I submit the above two animated gifs (made in GeoGebra) for the sake of those who think that ellipses and parabolas are too complicated to understand.
The 1st gif, yielding an ellipse, only requires that the red segment and the blue segment always have the same combined length.
The 2nd gif, the parabola, only requires that the two dotted lines are always equal in length.
That's it. The rest comes from there.