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
Assume the center of $C$ is $(a, a)$. Then its radius is also $a$.
A circle with center $C$ is tangent to the positive $x,y$-axes and externally tangent to the circle centered at $(5,0)$ with radius $2$. What is the sum of all possible radii of the circle centered at $C$?
