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

Explanation

Let the sides of reactangle be p and q.
Area or reactangle `=pq=200" "(1)`

Area of ellipse `=piab=200pi`
`:. ab=200" (2)`
Perimeter of rectangle =2(p+q)
From triangle ABD,
Distance `BD=sqrt(p^(2)+q^(2))`= Distance between foci
`or p^(2)+q^(2)=4a^(2)e^(2)`
`or (p+q)^(2)-2pq=4(a^(2)-b^(2))" "(3)`
Also, from the definition of ellipse, the sum of focal lengths is 2a. Then,
`AB+AD=p+q=2a " "(4)`
Putting the value of (p+q) in (3) from (4), we have
`(2a)^(2)-2pq=4a^(2)-4b^(2)" "`[Using (1)]
`or 4a^(2)-2xx200xx=4(a^(2)-b^(2))`
`or a^(2)-100=a^(2)-b^(2)`
`or b=10`
From (2),
`ab=200 or a=20`
Since `p+q=2a" "` [ From (4)]
Perimete`=2(p+q)=4a=4xx20=80`