To find: The locus of the centers of all circles passing through two given points.

Question

gotovub · Accepted Answer

Consider the figure:

Let the two points be A and B.
So, any circle passing through both A and B will have AB as its chord.
Let there be two circles 1 and 2 with centers $C_{1} and C_{2}$ respectively, satisfying above condition.
By theorem that perpendicular bisector of a chord of a circle passes through its center.
Therefore, perpendicular bisector of chord AB with respect to circle -1, passes through $C_{1}$
Similarly, perpendicular bisector of chord AB with respect to circle -2, passes through $C_{2}$ .
Thus, the perpendicular bisector passes though both $C_{1}$ and $C_{2}$ in other words, both centers $C_{1} and C_{2}$ lie on the perpendicular bisector.
So the locus of centers of circles (passing though both A and B) is the perpendicular bisector of line segment AB.
Therefore, the locus of centers of all circles passing through two given points is the perpendicular bisector of the line segment joining the two given points.
Final Statement:
The locus of centers of all circles passing through two given points is the perpendicular bisector of the line segment joining the two given points.

To find: The locus of the centers of all circles

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