1952 AMC 12 第 8 题

Question

同一平面内两个半径相等的圆，其公切线条数不可能是：

Two equal circles in the same plane cannot have the following number of common tangents:

1

2

3

4

以上答案均不正确

None of these

答案：A

知识点：圆切线分类讨论

难度评级：1430

小提示：

分别考虑两个等圆相交、外切和相离的情形

Consider intersecting, externally tangent, and disjoint equal circles

大提示：

除非两圆重合，否则内切要求两圆半径不相等

An internal tangency would require unequal radii unless the circles coincide

解答：

两个不同的等圆相交时有

2

条公切线，外切时有

3

条，相离时有

4

条。它们不能内切，因为内切要求圆心距等于两半径之差，即

0

。圆心距为

0

时，两个等圆重合，并有无穷多条公切线。

因此，恰有

1

条公切线是不可能的，所以正确答案是 A。

Two distinct equal circles have

2

common tangents when they intersect,

3

when externally tangent, and

4

when disjoint. They cannot be internally tangent, because internal tangency requires the center distance to equal the difference of the radii, which is

0.

At center distance

0

the equal circles coincide and have infinitely many common tangents.

Thus exactly

1

common tangent is impossible, so the correct answer is A.

1952 AMC 12 第 8 题

答案：A

难度评级：1430

小提示：

大提示：

解答：

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