The general equation ax^{2+2hxy+by}^{2+2gx+2fy+c}=0 represents:

• Pair of lines if Delta = 0 (where Delta = abc+2fgh-af^{2-bg}^{2-ch}^2)
• Ellipse if Delta != 0 and $h^{2-ab}$ < 0
• Parabola if Delta != 0 and $h^{2-ab}$ = 0
• Hyperbola if Delta != 0 and $h^{2-ab}$ > 0
• Circle if a=b and h=0

Invariants under rotation: a+b, $h^{2-ab}$, Delta. These don't change when axes rotate.

To reduce the general equation to standard form:

1. Rotate to eliminate xy-term (using tan(2*theta)=2$\frac{h}{a-b}$).
2. Translate to move center/vertex to origin.