Using polar
coordinates, there is an
alternate way to define the conics. Rectangular
coordinates place the
most importance on the location of the center of the conic, but polar
coordinates place more importance on where the focus of a conic is. In
certain situations, this makes more sense (the reflective property of a parabola
depends more on the location of the focus than the center).
Now we will define a conic this way: a conic is a set of points such that the
distance between a point on the conic and a fixed point is related to the
distance from that point to a fixed line by a constant ratio. The fixed point
is the focus, and the fixed line is the directrix. This constant ratio is
the eccentricity e of the conic. e tells us which kind of conic it is.
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.
In a polar equation for a conic, the
pole is the focus of
the conic, and the polar
axis lies along the
positive xaxis, as is conventional. Let p be the distance between the
focus (pole) and the directrix of a given conic. Then the polar equation for a
conic takes one of the following two forms:
r = 

r = 

When r = , the directrix is horizontal and p
units above the pole; the axis, major axis, or transverse axis of the conic
(depending on which type it is) is vertical, on the line θ = .
When r = , the directrix is horizontal and p
units below the pole; the "main" axis (term varies depending on which type of
conic it is) is vertical, on the line θ = .
When r = , the directrix is vertical and p units
to the right of the pole; the axis is horizontal, on the line θ = 0.
When r = , the directrix is vertical and p units
to the left of the pole; the axis is horizontal, on the line θ = 0.
This information is enough to analyze any conic in polar form. First, find e
and decide which type of conic it is. Then, based on the form of the conic,
decide where the directrix is and find p. Finally, plugging in different
values for θ based on whether the main axis of the conic is vertical or
horizontal, you can find the vertices of the conic, and find values for a,
b, and c.