Explorations with Polar Equations

by: Lauren Wright

In this investigation, we look at the following equations with different values of p.

, ,, and .

for k > 1, k = 1, and k < 1.

Note: The parameter k is called the "eccentricity" of these conics. It is usually called "e" but for many software programs e is a constant and can not be set as a variable.

For notes on a derivation of these formulas, see Dr. Jim Wilson's page by clicking here.

First, let's take a look at for k < 1 and p = -4.

 

For p = 2, we get the following:

To investigate these graphs on your own, click here to download a Graphing Calculator file.

*I chose p = -4 and p = 2 arbitrarily for illustration purposes. You can download the files to vary p on your own.

Now let's do the same thing for .

for k < 1 and p = -4

for k < 1 and p = 2

To investigate this on your own, click here.

For these illustrations, we see that if , an ellipse is formed. If k = -1, a parabola is formed. And, if k < -1, a hyperbola is formed. All of these conic sections have focal points lying on the x-axis.

**NOTE: The straight lines on the gray graphs are asymptotes - they are not part of the actual graph.

Next, let's take a look at for k < 1.

 

for k < 1 and p = -4

 

for k < 1 and p = 2

To investigate this on your own, click here.

Repeating the process for gives us the following...

 

for k < 1 and p = -4

 

for k < 1 and p = 2

To investigate this on your own, click here.

We see that the same thing occurs with and as did before. The only difference is that the focal points lie on the y-axis.

NOW, let's look at what happens when k > 1.

 

For and p = -4

 

For and p = 2

 

For and p = -4

 

For and p = 2

 

We can conclude that when k > 2, hyperbolas are formed. When there is a cosine in the denominator, the focal points lie on the x-axis. When there is a sine in the denominator, the focal points lie on the y-axis.

For our last and final case, we will explore what happens when k = 1.

 

 

 

 

 

Finally, when k = 1, we get only parabolas for varying values of p. When there is a cosine in the denominator, the focal point lies on the x-axis. When there is a sine in the denominator, the focal point lies on the y-axis.

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