HMMT 二月 2001 · CALC 赛 · 第 6 题

HMMT February 2001 — CALC Round — Problem 6

The graph of x − ( y − 1) = 1 has one tangent line with positive slope that passes − 1 a through ( x, y ) = (0 , 0). If the point of tangency is ( a, b ), find sin ( ) in radians. b 12

解析

The graph of x − ( y − 1) = 1 has one tangent line with positive slope that passes − 1 a through ( x, y ) = (0 , 0). If the point of tangency is ( a, b ), find sin ( ) in radians. b d y Solution: Differentiating both sides of the equation, we find that 2 x − 2( y − 1) = 0, d x d y x a b b a and so = = . The line passing through (0 , 0) and ( a, b ) has slope , so = . d x y − 1 b − 1 a a b − 1 2 2 2 2 Solving simultaneously with a − ( b − 1) = 1, we get b − b − [( b − 1) + 1] = 0, and so √ − 1 a π b = 2, a = (2). Finally, sin ( ) = . b 4 12