Question

What is the locus of centers of all circles passing through point $A$ and Point $B$, which are two fixed points in a plane?

Answer 1

The explanation for two fixed points in a plane.

The center of a circle is at an equal distance from all points on the circle. Thus, the center ($P$) of a circle that passes through two given points $A$, and $B$, is equidistant from the two points $A$, and $B$. In other words, $PA=PB$.

The locus of a point equidistant from two given points is the perpendicular bisector of the line segment joining the two points. Thus, the locus of the center $P$ of circles passing through the given points $A$, and $B$, is the perpendicular bisector of the line segment $AB$.

Hence, the required locus is the perpendicular bisector of the line segment $\mathit{A}\mathit{B}$.