Conic Challenge #11

Let the given circles have centers $C_1$ and $C_2$ and radii $r_1$ and $r_2$ respectively. Also let the center of the moving circle be $C$ and radius be $r$ .

So we have the following equations:
$CC_1 = r+r_1 \quad \dots (1)$
$CC_2 = r + r_2 \quad \dots (2)$
We also know, $C_1 C_2 > r_1 + r_2$ . Now subtracting the first two equations we get, $\displaystyle |CC_1 - CC_2| = | r_1 - r_2 | \quad \dots (3)$ .

The third equation and the inequality together can be read as " the locus of the center of the moving circle is such that the difference of its distances from two fixed points is constant and less than the distance between the fixed points"; which implies that the locus is a Hyperbola.