Rectangle has area 200. An ellipse with area  passes through and and has foci at and . If the perimeter of the rectangle is , then the value of  is

(4)

Let sides of rectangle be and

Area of rectangle     (1)

Area of ellipse

   (2)

We have to find the perimeter of rectangle

From triangle

Distance  distance between foci

or

or    (3)

Also from the definition of ellipse sum of focal length is ,

Then     (4)

Putting value of  in equation (3) and (4)

We have  (using equation (1))

From equation (2),

Since  (from equation (4))

Therefore perimeter