Write 'True' or 'False' and give reasons for your answer.
If a number of circles pass through the end points $P$ and $Q$ of a line segment $PQ$,then their centres lie on the perpendicular bisector of $PQ$.
If a number of circles pass through the end points $P$ and $Q$ of a line segment $PQ$,then their centres lie on the perpendicular bisector of $PQ$.
(TRUE) True.
Let there be a number of circles passing through the end points $P$ and $Q$ of a line segment $PQ$.
Since $PQ$ is a common chord for all these circles,the centre of any circle passing through $P$ and $Q$ must be equidistant from $P$ and $Q$.
We know that the locus of points equidistant from two fixed points $P$ and $Q$ is the perpendicular bisector of the line segment $PQ$.
Therefore,the centres of all such circles must lie on the perpendicular bisector of $PQ$.
Let there be a number of circles passing through the end points $P$ and $Q$ of a line segment $PQ$.
Since $PQ$ is a common chord for all these circles,the centre of any circle passing through $P$ and $Q$ must be equidistant from $P$ and $Q$.
We know that the locus of points equidistant from two fixed points $P$ and $Q$ is the perpendicular bisector of the line segment $PQ$.
Therefore,the centres of all such circles must lie on the perpendicular bisector of $PQ$.
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