Degenerate conic ($\det Q=0$)

For $A{x}^2+Bxy+C{y}^2+Dx+Ey+F=0$, the homogeneous quadratic coefficient matrix isThe conic is degenerate when $Q$ is a singular matrix. Examples over the real numbers include the point $x^2+y^2=0$, the repeated line $y^2=0$, and the two straight lines in $xy=0$. The real solution set alone need not reveal the algebraic distinction: $x^2+y^2=-1$ has no real points, but its matrix is nonsingular. This determinant test is different from the conic type discriminant, which only uses the quadratic part in $x,y$.