Question

Two circles with centres $O$ and $O^{\prime }$ intersect at two points $A$ and $B$. $A$ line $PQ$ is drawn parallel to $O{O}^{\prime }$ through $A$ or $B$, intersecting the circles at $P$ and $Q.$ Prove that $PQ=2O{O}^{\prime }.$

Answer 1

Given: Two circles with centres $O$ and $O^{\prime }$ intersect at two points $A$ and $B.$
Draw a line $PQ$ parallel to $O{O}^{\prime }$ through $B$, $OX$ perpendicular to $PQ$, $O^{\prime }Y$ perpendicular to $PQ$, join all.

We know that perpendicular drawn from the centre to the chord, bisects the chord.

$\therefore PX=XB$ and $YQ=BY$
$\therefore PX+YQ=XB+BY$ [on adding above two equations]
On adding $XB+BY$ on both sides, we get
$PX+YQ+XB+BY=2(XB+BY)$

$PX+BX+BY+YQ=2(XB+BY)$

$PQ=2\left(XY\right)$

$\therefore PQ=2\left(O{O}^{\prime }\right)$ [$\because XY=O{O}^{\prime }$]

Hence, $PQ=2O{O}^{\prime }$