Let the latus rectum of the parabola y ^ 2 = 4x be the common chord...

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, $∠ LS{C}_{1} = 90 °$

Now, radius of circle $L{C}_{1} = 2\sqrt{5}$

Now, applying pythagoras theorem in $△ LS{C}_{1}$,

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

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

$⇒ {C}_{1}S = \sqrt{16} = 4$

$∴ {C}_{1}{C}_{2} = 2{C}_{1}S = 2 \times 4 = 8$.