Question

Assertion (A)
If the circumferences of two circles are in the ratio 2 : 3, then the ratio of their areas is 4 : 9.

Reason (R)
The circumference of a circle of radius r is 2πr.

(a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).
(b) Both Assertion (A) Reason (R) true but Reason (R) is not a correct explanation of Assertion (A).
(c) Assertion (A) is true and Reason (R) is false.
(d) Assertion (A) is false and Reason (R) is true.

Answer 1

(b) Both assertion (A) and reason (R) are true, but reason (R) is not the correct explanation of assertion (A).

Assertion (A):
Let $r_1\mathrm{and}{r}_2$ be the radii of two circles.
Now,
Circumference of the first circle$=2\mathrm{\pi }{r}_1$
Circumference of the second circle$=2\mathrm{\pi }{r}_2$
Thus, we have:
$$\begin{gathered} \frac{2\mathrm{\pi }{r}_1}{2\mathrm{\pi }{r}_2}=\frac{2}{3} \\ \Rightarrow \frac{r_1}{r_2}=\frac{2}{3} \end{gathered}$$
Also,
Area of the first circle$=\mathrm{\pi }{r_1}^2$
Area of the second circle$=\mathrm{\pi }{r_2}^2$
Thus, we have:
$\frac{\mathrm{\pi }{r_1}^2}{\mathrm{\pi }{r_2}^2}=\frac{{r_1}^2}{{r_2}^2}$

$$\begin{gathered} =\frac{2^2}{3^2}\left[\because \frac{{\mathrm{r}}_1}{{\mathrm{r}}_2}=\frac{2}{3}\right] \\ =\frac{4}{9} \\ =4:9 \end{gathered}$$

Hence, the ratio of their areas is 4:9.
Hence, assertion (A) is true.

Reason (R):
The given statement is true.
Hence, both assertion (A) and reason (R) are true, but reason (R) is not the correct explanation of assertion (A).