Question

Show that the locus of the centres of all circles passing through two given points A and B, is the perpendicular bisector of the line segment AB.

Answer 1

Let  P and Q be the centres of two circles S and S', each passing through two given points A and B. Then,
PA = PB    ...[Radii of the same circle]
⇒ P lies on the perpendicular bisector of AB  ...(i)
Again, QA = QB    ...[Radii of the same circle]
⇒ Q lies on the perpendicular bisector of AB  ...(ii)

From (i) and (ii), it follows that P and Q both lies on the perpendicular bisector of AB.
Hence, the locus of the centres of all the circles passing through A and B is the perpendicular bisector of AB.