For a GSP script that constructs a circle that is tangent to 2 given circles and passes through a given point on one of the circles, click here.
The following is a construction of a tangent circle (the blue circle) where the given point is on the smaller circle and the smaller circle is external to the tangent circle.
The following is a construction of a tangent circle (the pink circle) where the given point is on the smaller given circle and the smaller circle is internal to the tangent circle.
Let's look at the loci of the centers (in red below) of the tangent circles for the above 2 cases. In both cases, they look like an ellipse.
Case I: The tangent circles (in blue) are internal to one circle and external to the other circle.
Case II: The tangent circle (in blue) is either external to both given circles or internal to both given circles.
Let's look at the loci of the centers (in red below) of the tangent circles for the above 2 cases. In the first case, it looks like an ellipse.
In the second case, it looks like a hyperbola. Click here for a GSP sketch of this case.