Question

设 $S$ 为坐标平面中所有同时与下面三个圆相切的圆的集合：求 $S$ 中所有圆的面积之和。 $\begin{gathered}x^{2}+y^{2}=4,\qquad x^{2}+y^{2}=64,\\ \text{and}\qquad (x-5)^{2}+y^{2}=3.\end{gathered}$

Let $S$ be the set of circles in the coordinate plane that are tangent to each of the three circles with equations $\begin{gathered}x^{2}+y^{2}=4,\qquad x^{2}+y^{2}=64,\\ \text{and}\qquad (x-5)^{2}+y^{2}=3.\end{gathered}$ What is the sum of the areas of all circles in $S?$

$\ 48 \pi$

$\ 68 \pi$

$\ 96 \pi$

$\ 102 \pi$

$\ 136 \pi$

答案：E

知识点：相切圆圆面积分类讨论

难度评级：2390

解答：

记 $8$ 为圆 $r$ ， $d$ 为圆 $d+r=8$ ，为圆。首先注意，中任意圆都必须与内切。接着按照与和的相切方式分类。情况这对应粉色圆。此时和都与内切。情况 $2$ 这对应蓝色圆。此时和都与外切。

情况 $3$ 这对应绿色圆。此时 $d-r=2$ 外切、 $(r,d)=(3,5)$ 内切于 $r-d=2$ 。情况 $(r,d)=(5,3)$ 这对应红色圆。此时内切、外切于。可以把情况和放在一起考虑。注意和同心。连接与圆心的直线会经过与的切点以及与的切点。这条线是的直径，其长度为因此的半径是 $5$ 。

再把情况 $\sqrt3$ 和 $(5,0)$ 放在一起考虑。与上面类似，连接 $r$ 与 $r-\sqrt3$ 圆心的直线会经过切点。不过这一次， $r+\sqrt3$ 的直径为因此 $q$ 的半径是 $|d-q|<5<d+q$ 。 $d$ 中共有 $x$ 个圆：其中 $4$ 个半径为，另外个半径为；这是因为把图中的所有圆关于横轴翻折，还能得到另外个圆。

中所有圆的总面积为所以正确答案是 E。 $4(5^2\pi+3^2\pi)=136\pi.$

Call the concentric circles of radii $2$ and $8$ the inner and outer circles. Let a desired circle have radius $r$ and let its center be distance $d$ from the origin. It must be internally tangent to the outer circle, so $d+r=8.$

If it is externally tangent to the inner circle, then $d-r=2,$ giving $(r,d)=(3,5).$ If it contains the inner circle, then $r-d=2,$ giving $(r,d)=(5,3).$ Thus every desired circle has radius $3$ or $5.$

The third given circle has radius $\sqrt3$ and center $(5,0).$ For either value of $r,$ a desired circle may be internally or externally tangent to it, so the distance between their centers is $r-\sqrt3$ or $r+\sqrt3.$ In all four cases, if this distance is $q,$ then $|d-q|<5<d+q,$ so the circle of possible centers intersects the circle of radius $d$ about the origin in two points, symmetric across the $x$ -axis, as shown. Hence there are $4$ desired circles of each radius.

The total area is therefore $4(5^2\pi+3^2\pi)=136\pi.$ Thus, E is the correct answer.

2022 AMC 10B 第 22 题

答案：E

难度评级：2390

解答：

其他年份的第 22 题