Hello, Beggarsbelief!
$\displaystyle \text{Two equal circles intersect at }P(2,0)\text{ and }Q(2,8).$
$\displaystyle \text{The distance from the centre of each circle}$
. . $\displaystyle \text{to the common chord }PQ\text{ is }2\sqrt{5}$
$\displaystyle \text{Find the equations of the two circles.}$
Code:

 * * *
* *
*  *
*  *

Q*  A *
* o *  o *
* *  * *
*  *
* * oM * *
 *
* * * *
*o+*o*
* B  *P

*  *
*  *
*  *
* * * 

The centers of the circles are $\displaystyle \,A$ and $\displaystyle \,B.$
The circles intersect at $\displaystyle P(2,0)$ and $\displaystyle Q(2,8).$
Draw chord $\displaystyle \,PQ.$
Draw segment $\displaystyle \,AB.$
It can be shown that they intersect at $\displaystyle M(0,4).$
Code:
Q 
o 
*  A
*  o
*  * :
*  * :2
4 *M * :
+      o      +
: * * 4
2: *  *
: * 4 *
o  *
B  *
+o
 2 P

Note that: .$\displaystyle AB \perp PQ$ . . . Quadrilateral $\displaystyle APBQ$ is a rhombus.
We are told that: .$\displaystyle AM = BM = 2\sqrt{5}$
. . .We find that: .$\displaystyle PM = QN = 2\sqrt{5}$ . . . $\displaystyle APBQ$ is a square.
To go from from $\displaystyle \,P$ to $\displaystyle \,M$, we go left 2, up 4.
. . Then, to go from $\displaystyle \,M$ to $\displaystyle \,A$, we got right 4, up 2.
. . Hence, center $\displaystyle \,A$ is at $\displaystyle (4,6).$
. . To go from $\displaystyle \,M$ to $\displaystyle \,B$, we go left 4, down 2.
. . Hence, center $\displaystyle \,B$ is at $\displaystyle (4,2).$
Now you can write the equations of the circles.