As shown in the figure, three circles which have the same radius $r$ have centrers at $(0,0),(1,1)$ and $(2,1).$ If they have a common tangent line, as shown, then the value of $20\sqrt{5}r$ is_________

 

The equation of the line joining the origin and the centre of circle $C_2,(2,1),$ is $y=\frac{x}{2}$ or $x-2y=0$

Let the equation of common tangent be 

$x-2y+c=0$ …………. (1)

$\therefore$Perpendicular distance from $(0,0)$ on this line =perpendicular distance from $(1,1)$ or $\left|\frac{c}{\sqrt{5}}\right|=\left|\frac{c-1}{\sqrt{5}}\right|$ or $c=1-c$ or $c=\frac{1}{2}$

The equation of common tangent is 

$x-2y+\frac{1}{2}=0or2x-4y+1=0$  ………….. (2)

The length of perpendicular from $(2,1)$ on the line $\left(2\right)$ is $r=\left|\frac{4-4+1}{\sqrt{20}}\right|=\frac{1}{2\sqrt{5}}=\frac{\sqrt{5}}{10}$