Write-up # 11

Exploring Polar Coordinates

by Mark Cowart and Beverly Hales

To form a polar coordinate system in a plane, pick a fixed point O called the origin or pole. Using the origin as an endpoint construct a ray called the polar axis. After selecting a unit of measurement, we may associate with any point P in the plane a pair of polar coordinates () where r is the distance from P to the origin, and measures the angle from the polar axis to the line segment OP (see figure). The number r is called the radial distance of P and is called a polar angle of P.

The distance from a point on a conic to a vertical line p can be expressed as . If a conic graph is a distance r from the focus and a distance kr from a directrix, this yields . Furthermore, solving for r gives the following polar form of a conic equation:

Explore the basic polar form of conics by graphing several equations using the graphing software tools Algebra Xpressor or Graphing Calculator 2.0. Also investigate with different values of p for k>1, k=1, and k<1. For example, using the above polar conic form with p=1 and k=1 gives the following graph:

With k=1, the graph is a parabola. What would a different value for p, such as p= 5, do to the graph of the parabola?

Notice the parabola seems wider. What about when p=1/4? Can you predict what the graph might look like now?

The graph has indeed become "skinnier". What do you think will happen if p is a negative number?

Obviously a reflection across the y-axis has occurred. Next, explore the graphs when k<1 and k>1. Describe your findings, and be sure to tell what happens as the values of p are varied.

1) a. k<1 and p=1 is the graph of __________________________.

b. Describe changes in the graph when p=5, p=1/4, and when p is negative.

p=5 _________________________________________

p=1/4 ________________________________________

p=-1 __________________________________________

2) a. k>1 and p=1 is the graph of _________________________.

b. Describe changes in the graph when p=5, p=1/4, and when p is negative.

p=5 _________________________________________

p=1/4 ________________________________________

p=-1 __________________________________________

Another interesting investigation could result from substituting sine in for cosine in the equations. How does this affect the graph of the conics?

The sine value creates a parabola that opens vertically up!!! Once again the exploration of p values and k values might yield patterns and observatons that are interesting. How does this graph for p=1 and k=1 compare to the graph of the more familiar equation Can you draw any conclusions or make any connections relating the polar form of conics with the two varible x and y form?