Mathematical students are usually familiar with the parametric coordinates (at2, 2at) of a point on a parabola, and with the eccentric-angle coordinates of a point on an ellipse, and with similar coordinates for a hyperbola. Such coordinates are probably welcomed by the student just because they reduce by one the number of letters and the number of equations with which he has to deal.

If at2 is substituted for x1, and 2at for y1, but everything else is done in just the same way as before, parametric methods are not really being used. Parametric equations are something more than convenient coordinates.