Determine the locus

Question

Let $0<a<b$. Consider two circles with radii $a$ and $b$ and centres $(a, 0)$ and $(b,0)$ respectively with $0<a<b$. Let $c$ be the center of any circle in the crescent shaped region $M$ between the two circles and tangent to both (see figure below). Determine the locus of $c$ as the circle c traverses through region $M$ maintaining tangency to both the circles.

Answer 1

Let's call the center of one fixed circle $O_1$ and the other $O_2$. Say the radius of the variable circle is $r$.

Because the variable circle maintains tangency with the fixed circles we have the following equations: $$\overline{CO_1}=a+r$$ $$\overline{CO_2}=b-r$$ And adding the two we have:$$\overline{CO_1}+\overline{CO_2}=a+b$$

Hence we conclude that the locus of $C$ is the ellipse with foci at $O_1$ and $O_2$ and length of major axis $a+b$