HMMT 二月 2008 · 冲刺赛 · 第 26 题

英文解析

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 Qis also a parabola, and let V and F denote its vertex and focus, respectively. Determine the ratio 1
    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
    2 2 ° 2 28
    A = ( a, a ) and B = ( b, b ), the condition that ∠ AV B = 90 gives ab + a b = 0, or ab = − 1. Then,
    the midpoint of AB is 1
    ( ) ( ) ( )
    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 thelocus of the midpoint of AB is the curve y = 2 x + 1.2
    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
    °