2006 AMC 12B Problems/Problem 21
Problem
Rectangle has area . An ellipse with area passes through and and has foci at and . What is the perimeter of the rectangle? (The area of an ellipse is where and are the lengths of the axes.)
Solution
Let the rectangle have side lengths and . Let the axis of the ellipse on which the foci lie have length , and let the other axis have length . We have From the definition of an ellipse, . Also, the diagonal of the rectangle has length . Comparing the lengths of the axes and the distance from the foci to the center, we have Since , we now know and because , or one-fourth of the rectangle's perimeter, we multiply by four to get an answer of .
See also
2006 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 20 |
Followed by Problem 22 |
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All AMC 12 Problems and Solutions |
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