HMMT 二月 2009 · 冲刺赛 · 第 32 题
HMMT February 2009 — Guts Round — Problem 32
题目详情
英文原题
- [ 18 ] Circle Ω has radius 13. Circle ω has radius 14 and its center P lies on the boundary of circle
Ω. Points A and B lie on Ω such that chord AB has length 24 and is tangent to ω at point T. Find
AT · BT.
解析
英文解析
- [ 18 ] Circle Ω has radius 13. Circle ω has radius 14 and its center P lies on the boundary of circle
Ω. Points A and B lie on Ω such that chord AB has length 24 and is tangent to ω at point T. Find
AT · BT.
Answer: 56 Solution: Let M be the midpoint of chord AB ; then AM = BM = 12 and Pythagorason triangle AM O gives M O = 5 . Note that ∠ AOM = ∠ AOB/ 2 = ∠ AP B = ∠ AP T + ∠ T P B ortan ( ∠ AOM ) = tan ( ∠ AP T + ∠ T P B ). Applying the tangent addition formula,
AT BT
AM+
T P T P
AT BT=
M O
1 − ·
T P T P
AB · T P = ,
T P − AT · BT2
2 2
from which AT · BT = T P − AB · T P · M O/AM = 14 − 24 · 14 · 5 / 12 = 56 .