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 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
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