A parabola is the locus of all points equidistant from a given point, the focus, and a line, the directrix.
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.
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