Polar Equations

by

Ralph Hickman

This is an investigation of the following equations:

r = kp/[1 + k(cos(µ))]

r = kp/[1 - k(cos(µ))]

r = kp/[1 + k(sin(µ))]

r = kp/[1 - k(sin(µ))]

r = kp/[1 + k(cos(µ))] is the polar equation of a conic with eccentricity k, focus at the polar axis (pole), and vertical directrix p units to the right of the pole. The major axis of the conic is horizontal.

When 0 < k < 1, the conic is an ellipse. Consider the equation r = (.7)(3)/[1 + .7(cos(µ))]. The eccentricity is .7; the focus is at the pole; the directrix is vertical and is 3 units to the right of the pole.

r = (.7)(3)/[1 - .7(cos(µ))] is the equation of an ellipse with eccentricity .7, focus at the pole, and vertical directrix 3 units to the left of the pole.

When k = 1, the conic is a parabola. r = (1)(2)/[1 + 1(cos(µ))] is the polar equation of a parabola with eccentricity 1, focus at the pole, and vertical directrix 2 units to the right of the pole.

The parabola represented by r = (1)(2)/[1 - 1(cos(µ))] has a vertical directrix 2 units to the left of the pole.

When k > 1, the conic is a hyperbola. r = (1.5)(1.5)/[1 + 1.5(cos(µ))] is the equation of a hyperbola with eccentricity 1.5. Its focus is at the pole, and its directrix is 1.5 units to the right of the polar axis.

r = (1.5)(1.5)/[1 - 1.5(cos(µ))] is the polar equation of a hyperbola whose directrix is 1.5 units to the left of the pole.