HMMT 十一月 2009 · 冲刺赛 · 第 17 题

HMMT November 2009 — Guts Round — Problem 17

[ 9 ] A circle passes through the points (2 , 0) and (4 , 0) and is tangent to the line y = x . Find the sum of all possible values for the y -coordinate of the center of the circle.

解析

[ 9 ] A circle passes through the points (2 , 0) and (4 , 0) and is tangent to the line y = x . Find the sum of all possible values for the y -coordinate of the center of the circle. Answer: − 6 First, we see that the x coordinate must be 3. Let the y coordinate be y. Now, we see √ 2 that the radius is r = 1 + y . The line from the center of the circle to the point of tangency with the line x = y is perpendicular to the line x = y . Hence, the distance from the center of the circle to √ the line x = y is d = 2(3 − y ). Letting r = d we see that the two possible values for y are 1 and − 7, which sum to − 6.