10
,\\
|AD|&=4\cdot3\cdot\frac 35=\frac{36}5
,\\
r_1&=\frac{16}5
,\\
r_2&=\frac{24}5
,\\
|PQ|&=8
.
\end{align}
