Question

一个圆的圆心为 $C$ 与正 $x$ 轴和正 $y$ 轴都相切，并与圆心在 $(3, 0)$ 、半径为 $1$ 的圆外切。圆心为 $C$ 的圆所有可能半径之和是多少？

A circle with center $C$ is tangent to the positive $x$ - and $y$ -axes and externally tangent to the circle centered at $(3, 0)$ with radius $1.$ What is the sum of all possible radii of the circle with center $C?$

$3$

$4$

$6$

$8$

$9$

答案：D

知识点：相切圆坐标几何韦达定理

难度评级：1910

解答：

一个与两条正坐标轴都相切且半径为 $r$ 的圆，圆心为 $(r, r)$ 。与圆心 $(3, 0)$ 、半径 $1$ 的圆外切，意味着两圆心距离为 $r + 1$ ： $(r - 3)^2 + r^2 = (r + 1)^2.$

展开得 $r^2 - 8r + 8 = 0$ 。两个根 $r = 4 \pm 2\sqrt{2}$ 都为正，由韦达定理，它们的和为 $8$ 。

因此，正确答案是 D。

A circle tangent to both positive axes with radius $r$ has center $(r, r).$ External tangency to the circle at $(3, 0)$ of radius $1$ means the distance between centers is $r + 1$ : $(r - 3)^2 + r^2 = (r + 1)^2.$

Expanding gives $r^2 - 8r + 8 = 0.$ Both roots $r = 4 \pm 2\sqrt{2}$ are positive, and by Vieta's formulas their sum is $8.$

Thus, the correct answer is D.

2009 AMC 12A 第 16 题

答案：D

难度评级：1910

解答：

其他年份的第 16 题