1. Tracing the center of our tangent circle ...
The first thing we want to do is look for patterns when we trace the center of our circle. We can break that down into six subcases:
a) Smaller circle, circles intersect:
Let's start with two circles and and the smaller of the two tangent circles when the two circles intersect:

b) Smaller circle, circles are disjoint, and one is not inside the other:

c) Smaller circle, circles are disjoint, and one is inside the other:



d) Bigger circle, circles intersect:



e) Bigger circle, circles are disjoint, and one is not inside the other:



c) Bigger circle, circles are disjoint, and one is inside the other:



From these observations, we can see that from three of our cases, the result is three ellipses and three hyperbolas. Here is the breakdown table:
Circles Intersect  One outside the other  One inside the other  
Big Circle  Hyperbola  Hyperbola  Ellipse 
Small Circle  Ellipse  Hyperbola  Ellipse 
2. What happens when one has a circle tangent to a line?
