Two circles with centres O and O’ intersect at two points A and B. A line PQ is drawn parallel to OO’ through A or B, intersecting the circles at P and Q.
Question:

Two circles with centres O and O‘ intersect at two points A and B. A line PQ is drawn parallel to OO‘ through A or B, intersecting the circles at P and Q. Prove that PQ = 2OO‘.

Solution:

Given: Two circles with centres O and O‘ intersect at two points A and B.

Draw a line PQ parallel to OO‘ through BOX perpendicular to PQOY perpendicular to PQ, join all.

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

∴ PX = XB and YQ = BY

∴ PX + YQ = XB + BY

On adding XB + BY on both sides, we get

PX + YQ + XB + BY = 2(XB + BY)
⇒ PQ = 2(XY)
⇒ PQ = 2(OO‘)

Hence, PQ = 2OO‘.

 

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