I wish to examine the locus of points equidistant from a fixed point F and a circle. The locus of points equisdistant from these two points will lie on the perpendicular bisector of the segment joining F and the point on the circle.

Click to explore a GSP file (with script tool!!) demonstrating the construction and results: Conic!

When point F is outside the circle, the locus of the points equidistant between F and a point on the circle is a (ugly pdf file) hyperbola.

When point F is outside the circle, the locus of the points equidistant between F and a point on the circle is an (ugly pdf file) ellipse.