2001 AIME I Problems/Problem 5
Problem
An equilateral triangle is inscribed in the ellipse whose equation is . One vertex of the triangle is
, one altitude is contained in the y-axis, and the length of each side is
, where
and
are relatively prime positive integers. Find
.
Contents
[hide]Solution
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Solution 1
Denote the vertices of the triangle and
where
is in quadrant 4 and
is in quadrant
Note that the slope of is
Hence, the equation of the line containing
is
This will intersect the ellipse when
Since the triangle is symmetric with respect to the y-axis, the coordinates of
and
are now
and
respectively, for some value of
Since we're going to use the distance formula, the value of is irrelevant. Our answer is
Solution 2
Solving for in terms of
gives
, so the two other points of the triangle are
and
, which are a distance of
apart. Thus
equals the distance between
and
, so by the distance formula we have
Squaring both sides and simplifying through algebra yields , so
and the answer is
.
See also
2001 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 4 |
Followed by Problem 6 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |