Question
Download Solution PDFThe length of the common chord of two circles of radii 15 cm and 13 cm, whose centres are 14 cm apart, is:
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFGiven:
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:
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 - PM2 = 169 - MQ2
⇒ PM2 - 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
⇒ AM2 = 225 - 81
⇒ AM2 = 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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