If a number of circles pass through the end points P and Q of a line segment PO, then their centers lie on the perpendicular bisector of PQ.

True

Let as draw a line segment PQ and S1 S2 S3 …. Sn are n circles with center C1 C2 C3 … Cn respectively passing through p and Q

Here PQ either will be a chord to a circle or the diameter (possible only once when diameter length is equal to the length of PQ.

Now, if PQ is diameter then clearly center lies on the half of diameter i.e. on the perpendicular bisector of PQ.

But if PQ is a chord then we know that perpendicular from the center to the chord of circle bisect the chord implies that center lies on the perpendicular bisector of the chord.

So for each circle center lies on the perpendicular bisector of PQ.

So above statement is true .

9