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 $C D = 2 O O^{'}$ .

True

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False

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Solution

The correct option is A
True

Construction:

Draw perpendiculars $O A$ and $O^{'} B$ to $C D$ from $O$ and $O^{'}$ , respectively.

Since $O A ⊥ C D$ , $O A$ bisects the chord $C P$ . ( $∵$ Perpendicular from the centre of the chord bisects the chord)

$∴ A P = \frac{1}{2} C P$

or $C P = 2 A P$ - - - - - (i)

Similarly since $O^{'} B ⊥ P D$ , we must have $B P = \frac{1}{2} P D$

or $P D = 2 B P$ - - - - - (ii)

Now, $C D = C P + D P$

$⟹ C D = 2 A P + A B P = 2 (A P + B P)$ [from (i) and (ii)]

$⟹ C D = 2 (A B)$ - - - - - (iii)

In quadrilateral $A B O^{'} O$ , $O O^{'} ∥ A B$ and $∠ O A B = ∠ O^{'} B A = 90^{\circ}$

So, $O A ∥ O^{'} B$

Thus $A B O^{'} O$ is a rectangle.

$∴ A B = O O^{'}$

Hence $C D = 2 A B = 2 O O^{'}$

Therefore the given statement is true.

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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'.

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

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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Two circles whose centres are O and O′ intersect at P. Through P, a line parallel to OO′, intersecting the circle at C and D is drawn as shown. Then CD=2OO′.

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 $C D = 2 O O^{'}$ .