HMMT 二月 2007 · CALC 赛 · 第 6 题

英文解析

6. [ 5 ] The elliptic curve y = x +1 is tangent to a circle centered at (4 , 0) at the point ( x , y ). Determine
   0 0
   the sum of all possible values of x .
   2 3 2 210
   Answer: . Note that y ≥ 0, so x ≥ − 1 and x ≥ − 1. Let the circle be defined by ( x − 4) + y = cd y 3
   for some c ≥ 0. Now differentiate the equations with respect to x , obtaining 2 y = 3 x from the 2
   d xd y d ygiven and 2 y = − 2 x + 8 from the circle. For tangency, the two expressions must be equal if theyd x d xare well-defined, and this is almost always the case. Thus, − 2 x + 8 = 3 x so x = − 2 or x = 4 / 3,2
   0 0 0
   2 30
   but only the latter corresponds to a point on y = x + 1. Otherwise, y = 0, and this gives the trivialsolution x = − 1.0 0