Common Locus Results

Equidistant from two points A,B: perpendicular bisector of AB.
Equidistant from a point F and line l: parabola with focus F, directrix l.
Sum of distances from two points = constant: ellipse.
|Difference of distances from two points| = constant: hyperbola.
Constant distance from a fixed point: circle.
PA/PB = constant k (!=1): circle (Apollonius circle).
PA * PB = constant with angle between PA, PB fixed: related to conics.