Critique of polar equation

Great as it is to have polar coordinate equations for the ellipse, it's not clear what they mean in terms of the cylinder or cone. Also, what happens to the equation if the figure is translated or rotated? For example, what if we'd placed the origin at the right focus instead of the left?

The fractional form of the equation's right side might make us wonder about the figure whose equation is the reciprocal of this one.