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HMMT 十一月 2012 · 冲刺赛 · 第 15 题

HMMT November 2012 — Guts Round — Problem 15

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

  1. [ 9 ] Find the area of the region in the xy -plane consisting of all points ( a, b ) such that the quadraticax + 2( a + b − 7) x + 2 b = 0 has fewer than two real solutions for x .2
    HMMT NOVEMBER 2012, 10 NOVEMBER 2012 — GUTS ROUND

英文原题

[ 9 ] Find the area of the region in the xy -plane consisting of all points ( a, b ) such that the quadratic
ax 2 + 2( a + b − 7) x + 2 b = 0 has fewer than two real solutions for x .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
HMMT NOVEMBER 2012, 10 NOVEMBER 2012 — GUTS ROUND

解析

英文解析

  1. [ 9 ]
    Answer: 49 π To find the region in question, we want to find ( a, b ) such that the discriminant ofthe quadratic is not positive. In other words, we want
    2 2 2 2 2
    4( a + b − 7) − 4( a )(2 b ) ≤ 0 ⇔ a + b − 7 a − 7 b + 49 ≤ 0 ⇔ ( a − 7) + ( b − 7) ≤ 49 ,
    which is a circle of radius 7 centered at (7 , 7) and hence has area 49 π .