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Prove that locus of centers of circles passing through  points A and B is perpendicular bisector of line segment AB.

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Let P and Q be the centers of two circle C and C, each passing through two given points A and B.

Then,

PA = PB (radii of the circle C)

⇒ P lies on the perpendicular bisector of AB ….(i)

Again QA = QB (radii of the circle C)

⇒ Q lies on the perpendicular bisector of AB … (ii)

From (i) and (ii) it follows that P and Q both. lie on the perpendicular bisector of AB.

Hence, the locus of the centers of all the circles passing through A and B is the perpendicular bisector of AB.

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