Since we know the circles are tangent to the $y$-axis at the same point, we know that the top edge of the rectangle is as long as both radii of the two circles added together. This means that this edge has a length of 9 . The other side is equal to the radius of the larger circle. Let's set $a=$ longer radius and $b=$ shorter radius. The ratios of the areas of the circles would be:

$$
\(\frac{\pi a^2}{\pi b^2}=\frac{a^2}{b^2}=\left(\frac{a}{b}\right)^2\)
$$


Since that ratio is 4 , we can further state:

$$
\(\begin{aligned}
\left(\frac{a}{b}\right)^2 & =4 \\
\frac{a}{b} & =2
\end{aligned}\)
$$


This means $\(a=2 b\)$. Since the distance between the centers is equal to the combined radii of the two circles, we can set up another equation, $\(a+b=9\)$. Through substitution, this gives us:

$$
\(\begin{aligned}
2 b+b & =9 \\
3 b & =9 \\
b & =3 \\
a & =6
\end{aligned}\)
$$


With length $\(=9\)$ and $\(a=\)$ width $=6$, length $\(\times\)$ width $\(=9 \times 6=54\)$.

Therefore, the area of the rectangle is 54 .