HMMT 二月 2008 · 冲刺赛 · 第 16 题

HMMT February 2008 — Guts Round — Problem 16

[ 9 ] Point A lies at (0 , 4) and point B lies at (3 , 8). Find the x -coordinate of the point X on the x -axis maximizing ∠ AXB .

解析

[ 9 ] Point A lies at (0 , 4) and point B lies at (3 , 8). Find the x -coordinate of the point X on the x -axis maximizing ∠ AXB . √ Answer: 5 2 − 3 Let X be a point on the x -axis and let θ = ∠ AXB . We can easily see that the circle with diameter AB does not meet the x -axis, so θ ≤ π . Thus, maximizing θ is equivalent to maximizing sin θ . By the Law of Sines, this in turn is equivalent to minimizing the circumradius of triangle ABX . This will occur when the circumcircle of ABX is the smaller of the two circles through A and B tangent to the x -axis. So let X now be this point of tangency. Extend line AB to meet the √ 2 x -axis at C = ( − 3 , 0); by Power of a Point CX = CA · CB = 50 so CX = 5 2. Clearly X has larger √ x -coordinate than C , so the x -coordinate of X is 5 2 − 3.