Question

The areas of two circles are in the ratio 4 : 9. The ratio of their circumferences is
(a) 2 : 3
(b) 3 : 2
(c) 4 : 9
(d) 9 : 4

Answer 1

(a) 2:3
Let $r_1\mathrm{and}{r}_2$ be the radii of the two circles.
Now,
Area of the first circle$=\mathrm{\pi }{r_1}^2$
And,
Area of the second circle$=\mathrm{\pi }{r_2}^2$
Thus, we have:
$$\begin{gathered} \frac{\mathrm{\pi }{r_1}^2}{\mathrm{\pi }{r_2}^2}==\frac{4}{9} \\ \Rightarrow \frac{{r_1}^2}{{r_2}^2}=\frac{4}{9} \end{gathered}$$
Also,
Perimeter of the first circle$=2\mathrm{\pi }{r}_1$
And,
Perimeter of the second circle $=2\mathrm{\pi }{r}_2$
Thus, we have:
$$\begin{gathered} \frac{2\mathrm{\pi }{r}_1}{2\mathrm{\pi }{r}_2}=\frac{r_1}{r_2} \\ \Rightarrow \frac{r_1}{r_2}=\frac{2}{3}\left[\mathrm{Since}\frac{{{\mathrm{r}}_1}^2}{{{\mathrm{r}}_2}^2}=\frac{4}{9}\right] \end{gathered}$$
=2:3

Hence, the ratio of their circumferences is 2:3.