Question

Rectangle $ABCD$ has area 200. An ellipse with area $200\pi$ passes through $A$ and $C$ and has foci at $B$ and $D$. If the perimeter of the rectangle is $P$, then the value of $\frac{P}{20}$ is

Answer 1

Let the sides of rectangle be $p$ and $q$.
Area of rectangle $=pq=200\cdots \left(i\right)$

Area of ellipse $=\pi ab=200\pi$
$\therefore ab=200\cdots \left(ii\right)$
Perimeter of rectangle $=2(p+q)$
From triangle ABD,
$BD=$ Distance between foci
$\Rightarrow p^2+q^2=4a^2{e}^2$
$\Rightarrow (p+q{)}^2-2pq=4(a^2-b^2)\cdots (iii)$
Also, from the definition of ellipse, the sum of focal lengths is $2a$. Then,
$AB+AD=p+q=2a\cdots \left(iv\right)$
Putting the value of $(p+q)$ in (iii),
$(2a{)}^2-2pq=4a^2-4b^2$
$\Rightarrow 4a^2-2\times 200=4(a^2-b^2)$
$\Rightarrow a^2-100=a^2-b^2$
$\Rightarrow b=10$
From (ii),
$ab=200\Rightarrow a=20$
We know that,
$p+q=2a$
So the perimeter will be
$2(p+q)=4\times 20=80$
Hence $\frac{P}{20}=4$