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 
.
Solution
Solving for y in terms of x gives 
, so the two other points of the triangle are 
and 
, which are a distance of 2x apart. Thus 2x equals the distance between and (0,1), so by the distance formula we have 
. Squaring both sides, letting 
, and simplifying gives 
, so 
and the answer is 937.
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 | ||
