The September 2011 issue of The College Mathematics Journal (published by MAA) contains an article by Dan Joseph, Gregory Hartman and Caleb Gibson titled Generalized Parabolas (available online if a member/subscriber or through jstor: http://www.jstor.org/pss/10.4169/college.math.j.42.4.275 if you have access to jstor). In their article they investigate what happens if you change the directrix in the definition above to a general curve, for example, a parabola (see example 3 below). The authors took an analytical approach, using Mathematica to find the equation of each generalized parabola.
I recognized GeoGebra could be used for a purely geometrical investigation. Go to http://www.burke-consulting.com/GeneralizedParabolas.html to see some way cool examples and then try it out for yourself IMNSHO.