One distance-ratio rule produces ellipses, parabolas, and hyperbolas.
With focus at the origin and directrix x=d, squaring the distance relation gives x²+y²=e²(x−d)². For 0<e<1 the locus is an ellipse, for e=1 a parabola, and for e>1 a hyperbola. The visible vertex x=ed/(1+e) satisfies the unsquared nonnegative-distance relation.
The circle is a limiting conic case; setting e=0 with a fixed finite directrix collapses this particular focus-directrix equation to a point.