The polar equation of a conic section becomes much easier to work with if a focus is placed at the origin. A directrix (either vertical or horizontal) is then placed so that it does not go through the focus.
This interactive figure dynamically illustrates the polar equation of such a conic with focus F and given directrix. The eccentricity, e, is defined to be the ratio of the distance from the focus to any point on this conic to the distance from this point to its directrix. You can move point P anywhere on the graph of this conic.
Note:

If 0 < e < 1, the conic is an ellipse.

If e = 1, the conic is a parabola.

If e > 1, the conic is a hyperbola.

Developed for use with Thomas' Calculus, published by Pearson.