P and Q are points on intersection of two circles with centre O and O'. If straight line APB and CQD are parallel to OO' then AB = ___

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Solution

Join PQ,

Draw OM' $⊥$ CQ, OM $⊥$ AP, O'N' $⊥$ QD and O'N $⊥$ PB.

MP = $\frac{1}{2}$ AP, NP = $\frac{1}{2}$ BP, M'Q = $\frac{1}{2}$ CQ, and N'Q = $\frac{1}{2}$ QD.
[ $∵$ A perpendicular drawn from the centre of a circle to its chord bisects the chord]

$\Rightarrow$ OO' = MN = MP + PN= $\frac{1}{2}$ ( AP + BP) = $\frac{1}{2}$ AB.......(i)

Similarly,

OO' = M'N' = M'Q + QN' = $\frac{1}{2}$ ( CQ + QD) = $\frac{1}{2}$ CD.......(ii)

From (i) and (ii), we can see that AB = CD.

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Similar questions

Q. In the following figure: P and Q are the points of intersection of two circles with centres O and O’. If straight lines APB and CQD are parallel to OO’. Prove that
(i) OO’ = ½ AB
(ii) AB = CD

In the figure; P and Q are the points of intersection of two circles with centres O & O'. If straight line APB and CQD are parallel to OO', then what is the ratio of OO' and AB?

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

Q. Two circles with centres

O

and

O^{'}

intersect at two points

A

and

B

A

line

P Q

is drawn parallel to

O O^{'}

through

A

(o r B)

intersecting the circles at

P

and

Q

Prove that

P Q = 2 O O^{'}

Question 18
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 = 2 OO'.

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