The graph of a quadratic equation in two variables is always a conic section.
In the [Cartesian Coordinate System]?, a conic is a curve that has an equation of the 2nd degree, i.e., of the form:
ax2+2hxy+by2+2gx+2gy+c=0.
If h2=ab, this equation represents a parabola.
If h2<ab, this equation represents an ellipse.
If h2>ab, this equation represents a hyperbola.
If a=b and h=0, it represents a circle.
If a+b=0, it represents a [rectangular hyperbola]?.
Finally, if the following determinant,
| a h g | | h b f | = 0 | g h e |
it represents a pair of straight lines, that may not coincide.
Again this page really needs a visual and should be written in a way accessible to all readers. This is not complex material. And a revision should be fairly easy...before this whole topic becomes esoteric.
I recommend the following link for graphics on [[|conics]].
Very good visuals, thank you. RoseParks