Polar Equations of Conics

A conic section can be defined in polar coordinates as the set of all points whose distance from a fixed point, known as the focus, bears a constant ratio to their distance from a fixed line, known as the directrix. This constant ratio is called the eccentricity. This definition unifies all types of conic sections—ellipses, parabolas, and hyperbolas—under a single framework. When the focus is positioned at the origin of the polar coordinate system, a single polar equation can describe any conic section, simplifying both their representation and analysis.

The general polar equation of a conic is:

Here, the variable r is the distance from the origin to a point on the conic, 𝜃 is the polar angle, d is the distance from the focus to the directrix, and e is the eccentricity. The choice between cosine and sine depends on the orientation of the directrix, while the sign indicates its direction relative to the focus.

The specific nature of the conic section depends entirely on the value of the eccentricity. When the eccentricity is equal to one, the conic is called a parabola. When the eccentricity is greater than zero but less than one, the conic is an ellipse. When the eccentricity is greater than one, the conic is a hyperbola.

If the directrix is vertical, the equation involves the cosine function. A directrix to the right of the focus results in a plus sign with cosine, while a directrix to the left uses a minus sign. When the directrix is horizontal, the equation uses the sine function instead. A directrix above the focus leads to a plus sign with sine, and one below corresponds to a minus sign. These variations help specify the orientation and placement of the conic in the plane.