Circles- finding radii of smallest and largest circle

Question

If $r_1$ and $r_2$ are the radii of smallest and largest circle which passes through $(5,6)$ and touches the circle $(x-2)^2+y^2=4$. Then $r_1r_2$ = ??

Answer 1

Let $A=(5,6)$ and $C=(2,0)$ (center of the given circle). Then the radius of the smaller circle touching the given circle and passing through $A$ will be $(AC-2)/2$ (where $2$ is the radius of the given circle). Likewise the radius of the bigger circle will be $(AC+2)/2$. Thus $$r_1r_2=(AC^2-4)/4=\frac{[(5-2)^2+(6-0)^2]-4}{4}=41/4.$$