Ans: (D)
Hint: Let, centre \(\equiv(\mathrm{h}, \mathrm{k})\) and radius \(=\mathrm{r}\) for the variable circle
So, using \(C_{1} C_{2}=r_{1}+r_{2}\) for both cases we have:
$$
\mathrm{h}^{2}+\mathrm{k}^{2}=(\mathrm{r}+\mathrm{a})^{2} \rightarrow(1) \text { and }(\mathrm{h}-2 \mathrm{a})^{2}+\mathrm{k}^{2}=(\mathrm{r}+2 \mathrm{a})^{2} \rightarrow(2)
$$
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
