Quadratic curve classification- case of a circle?

Question

This link gives a table providing classification of conics in terms of $\Delta, I$ and $J$. If $\Delta\ne 0,J>0,\frac{\Delta}{I}<0$, then the curve is a real ellipse.

I want to know about the classification where this ellipse is a circle?

Answer 1

The values $Δ,I,J,K$ don't allow us to characterize a circle. Circles are precisely those ellipses where the value $a = c$. None of the quantities describe this nicely.

Answer 2

A circle is just a special case of an ellipse where the lengths of the semi-major and semi-minor axes are the same. Thus, the classification for circles is the same as that for ellipses. At the most, a circle might limit the ranges of these values (i.e., $\Delta, I$ and $J$) within those regions (i.e., non-zero, positive or negative) compared to what an ellipse may have.

Note the page you linked to even explicitly says this

... (the ellipse and its special case the circle, ...