2009 AMC 12A Problems/Problem 16

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Problem

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 $C$ ?

$\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 9$

Solution

Let $r$ be the radius of our circle. For it to be tangent to the positive $x$ and $y$ axes, we must have $C=(r,r)$ . For the circle to be externally tangent to the circle centered at $(3,0)$ with radius $1$ , the distance between $C$ and $(3,0)$ must be exactly $r+1$ .

By the Pythagorean theorem the distance between $(r,r)$ and $(3,0)$ is $\sqrt{ (r-3)^2 + r^2 }$ , hence we get the equation $(r-3)^2 + r^2 = (r+1)^2$ .

Simplifying, we obtain $r^2 - 8r + 8 = 0$ . By Vieta's formulas the sum of the two roots of this equation is $\boxed{8}$ .

(We should actually solve for $r$ to verify that there are two distinct positive roots. In this case we get $r=4\pm 2\sqrt 2$ .)

2009 AMC 12A (Problems • Resources)
Preceded by Problem 15	Followed by Problem 17
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2009 AMC 12A Problems/Problem 16

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