1987 AIME 第 2 题

Question

一个点在以

(-2,-10,5)

为球心、半径为

19

的球面上，另一个点在以

(12,8,-16)

为球心、半径为

87

的球面上。两点间距离的最大可能值是多少？

What is the largest possible distance between two points, one on the sphere of radius

19

centered at

(-2,-10,5)

and the other on the sphere of radius

87

centered at

(12,8,-16)?

答案

先求两个球心之间的距离

First find the distance between the two centers

最大值在两点分别位于球心连线向外延伸的两侧时取得

The maximum occurs along the line of centers, on the two outward sides

两球心间距离的平方为

14^2+18^2+(-21)^2=961

，所以两球心相距

\sqrt{961}=31

。由三角不等式，两点间的最大距离在球心连线上取得，等于

19+31+87=137

。

The squared distance between the centers is

14^2+18^2+(-21)^2=961,

so they are

\sqrt{961}=31

apart. By the triangle inequality, the greatest point-to-point distance is obtained on the line of centers and equals

19+31+87=137.

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