Question

Three circles of radii $r_1$, $r_2$ and $r_3$ are drawn concentric to each other. The radii $r_1$ and $r_2$ are such that the area of the circle with radius $r_1$ is equal to the area between the circles of radii $r_2$ and $r_1$. The area between the circles of radii $r_3$ and $r_2$ is equal to the area between the circles of radii $r_2$ and $r_1$ What is the value of $r_1:r_2:r_3$?

• A. $1:\sqrt{2}:\sqrt{3}.$
• B. $2:3:5$
• C. $3:4:5$
• D. $4:8:9$

Answer 1

The correct option is A $1:\sqrt{2}:\sqrt{3}.$

We have $\pi r_1^2=\pi (r_2^2-r_1^2)\Rightarrow 2r_1^2=r_2^2\Rightarrow \frac{r_1^2}{r_2^2}=\frac{1}{2}$
$\Rightarrow \frac{r_1}{r_2}=\frac{1}{\sqrt{2}}\Rightarrow r_1:r_2=1:\sqrt{2}..........\left(1\right)$ Also $\pi (r_3^2-r_2^2)=\pi (r_2^2-r_1^2)$
$\Rightarrow r_3^2=2r_2^2-r_1^2=2r_2^2-\frac{r_2^2}{2}=\frac{3r_2^2}{2}\Rightarrow (\therefore \frac{r_1}{r_2}=\frac{1}{\sqrt{2}}\Rightarrow r_1=\frac{r_2}{\sqrt{2}})$
$\Rightarrow \frac{r_2^2}{r_3^2}=\frac{2}{3}\Rightarrow \frac{r_2}{r_3}=\frac{\sqrt{2}}{\sqrt{3}}\Rightarrow r_2:r_3=\sqrt{2}:\sqrt{3}........\left(2\right)$
From (1) and (2) $r_1:r_2:r_3=1:\sqrt{2}:\sqrt{3}.$