HMMT 二月 2008 · 冲刺赛 · 第 26 题

HMMT February 2008 — Guts Round — Problem 26

[ 12 ] Let P be a parabola, and let V and F be its vertex and focus, respectively. Let A and B be 1 1 ◦ points on P so that ∠ AV B = 90 . Let Q be the locus of the midpoint of AB . It turns out that Q 1 is also a parabola, and let V and F denote its vertex and focus, respectively. Determine the ratio 2 2 F F /V V . 1 2 1 2 ◦

解析

[ 12 ] Let P be a parabola, and let V and F be its vertex and focus, respectively. Let A and B be 1 1 ◦ points on P so that ∠ AV B = 90 . Let Q be the locus of the midpoint of AB . It turns out that Q 1 is also a parabola, and let V and F denote its vertex and focus, respectively. Determine the ratio 2 2 F F /V V . 1 2 1 2 7 2 Answer: Since all parabolas are similar, we may assume that P is the curve y = x . Then, if 8 2 2 ◦ 2 2 A = ( a, a ) and B = ( b, b ), the condition that ∠ AV B = 90 gives ab + a b = 0, or ab = − 1. Then, 1 the midpoint of AB is ( ) ( ) ( ) 2 2 2 2 A + B a + b a + b a + b ( a + b ) − 2 ab a + b ( a + b ) = , = , = , + 1 . 2 2 2 2 2 2 2 (Note that a + b can range over all real numbers under the constraint ab = − 1.) It follows that the 2 locus of the midpoint of AB is the curve y = 2 x + 1. 2 1 1 Recall that the focus of y = ax is (0 , ). We find that V = (0 , 0), V = (0 , 1), F = (0 , ), 1 2 1 4 a 4 1 7 F = (0 , 1 + ). Therefore, F F /V V = . 2 1 2 1 2 8 8 ◦