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.