This is an investigation of the following equations:
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.
Click here to view these ellipses, using a Geometer's Sketchpad animation.
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.
Click here to view these parabolas.
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.
Click here to view these hyperbolas.
When the sine function is substituted for the cosine function in the polar equation for a conic, the graph of the conic has a horizontal directrix above (r = kp/[1 + k(sin(µ))]) or below (r = kp/[1 - k(sin(µ))]) the polar axis. Thus, the major axis of the conic is vertical. The eccentricity is k. The focus is the polar axis.
Click on the following to view graphs of these polar equations:
ellipses: r = (.7)(3)/[1 + .7(sin(µ))] and r = (.7)(3)/[1 - .7(sin(µ))]
parabolas: r = (1)(2)/[1 + 1(sin(µ))] and r = (1)(2)/[1 - 1(sin(µ))]
hyperbolas: r = (1.5)(1.5)/[1 + 1.5(sin(µ))] and r = (1.5)(1.5)/[1 - 1.5(sin(µ))]