Given x

^{2}- Dy

^{2}= 1, in rationals:

y = 2t/(D - t

^{2})

and

x = (D + t

^{2})/(D - t

^{2})

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.

James Harris