The length of the common chord of two circles of radii 15 cm and 13 cm, whose centres are 14 cm apart, is:

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SSC CGL 2023 Tier-I Official Paper (Held On: 24 Jul 2023 Shift 2)
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  1. 14 cm
  2. 12 cm
  3. 15 cm
  4. 24 cm

Answer (Detailed Solution Below)

Option 4 : 24 cm
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Detailed Solution

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Given:

Radii of circles are 13 cm and 15 cm 

Distance between centres = 14 cm 

Concept Used:

The line joining the centers of two circles is the perpendicular bisector of the common chord of the two circles.

⇒ AB = 2AM

Calculation:

F1 SSC  PriyaSS 14-2-24 D58

From the figure, P and Q are the centers of two circles of radius 15 cm and 13cm respectively and AB is the common chord

In ΔAMP, PA = 15 cm

By Pythagoras theorem,

AM2 = PA2 - PM2

⇒ AM2 = 152 - PM2

⇒ AM2 = 225 - PM2     -----(1)

In ΔAMQ, QA = 13 cm

By Pythagoras theorem,

AM2 = QA2 - MQ2

⇒ AM2 = 132 - MQ2

⇒ AM2 = 169 - MQ2     -----(2)

From (1) and (2)

225 - PM169 - MQ2 

⇒ PM - MQ2 = 56

⇒ (PM + MQ) (PM - MQ) = 56

According to the question,

(PM + MQ) = PQ = Distance between centres = 14 cm ----(3)

⇒ 14 (PM - MQ) = 56

⇒ (PM - MQ) = 56/14 = 4                                               ----(4)

From (3) and (4)

PM = 9 cm

Putting value of PM in (1)

AM2 = 225 - 92

⇒ AM = 225 - 81

⇒ AM = 144

⇒ AM = 12 cm

Thus, length of common chord = AB = 2AM = 24 cm

∴ The length of the common chord is 24 cm.

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