PUMaC 2012 · 几何（A 组） · 第 4 题

PUMaC 2012 — Geometry (Division A) — Problem 4

[ 4 ] A square is inscribed in an ellipse such that two sides of the square respectively pass through the two foci of the ellipse. The square has a side length of 4. The square of the length of the √ minor axis of the ellipse can be written in the form a + b c where a , b , and c are integers, and c is not divisible by the square of any prime. Find the sum a + b + c . ◦ ◦

解析

[ 4 ] A square is inscribed in an ellipse such that two sides of the square respectively pass through the two foci of the ellipse. The square has a side length of 4. The square of the length of the √ minor axis of the ellipse can be written in the form a + b c where a , b , and c are integers, and c is not divisible by the square of any prime. Find the sum a + b + c . Solution: Let a be the length of the major axis, b be the length of the minor axis, and c be the distance from the foci to the center of the ellipse. Since the sum of the distances from any point on the ellipse to the foci is 2 a , we can use a vertex of the square to calculate 2 a . We have √ 2 a = 2 + 2 5 √ 2 2 2 so a = 1 + 5. Now using the relation b = a − c , we have √ √ 2 b = 1 + 2 5 + 5 − 4 = 2 + 2 5 √ √ √ √ 2 b = 2 2 + 2 5 = 8 + 8 5 √ So the square of this is 8 + 8 5, so a + b + c = 8 + 8 + 5 = 21 Problem contributed by Elizabeth Yang ◦ ◦