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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. - CBSE Class 9 - Mathematics

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Question

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.

Solution

Let the radius of the circle centered at O and O' be 5 cm and 3 cm respectively.

OA = OB = 5 cm

O'A = O'B = 3 cm

OO' will be the perpendicular bisector of chord AB.

∴ AC = CB

It is given that, OO' = 4 cm

Let OC be x. Therefore, O'C will be x − 4

In ΔOAC,

OA2 = AC2 + OC2

⇒ 52 = AC2 + x2

⇒ 25 − x2 = AC2 ... (1)

In ΔO'AC,

O'A2 = AC2 + O'C2

⇒ 32 = AC2 + (x − 4)2

⇒ 9 = AC2 + x2 + 16 − 8x

⇒ AC2 = − x2 − 7 + 8x ... (2)

From equations (1) and (2), we obtain

25 − x= − x2 − 7 + 8x

8x = 32

x = 4

Therefore, the common chord will pass through the centre of the smaller circle i.e., O' and hence, it will be the diameter of the smaller circle.

Length of the common chord AB = 2 O'A = (2 × 3) cm = 6 cm

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APPEARS IN

 NCERT Solution for Mathematics Class 9 (2018 to Current)
Chapter 10: Circles
Ex. 10.40 | Q: 1 | Page no. 179

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Solution 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. Concept: Equal Chords and Their Distances from the Centre.
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