1. Find the latus rectum of the parabola

The parabola is $y^2=4x$ which is of the form $y^2=4ax$ with $a=1$.

For this parabola, the latus rectum is the line $x=a=1$ Its endpoints on the parabola are obtained by putting $x=1$: $y^2=4 \Rightarrow y=\pm 2$ So the latus rectum is the segment joining $(1,2)$ and $(1,-2)$.

Hence, the length of the common chord is $L=4$

2. Use the chord-length formula in a circle

Each circle has radius $r=2\sqrt{5}$

If a chord of length $L$ is at perpendicular distance $d$ from the centre of a circle of radius $r$, then $L=2\sqrt{r^2-d^2}$

Substitute $L=4$ and $r=2\sqrt{5}$: $4=2\sqrt{(2\sqrt{5})^2-d^2}$ $2=\sqrt{20-d^2}$ $4=20-d^2$ $d^2=16$ $d=4$

So, in each circle, the centre is at distance $4$ from the common chord.

3. Find the distance between the centres

Since the latus rectum is the common chord of both circles and both circles have the same radius, their centres must lie on the perpendicular bisector of the chord, on opposite sides of the chord.

Therefore, the distance between the centres is $2d=2\times 4=8$

4. Check options

• A: $8$ ✅
• B: $12$ ❌
• C: $8\sqrt{5}$ ❌
• D: $4\sqrt{5}$ ❌

Thus, the correct answer is $\boxed{8}$