Rectangle $ABCD$ has area $2006$. An ellipse with area $2006\pi$ passes through $A$ and $C$ and has foci at $B$ and $D$. What is the perimeter of the rectangle? (The area of an ellipse is $ab\pi$ where $2a$ and $2b$ are the lengths of the axes.) $\mathrm{(A)}\ \frac {16\sqrt {2006}}{\pi} \qquad$ $ \mathrm{(B)}\ \frac {1003}4 \qquad$ $ \mathrm{(C)}\ 8\sqrt {1003} \qquad$ $ \mathrm{(D)}\ 6\sqrt {2006} \qquad$ $ \mathrm{(E)}\ \frac {32\sqrt {1003}}\pi$
Solution
Let the rectangle have sides $p=AB$ and $q=AD$, so $pq=2006$.
$A$ is on the ellipse with foci $B$ and $D$, so $2a=AB+AD=p+q$. The distance between the foci is the diagonal: $2c=BD=\sqrt{p^2+q^2}$.
Rectangle $ABCD$ has $AB=4$ and $BC=2$. An ellipse passes through $A$ and $C$ and has foci at $B$ and $D$. What is the area of the ellipse? (The area of an ellipse is $ab\pi$ where $2a$ and $2b$ are the lengths of the axes.)
Let the rectangle have sides $p=AB$ and $q=AD$, so $pq=2006$.