# Rectangle $ABCD$ has area 200.An ellipse with area $200\pi$ passes through $A$ and $C$ and has foci at $B$ and $D$.

Rectangle $ABCD$ has area 200.An ellipse with area $200\pi$ passes through $A$ and $C$ and has foci at $B$ and $D$.Find the perimeter of the rectangle.

Let the side lengths of the rectangle ABCD be $l$ and $\frac{200}{l}$ and the foci of the rectangle be $(-c,0)$ and $(c,0)$

So $2c=\sqrt{l^2+\frac{40000}{l^2}}$
$4c^2=l^2+\frac{40000}{l^2}$

And the area of the ellipse is $200\pi=\pi ab$

where the $a$ and $b$ are the semi major axes and semi minor axes of the ellipse.
I cannot complete the solution.I am confused where to take the origin and what are the coordinates of $A$ and $C.$

• you havent made use of property foci is at B,D – Archis Welankar Mar 17 '16 at 14:28

Make use of the fact that $2a=BA+DA=l+200/l$ and $a^2-b^2=c^2$. Combining these you get $b=10$.