HMMT 二月 2001 · CALC 赛 · 第 6 题

英文解析

6. The graph of x − ( y − 1) = 1 has one tangent line with positive slope that passes − 1 athrough ( x, y ) = (0 , 0). If the point of tangency is ( a, b ), find sin ( ) in radians.
   d yb
   Solution: Differentiating both sides of the equation, we find that 2 x − 2( y − 1) = 0,
   d xd yx a b b aand 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 πb = 2, a = (2). Finally, sin ( ) = .
   b 4
   12