2001 AIME I Problems/Problem 5
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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 |
Solution
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 • Resources) | ||
| Preceded by Problem 4 | Followed by Problem 6 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
