While number theorists consider Pell's Equation only with integers, it can be considered with rationals, parameterized and related then to all the conic sections except the parabola:
Given x^2 - Dy^2 = 1, in rationals:
y = 2t/(D - t^2)
and
x = (D + t^2)/(D - t^2)
and you get hyperbolas with D>0, the circle with D=-1, and ellipses in general 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