$\text{ Let the latus rectum of the parabola }{y}^2=4x\text{ be the common chord to the circles }{C}_1\text{ and }{C}_2\text{ each of them }$

$\text{ having radius }2\sqrt{5}\text{ . Then the distance between the centres of the circles }{C}_1\text{ and }{C}_2\text{ is: }$

$\text{ Equation of the parabola is }{y}^2=4x$

$\text{ Its latusrectum is }x=1\text{ and end points of the latursectum are }(1,2),(1,-2)$

$\text{ Its latusrectum is }x=1\text{ and end points of the latursectum are }(1,2),(1,-2)$

$\text{ Suppose that the general equation of circle is }{x}^2+y^2+2gx+2fy+c=0$

$\text{ It passes through two points }(1,2),(1,-2)$

Hence, 

$\begin{array}{l}2g+4f+c=-5\\ 2g-4f+c=-5\end{array}$

$\text{ Adding both equations }4g+2c=-10\Rightarrow c=-5-2g$

$\text{ By substituting }c=-5-2g\text{ in any of the above equation, }f=0$

$\text{ Given radius of the circle is }2\sqrt{5}\text{ units }$

            Hence, 

$\begin{array}{c}{g}^2+f^2-c=20\\ g^2+0+5+2g=20\\ g^2+2g-15=0\end{array}$

$\text{ It gives }g=3,-5$

$\text{ Hence, the centres of the circles are }(-3,0),(5,0)$

$\text{ Therefore, the distance between the centres is }|3+5|=8\text{ units }$