A conic section is the solution to the general quadratic equation in two variables,
Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 (1)
and can be defined in terms of three quantities, (1) fixed point (focus), (2) fixed line (directrix), and (3) eccentricity. The polar version of the above equation is
r = f (theta) = pe/ 1-e cos (theta) (2)
where r is the radius from the pole to the conic, p is the horizontal distance leftward from the pole to the directrix, e is the eccentricity, and the focus is assumed to be at the pole.
a. For p = 1, and a focus at the pole, transform the polar equation (2) to the Cartesian equation (1) using the transformation, and determine the constants A-F in (1) in terms of eccentricity (e).
b. Graph the solutions for e = 0.0, 0.5, 1.0, and 2.0