Quadratic curve classification- case of a circle? 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?

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, ...

