It's worth pulling out into a separate post a wonderful unifying result for 3 of the 4 conic sections that follows from the parameterization of Pell's Equation:
Given x2 - Dy2 = 1, in rationals:
y = 2t/(D - t2)
x = (D + t2)/(D - t2)
showing it more traditionally versus the way that results from my own derivation.
You get hyperbolas with D>0, the circle with D=-1, and ellipses with D<0.
You can see the D=-1 case from a well-known mainstream source at the following link:
See: http://mathworld.wolfram.com/Circle.html eqns. 16 & 17
I actually feel very honored to have my own derivation leading to such a beautiful result, which allows categorization of ellipses and hyperbolas by a single number, which then is probably connected to the eccentricity.
So what mathematicians call "Pell's Equation" is in other areas a one-stop equation for generating 3 of the 4 conic sections in rationals just by fiddling with one variable--D.