Question

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.

Answer 1

Le t the length of the rectangle be $x,y$. Let the axis of the ellipse on which the foci lie have the length $2a$, and the other axis have length $2b$. We have,

$xy=ab=200$

From the definition, we have $x+y=2a$. Also, the diagonal of the rectangle have a length of $\sqrt{x^2+y^2}$. Comparing the length of the axes and the distance from the foci to the center, we have

$a^2=\frac{x^2+y^2}{4}+b^2$

$(x+y{)}^2=x^2+y^2+4b^2$

$2xy=4b^2$

$b=10$

$a=20$

Perimeter = $2(a+b)=2\times 40=80$