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
.
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
It is clear that the value of is irrelevant to the length of
. 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
.
Solution 3
Since the altitude goes along the axis, this means that the base is a horizontal line, which means that the endpoints of the base are
and
, and WLOG, we can say that
is positive.
Now, since all sides of an equilateral triangle are the same, we can do this (distance from one of the endpoints of the base to the vertex and the length of the base):
Square both sides,
Now, with the equation of the ellipse:
Substituting,
Moving stuff around and solving:
The second is found to be extraneous, so, when we go back and figure out and then
(which is the side length), we find it to be:
and so we get the desired answer of .
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 |
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