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 $C$?

1. As in the problem, the circle centered at $C=(a,a)$ has radius $a$, and external tangency to the circle at $(d,0)$ with radius $k$ gives $(a-d)^2+a^2=(a+k)^2$, i.e. $a^2-2a(d+k)+(d^2-k^2)=0$. Both roots are positive, so by Vieta's formulas the sum of the possible radii is $2(d+k)$.
2. With $d=5,k=2$: sum $=2(7)=\boxed{14}$.