10.2. Polar Equations of Conics http://www.ck12.org
There are other ways of representing a circle like this using cofunction identities and coterminal angles.
Ellipses, parabolas and hyperbolas have a common general polar equation. Just like with the circle, there are other
ways of representing these relations using cofunction and coterminal angles; however, this general form is easiest to
use because each parameter can be immediately interpreted in a graph.
r= 1 −e·cosk·e(θ−β)
One of the focus points of a conic written in this way is always at the pole (the origin). The angleβindicates the
angle towards the center if the conic is an ellipse, the opening direction if the conic is a parabola and the angle away
from the center if the conic is a hyperbola. The eccentricityeshould tell you what conic it is. The constantkis the
distance from the focus at the pole to the nearest directrix. This directrix lies in the opposite direction indicated by
β.
There are many opportunities for questions involving partial information with polar conics. A few relationships that
are often useful for solving these questions are:
- e=ca=PFPD→PF=e·PD
- Ellipses:k=ac^2 −c
- Hyperbolas:k=c−ac^2
Example A
A great way to discover new types of graphs in polar coordinates is to experiment on your own with your calcula-
tor. Try to come up with equations and graphs that look similar to the following two polar functions.