2001 AIME I Problems/Problem 5
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 .
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
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 .
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