The locus of points equidistant from two circles.
There are two solutions: a single branch of a hyperbola (orange), and an ellipse (blue). Case I: If one circle completely encloses the other, there is one solution: an ellipse. Case II: If the two circles are entirely outside one another, there is one solution: the branch of the hyperbola closer to the midpoint of the smaller circle. Case III: If the two circles intersect, both solutions occur simultaneously. Please let me know if you would like more information about the construction; some of the details are not obvious, even from the full worksheet.