9. In the figure shown there are three concentric circles and two perpendicular diameters. The three shaded regions have equal area. The radius of the small circle is 2 . The product of the three radii is $Y$. What is the value of $Y^{2}$ ?
SOLUTION
384

We note that the central circle, the inner hoop and the outer hoop are also equal in area.
Let the radii of the middle circle and outer circle be $a$ and $b$ respectively.
The area of the central circle is $4 \pi$.
The area of the inner hoop is $\pi a^{2}-4 \pi$. Therefore, $\pi a^{2}-4 \pi=4 \pi$. This simplifies to $a^{2}=8$.
The area of the outer hoop is $\pi b^{2}-\pi a^{2}=\pi b^{2}-8 \pi$.
Therefore, $\pi b^{2}-8 \pi=4 \pi$. This simplifies to $b^{2}=12$.
The product of the radii, $Y$, is $2 \times a \times b$. Therefore, $Y^{2}=4 \times a^{2} \times b^{2}=4 \times 8 \times 12=384$.