Fastest way to determine if a conic section is an ellipse?

Question

Given an arbitrary conic section in the form $$Ax^2 + Bxy + Cy^2 + Dx + Ey +F=0$$ (Where the coefficients are real valued) is there a simple test which can determine whether or not a particular conic is an ellipse? I know that if a conic section is an ellipse, then $A$ and $C$ will have the same sign, however I am not sure if this is a sufficient condition as well.

Edit: Forgot to include the "$...+F=0$".

Answer 1

Here is the answer for the normalised equation of a non-degenerate conic $$Ax^2+2Bxy+Cy^2+2Dx+2Ey+T=0.$$

Consider the matrix $$M=\begin{bmatrix} A & B & D \\ B & C &E \\ D &E & F \end{bmatrix} $$ and the matrix of the quadratic part of the equation $$Q=\begin{bmatrix} A & B \\ B & C \end{bmatrix}. $$ The conic is non-degenerate if and only if $\det M\ne 0$. Further, the conic is an ellipse if and only if:

1. The quadratic part of the equation (associated to the matrix $Q$) has signature $(2,0)$;
2. The quadratic form on $\mathbf R^3$ associated to $M$ has signature $(2,1)$.

Answer 2

If $ I_2=B^2-4AC$ is negative then it is an ellipse whether or not (RHS being zero ) it passes through origin as special case.