In the figure shown alongside, side length of the square is 2a\ units. Assertion (A): Area of smaller circle = Area enclosed

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Solution

Q.
In the figure shown alongside, side length of the square is 2a units.

Assertion (A): Area of smaller circle = Area enclosed between two circles.

Reason (R):Radius of outer circle $= \sqrt{2} \times$ (Radius of inner circle).

1 Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

2 Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

3 Assertion (A) is true but Reason (R) is false.

4 Assertion (A) is false but Reason (R) is true.

Ans.
(1)

Let R and r be the radii of outer and inner circle respectively.
Diameter of inner circle = 2a

$∴ r = a \Rightarrow R = \sqrt{a^{2} + a^{2}} = \sqrt{2} a A r e a e n c l o s e d b e t w e e n t w o c i r c l e s = Area of outer circle - Area of inner circle = {πR}^{2} - {πr}^{2} = π (\sqrt{2} a)^{2} - {πa}^{2} = 2 {πa}^{2} - {πa}^{2} = {πa}^{2} = Area of smaller circle$

Both Assertion (A) and Reason (R) are true and Reason (R) is correct explanation of Assertion (A).

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