If  $r_1$ and $r_2$  are radii of the smallest and  the largest circles, respectively, which pass through (5,6) and touch the circle ${(x-2)}^2+y^2=4$ , then  $r_1{r}_2$  is

The given circle is ${(x-2)}^2+y^2=4,$  the center is (2,0), and the radius is 2. Therefore, distance between (2,0) and (5,6) is  $\sqrt{9+36}=3\sqrt{5}.$ Then,

$r_1=\frac{3\sqrt{5}-2}{2}$
And $r_2=\frac{3\sqrt{5}+2}{2}$ 
$\therefore r_1{r}_2=\frac{41}{4}$