Let, centre $=( h , k )$ and radius
$= r$ for the variable circle So,
using $C_{1} C_{2}=r_{1}+r_{2}$
for both cases we have:
$h^{2}+k^{2}=(r+a)^{2} \rightarrow(1)$ and
$(h-2 a)^{2}+k^{2}=(r+2 a)^{2} \rightarrow(2)$
Eq. (2) - Eq. (1), gives :
$r=\frac{a-4 h}{2} \rightarrow$ (3)
Substitute (3) in (1) to get:
$12 h^{2}-4 k^{2}-24 a h+9 a^{2}=0$
$\therefore $ locus : $12 x^{2}-4 y^{2}-24 a x+9 a^{2}=0$ i.e. a hyperbola