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’.

Answer 1

Construct OM ⊥ PQ and O’N ⊥ PQ

So we get

OM ⊥ AP

We know that the perpendicular from the centre of a circle bisects the chord

AM = PM

It can be written as

AP = 2AM ………. (1)

We know that

O’N ⊥ AQ

We know that the perpendicular from the centre of a circle bisects the chord

AN = QN

It can be written as

AQ = 2AN ……. (2)

So we get

PQ = AP + PQ

By substituting the values

PQ = 2 (AM + AN)

We get

PQ = 2MN

From the figure we know that MNO’O is a rectangle

PQ = 2OO’

Therefore, it is proved that PQ = 2OO’.