HMMT 二月 2003 · 几何 · 第 4 题

HMMT February 2003 — Geometry — Problem 4

Farmer John is inside of an ellipse with reflective sides, given by the equation x /a + 2 2 y /b = 1, with a > b > 0. He is standing at the point (3 , 0), and he shines a laser pointer in the y -direciton. The light reflects off the ellipse and proceeds directly toward Farmer Brown, traveling a distance of 10 before reaching him. Farmer John then spins around in a circle; wherever he points the laser, the light reflects off the wall and hits Farmer Brown. What is the ordered pair ( a, b )? 1

解析

Farmer John is inside of an ellipse with reflective sides, given by the equation x /a + 2 2 y /b = 1, with a > b > 0. He is standing at the point (3 , 0), and he shines a laser pointer in the y -direciton. The light reflects off the ellipse and proceeds directly toward Farmer Brown, traveling a distance of 10 before reaching him. Farmer John then spins around in a circle; wherever he points the laser, the light reflects off the wall and hits Farmer Brown. What is the ordered pair ( a, b )? Solution: (5 , 4) The points where the farmers are standing must be the foci of the ellipse, so they are (3 , 0) and ( − 3 , 0). If the total distance traveled is 10, then a must be half of that, or 5, since the distance traveled by a ray reflecting off the wall from when it leaves one focus to when it reaches the other focus is 2 a , the length of the major axis. If a focus √ √ 2 2 2 is at x = 3, then we have 3 = a − b = 25 − b , yielding b = 4.