Q:

Find the Locus of The Circle Touching Two Circles

Here circle 'A' and 'B' have the radii $a$ and $b$ respectively. 'C' is a circle belonging to the mentioned family of circles. How to solve this problem?
(circle 'C' is drawn just for clarification)

A:

Hint.  Assume you have two fixed points $F_1$ and $F_2$ in the plane and a fixed positive number $d > |F_1F_2|$. What is the geometric locus of all points $X$ in the plane with the property that $|F_1X| + |XF_2| = d \,$? Do you have two such points on your picture and a number $d$ that you can use to form such a locus?   

