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Two circles of radii 5 cm and 3 cm intersect at two points and

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Two circles of radii $5 c m$ and $3 c m$ intersect at two points and the distance between their centres is $4 c m$ . Find the length of the common chord.

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Two circles of radii

5 c m

and

3 c m

intersect at two points and the distance between their centres is

4 c m

. Find the length of the common chord.

15 mins ago

Discuss this question LIVE

15 mins ago

Text solutionVerified

Let the common chord be

A B

and

P

and

Q

be the centers of the two circles.

∴ A P = 5

cm and

A Q = 3

cm.

PQ = 4

cm ....given

Now, seg

PQ ⊥

chord

A B

∴ A R = RB = \frac{1}{2} A B

....perpendicular from center to the chord, bisects the chord

Let

PR = x

cm, so

RQ = (4 - x)

△ A RP

A P^{2} = A R^{2} + P R^{2}

A R^{2} = 5^{2} - x^{2}

...(1)

△ A RQ

A Q^{2} = A R^{2} + Q R^{2}

A R^{2} = 3^{2} - (4 - x)^{2}

...(2)

∴ 5^{2} - x^{2} = 3^{2} - (4 - x)^{2}

....from (1) & (2)

25 - x^{2} = 9 - (16 - 8 x + x^{2})

25 - x^{2} = - 7 + 8 x - x^{2}

32 = 8 x

∴ x = 4

Substitute in eq(1) we get,

A R^{2} = 25 - 16 = 9

∴ A R = 3 c m

∴ A B = 2 \times A R = 2 \times 3

∴ A B = 6 c m .

So, length of common chord

A B

6 c m .

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Question Text	Two circles of radii $5 c m$ and $3 c m$ intersect at two points and the distance between their centres is $4 c m$ . Find the length of the common chord.
Answer Type	Text solution:1
Upvotes	150

Two circles of radii 5 cm and 3 cm intersect at two points and the distance between their centres is 4 cm. Find the length of the common chord.

Text solutionVerified

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Two circles of radii $5 c m$ and $3 c m$ intersect at two points and the distance between their centres is $4 c m$ . Find the length of the common chord.