Was fascinated to realize that an ancient expression gives quite a few conic sections as it in fact gives 2 of 3:
x2 - Dy2 = 1
Where depending on the sign of D, you can get an ellipse, which includes circle for D=-1, or a hyperbola for positive D. So I like to call it the two conics equation.
But have noted it as giving 3 conics as some consider the circle to be one in its own right, but circles are ellipses. And am sure is something I considered years ago before focusing as I do.
And that's with fields of course. The equation got lumped into the category of Diophantine where the hyperbolas got all the attention, though only looking at integer solutions, which is a historical oddity I think which traces back to Fermat as to how that emphasis stuck. The equation has been known for thousands of years though.
For a bit got excited as wondered if I should really go with three conics but I thought over so many things carefully years ago. And have learned the hard way to trust myself. There are times it takes me days to remember why things are a certain way. Luckily here was a matter of minutes.