AMC12 2015 A

AMC12 2015 A · Q11

AMC12 2015 A · Q11. It mainly tests Angle chasing, Circle theorems.

On a sheet of paper, Isabella draws a circle of radius 2, a circle of radius 3, and all possible lines simultaneously tangent to both circles. Isabella notices that she has drawn exactly $k \ge 0$ lines. How many different values of $k$ are possible?

在一张纸上，Isabella 画了一个半径为 2 的圆，一个半径为 3 的圆，以及所有同时与这两个圆相切的直线。Isabella 注意到她画了恰好 $k \ge 0$ 条直线。$k$ 有多少种不同的可能值？

(A) 2 2

(B) 3 3

(D) 5 5

(E) 6 6

Answer

Correct choice: (D)

正确答案：（D）

Solution

Answer (D): If the smaller circle is in the interior of the larger circle, there are no common tangent lines. If the smaller circle is internally tangent to the larger circle, there is exactly one common tangent line. If the circles intersect at two points, there are exactly two common tangent lines. If the circles are externally tangent, there are exactly three tangent lines. Finally, if the circles do not intersect, there are exactly four tangent lines. Therefore, $k$ can be any of the numbers 0, 1, 2, 3, or 4, which gives 5 possibilities.

答案（D）：如果小圆在大圆内部，则没有公共切线。如果小圆与大圆内切，则恰好有一条公共切线。如果两圆相交于两点，则恰好有两条公共切线。如果两圆外切，则恰好有三条公共切线。最后，如果两圆不相交，则恰好有四条切线。因此，$k$ 可以是 0、1、2、3 或 4 中的任意一个数，共有 5 种可能。

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