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AB is one of the direct common tangent of two circles of radii 12 cm and 4 cm respectively touching each other. Find the area of the region enclosed by the circles and the tangent.

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24.78

c m 2

18.53

c m 2

19.54

c m 2

14.8

c m 2

answer is B.

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Detailed Solution

It is given that the radii of two circles are PA=12cm and QB=4cm.
We have to find the area of the shaded region.
The area of a sector can be expressed as,

A = θ 360 ° π r 2

. Here

r

is the radius of a circle.
The required figure geometry is shown below,
Question Image

Then, by using the Pythagoras theorem from figure geometry,

P Q 2 = P R 2 + R Q 2 ⇒ 162 = 82 + R Q 2 ⇒ R Q 2 = 192 ⇒ R Q = 83 cm

The area of a trapezium, PABQ,

A = 12 A B P A + Q B ⇒ A = 1283 12 + 4 ⇒ A = 643 cm

In the triangle PQR,

∠ Q = 180 ° − ∠ P

⇒ ∠ Q = 180 ° − 60 °

⇒ ∠ Q = 120 °

Thus, the area of the region enclosed by the circles and the tangent is Question Image

The required area is

cm 2

.
Therefore, the correct option is 2.

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