Circle with External Tangency

Steps

1. The set of centers of all circles that have either two internal tangencies or two external tangencies with the given circles is a hyperbola. The foci of this hyperbola are the centers of the given circles. To construct the hyperbola, we need at least one point on it. One such point is the point of tangency of the two circles.
2. Draw the hyperbola using the point identified in the previous step.
3. The set of centers of circles that are externally tangent to one circle and internally tangent to the other is a straight line, which also passes through the centers of the given circles.
4. The required locus is the combination of the straight line and the hyperbola.