HMMT 十一月 2012 · 冲刺赛 · 第 15 题
HMMT November 2012 — Guts Round — Problem 15
题目详情
- [ 9 ] Find the area of the region in the xy -plane consisting of all points ( a, b ) such that the quadratic 2 ax + 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
- [ 9 ] Answer: 49 π To find the region in question, we want to find ( a, b ) such that the discriminant of the 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 .