Let the latus rectum of the parabola $y^2=4x$ be the common chord to the circles $C_1$ and $C_2$ each of them having radius $2\sqrt{5}$. Then, the distance between the centres of the circles $C_1$ and $C_2$ is :

Co-ordinates of focus of parabola will be $S=\left(a, 0\right)=\left(1, 0\right)$.

Now, length of latus-rectum $=4$.

So, length of semi latus-rectum $=2$.

Also, $\angle LS{C}_1=90°$

Now, radius of circle $L{C}_1=2\sqrt{5}$

Now, applying pythagoras theorem in $△LS{C}_1$,

${\left(L{C}_1\right)}^2={\left(C_{1}S\right)}^2+{\left(LS\right)}^2$

$\Rightarrow {\left(C_{1}S\right)}^2={\left(L{C}_1\right)}^2-{\left(LS\right)}^2={\left(2\sqrt{5}\right)}^2-{\left(2\right)}^2=16$

$\Rightarrow C_{1}S=\sqrt{16}=4$

$\therefore C_1{C}_2=2C_{1}S=2\times 4=8$.