Circle Tangent to Two Circles

The completed construction is in the applet below.  Be certain to drag all the points around. The Green circle is tangent to the grey circle AND the large circle. The center of the green circle traces a conic. Looks like an ellipse. Where are the foci?  Is it possible to get a hyperbola from this construction?

Give the construction a try!

You try it now.

Not sure?

Play through the construction below.