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)=2h/(a-b)).
2. Translate to move center/vertex to origin.