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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Solution

Given, draw two circles having centres O and O' intersect at points A and B.

Also, draw line PQ parallel to OO'
Construction : Join OO', OP, O'Q, OM and O'N
To prove that PQ = 2 OO’

Proof
In $Δ O P B$ , BM = MP [OM is the perpendicular bisector of PB]
And in $Δ O^{'} B Q$ , BN = NQ [O'N is the perpendicular bisector of BQ]
$∴$ BM + BN = PM + NQ
$\Rightarrow$ 2(BM + BN) = BM + BN + PM + NQ [adding both sides BM + BN]
$\Rightarrow$ 2 OO' = (BM + MP) + (BN + NQ)
$\Rightarrow$ 2 OO' = (PB + BQ)
$\Rightarrow$ 2 OO' = PQ, hence proved.

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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 $P Q = 2 O O^{'} .$

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'.

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^{'}

Two circles whose centres are $O$ and $O^{'}$ intersect at $P$ . Through $P$ , a line parallel to $O O^{'}$ , intersecting the circle at C and D is drawn as shown. Then $C D = 2 O O^{'}$ .

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

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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'.

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'.