Locus of the Centers of Circles. 
Find the locus of the centers of the circles that touch externally the circles $x^2+y^2=a^2$ and $x^2+y^2=4ax$.

I have tried many things like finding a relationship between the distance between the centers of the circles, but I am not getting it.
Any help is appreciated.

Let $S$ be the center of circle with a (variable) radius $r$ which touches both circles.
Since $F(0,0)$ is the center of the first and $F'(2a,0)$ is the center of the second circle we have:
$$SF' -SF = (r+2a)-(r+a) = a$$ so $S$ describe one branch of hyperbola with focus at $F$ and $F'$.
