Problem:
Rectangle ABCD has area 2006. An ellipse with area 2006π passes through A and C and has foci at B and D. What is the perimeter of the rectangle? (The area of an ellipse is πab, where 2a and 2b are the lengths of its axes.)
Answer Choices:
A. π162006​​
B. 41003​
C. 81003​
D. 62006​
E. π321003​​ Solution:
Let 2a and 2b, respectively, be the lengths of the major and minor axes of the ellipse, and let the dimensions of the rectangle be x and y. Then x+y is the sum of the distances from the foci to point A on the ellipse, which is 2a. The length of a diagonal of the rectangle is the distance between the foci of the ellipse, which is 2a2−b2​. Thus x+y=2a and x2+y2=4a2−4b2. The area of the rectangle is