GMAT. What is the ratio of the diameter vs radius of circles with areas in a specific ratio?

Question

Here is the image:

In the diagram above, point B is the center of Circle #1 and point D is the center of Circle #2. If the ratio of the area of Circle #2 to the area of Circle #1 is 3:2, what is the ratio CE:BC?

Can we try using two sizes as an example? Can we start out with circle 2 with an area of $30 \pi$ and circle 1 as $20\pi$?

This leads to the radius of circle 2 being $\sqrt[2]{30}$ and the radius of circle 1 as $\sqrt[2]{20}$?

Then 2 * the radius of circle 2 is $2 * \sqrt[2]{30}$

So CE/BC is $(2 * \sqrt[2]{30})$ / $\sqrt[2]{20}$

Anyway, the answer to the problem is $\sqrt[2]{6}$ but I can't seem to arrive at that answer independently.

Answer 1

Let $r_2$ be the radius of the bigger circle and $r_1$ that of the smaller one. We are told $r_2^2/r_1^2=1.5$ and we are asked to solve for $2r_2/(r_1)$. Then: $$ 2r_2/(r_1)=2(r_2/r_1)=2\sqrt{r_2^2/r_1^2}=2\sqrt{1.5}=\sqrt{6}. $$

Answer 2

The given equates to, if $R_1$ and $R_2$ are the radii of circle1 and circle2 respectively, $\frac{\pi R_2^2}{\pi R_1^2}=(\frac{R_2}{R_1})^2=\frac{3}{2}$, so $R_2=R_1\sqrt{\frac{3}{2}}$.

$|CE|=2R_2$ and $|BC|=R_1$, so $\frac{|CE|}{|BC|}=\frac{2R_2}{R_1}=\frac{2R_1\sqrt{\frac{3}{2}}}{R_1}=\sqrt{6}$