The general second-degree equation S: ax^2+2hxy+by^2+2gx+2fy+c=0 represents different conics depending on three invariants that remain unchanged under rotation and translation.

The discriminant Delta=abc+2fgh-af^2-bg^2-ch^2 determines degeneracy. If Delta=0, the equation represents a degenerate conic: a pair of lines (real or imaginary), a single point, or coincident lines. If Delta!=0, the conic is non-degenerate.

The quadratic discriminant h^2-ab determines the type: h^2-ab<0 gives an ellipse (or circle when a=b and h=0); h^2-ab=0 gives a parabola; h^2-ab>0 gives a hyperbola. A rectangular hyperbola has a+b=0.

Summary table for non-degenerate conics (Delta!=0):

• Circle: a=b, h=0
• Ellipse: h^2-ab<0 (and a!=b or h!=0)
• Parabola: h^2-ab=0
• Hyperbola: h^2-ab>0
• Rectangular hyperbola: h^2-ab>0 and a+b=0

For degenerate conics (Delta=0):

• Pair of real distinct lines: h^2-ab>0
• Pair of parallel lines: h^2-ab=0
• Pair of imaginary lines: h^2-ab<0 (or a point)

The invariant a+b equals the sum of the eigenvalues of the quadratic form matrix. Under rotation by any angle, a+b remains constant. Combined with h^2-ab (also invariant), we can determine the new coefficients after rotation without performing the substitution.

JEE strategy: for identification problems, compute h^2-ab first (determines conic type), then compute Delta (checks degeneracy). No need to perform the actual rotation or translation.