NCERT Solutions for Class 9 Maths Chapter 10

Important NCERT Questions

Circles

NCERT Books for Session 2022-2023

CBSE Board, UP Board and Other state Boards

EXERCISE 10.4

Page No:179

Questions No:1

# 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.

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Given : Circle C (P, 3) and circle C (Q, 5) are intersecting at points A and B.

Construction: Join PA and QA. Draw PM as bisector of chord AB.

Proof: AB is chord of circle C (P, 3) and PM is bisector of chord AB.

Therefore, PM ⊥AB

[∵ The line drawn through the center of a circle to bisect a chord is perpendicular to the

chord.]

Hence, ∠PMA = 90°

Let, PM = X, therefore, QM = 4 – x

In ΔAPM, using Pythagoras theorem

AM² = AP² – PM² …(1)

And in ΔAPM, using Pythagoras theorem

AM² = AQ² – QM² …(2)

From the equation (1) and (2), we get

AP² – PM² = AQ² – QM²

⇒ 3² – x² = 5² -(4 – x)² ⇒ 9 – x² = 25 – (16 + x² – 8x)

⇒ 9 – 9 = 8x ⇒ x = 0/8 = 0

from the equation (1), AM² = 3² – 0² = 9 ⇒ AM = 3

⇒ AB = 2AM = 6