Locus of the Centers of Circles.

Question

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.

Answer 1

Let $P=(x,y)$ be the center of a tangent circle, $A=(0,0)$ and $B=(2a,0)$. We have then: $$ PA-a=PB-2a, \quad\hbox{that is:}\quad \sqrt{x^2+y^2}+a=\sqrt{(x-2a)^2+y^2}. $$ Square the last equality, simplify the result and adjust it to have a single square root on the left hand side, then square again. You'll end up with the equation of the required locus.

Answer 2

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 $F'$.