Standard Polar Equations for Conic Sections

We have shown that each conic section (except a circle) has the following property, which we will refer to as the "Focus-Directrix" property:

For each conic section which is not a circle, there is a line L, and a point F not on L, and a real number e, for which each point P on the conic sections satisfies the relation PF/P L = e.

We call F a focus, L a directrix, and e the eccentricity of the conic section.

If we put the focus F at the origin, and directrix L at x = - d, then we can derive a polar equation for the conic section.

The graph of the conic section is only a single point if e = 0, so it is convenient to replace de with a non-zero constant A in the numerator. Now every conic section, including circles, can be represented indard form by the polar equation

Where

e is the eccentricity, and
whenever e is nonzero, d = A/e is the distance to the directrix.

Questions:

Graph conic sections for many different values of e. Describe the kinds of curves you get with different values of e.
What picture and formula do you get for conic sections if you choose the line y = -d for the directrix instead of the line x = -d? What if you choose x = d? How about y = d?
The point on a conic section corresponding to t = π is called a vertex, V. Find a formula for the distances VF and VL in terms of d and e.
When e is not 1, there is a second vertex W, corresponding to t = 0. Find a formula for the distances WF and VW in terms of d and e.
Find the distance from the focus F to the point on a conic section corresponding to t = π/2 in terms of d and e.

Last Update: September 19, 2002
Ronald K. Smith
Graceland University
Lamoni, IA 50140
rsmith@graceland.edu