Polar Form of the Conics
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 x -axis, 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:
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 .