Suppose we have a quadratic polynomial equation, $f(x,y)=0$. How do we determine if the conic, over $\mathbb{R}$, is degenerate? Over $\mathbb{C}$ the answer is much simpler. We can replace $f$ by $F(x,y,z) = z^2 f(x/z,y/z)$, then $F$ is a homogenous quadratic polynomial, therefore, $F(x,y,z) = \tfrac{1}{2}\mathbf{v}^t H \mathbf{v}$, where $\mathbf{v} = (x,y,z)$ and $H$ is the Hessian of $F(x,y,z)$. The conic section described by $F$ will therefore by degenerate if and only if $\det(H) = 0$.
The problem is this criterion does not work over $\mathbb{R}$, since a "degenerate" conic also includes the empty set, e.g. $x^2+y^2 + 1 = 0$, which is not degenerate over $\mathbb{C}$.