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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.
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Solution
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.
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