If $r_1$ and $r_2$ are radii of smallest and largest circles which passes though (5, 6) and touches the circle $(x-2{)}^2+y^2=4$ then find $r_1{r}_2$

Given $r_1,r_2$ are radii of smallest and largest circles passes through $\left(5,6\right)$ and touching the circle $(x-2{)}^2+y^2=4$

Here $C\left(2,0\right),r=2$

The shortest distance and longest distances from the point $\left(5,6\right)$ to the circle are diameters of required circles 

$2r_1=CP-r$ and $2r_2=CP+r$

Multiply, we get 

$\begin{array}{c}4r_1{r}_2=C{p}^2-r^2=\sqrt{11}\\ =9+36-4\\ =41\end{array}$

Hence, $r_1{r}_2=\frac{41}{4}=\boxed{10.25}$