HMMT 二月 2022 · 几何 · 第 8 题

HMMT February 2022 — Geometry — Problem 8

Let E be an ellipse with foci A and B . Suppose there exists a parabola P such that • P passes through A and B , • the focus F of P lies on E , • the orthocenter H of △ F AB lies on the directrix of P . 2 2 If the major and minor axes of E have lengths 50 and 14, respectively, compute AH + BH .

解析

Let E be an ellipse with foci A and B . Suppose there exists a parabola P such that • P passes through A and B , • the focus F of P lies on E , • the orthocenter H of △ F AB lies on the directrix of P . 2 2 If the major and minor axes of E have lengths 50 and 14, respectively, compute AH + BH . Proposed by: Jeffrey Lu Answer: 2402 Solution: Let D and E be the projections of A and B onto the directrix of P , respectively. Also, let ω be the circle centered at A with radius AD = AF , and define ω similarly. A B 2 2 If M is the midpoint of DE , then M lies on the radical axis of ω and ω since M D = M E . Since F A B lies on both ω and ω , it follows that M F is the radical axis of the two circles. Moreover, M F ⊥ AB , A B so we must have M = H . 1 Let N be the midpoint of AB . We compute that AD + BE = AF + F B = 50, so HN = ( AD + BE ) = 2 √ 2 2