HMMT 十一月 2009 · GEN2 赛 · 第 3 题

HMMT November 2009 — GEN2 Round — Problem 3

[ 5 ] Let C be the circle of radius 12 centered at (0 , 0). What is the length of the shortest path in the √ √ plane between (8 3 , 0) and (0 , 12 2) that does not pass through the interior of C ?

解析

[ 5 ] Let C be the circle of radius 12 centered at (0 , 0). What is the length of the shortest path in the √ √ plane between (8 3 , 0) and (0 , 12 2) that does not pass through the interior of C ? √ Answer: 12 + 4 3 + π The shortest path consists of a tangent to the circle, a circular arc, and √ √ then another tangent. The first tangent, from (8 3 , 0) to the circle, has length 4 3, because it is a leg √ 15 ◦ of a 30-60-90 right triangle. The 15 arc has length (24 π ), or π , and the final tangent, to (0 , 12 2), 360 has length 12.