HMMT 二月 2007 · CALC 赛 · 第 6 题

HMMT February 2007 — CALC Round — Problem 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 . 0

解析

[ 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 . 0 1 2 3 2 2 Answer: . Note that y ≥ 0, so x ≥ − 1 and x ≥ − 1. Let the circle be defined by ( x − 4) + y = c 3 d y 2 for some c ≥ 0. Now differentiate the equations with respect to x , obtaining 2 y = 3 x from the d x d y d y given and 2 y = − 2 x + 8 from the circle. For tangency, the two expressions must be equal if they d x d x 2 are well-defined, and this is almost always the case. Thus, − 2 x + 8 = 3 x so x = − 2 or x = 4 / 3, 0 0 0 0 2 3 but only the latter corresponds to a point on y = x + 1. Otherwise, y = 0, and this gives the trivial 0 solution x = − 1. 0