Classifying conic sections from quadratic coefficients

[Bruno Tolentino]

Is possible classify the quadric equation Axx + Bxy + Cyx + Dyy + Ex + Fy + G = 0 how straight, hyperbola, circle, ellipse, parabola, etc, in the same way that is did in the phase plan:

https://upload.wikimedia.org/wikipedia/commons/3/35/Phase_plane_nodes.svg

https://upload.wikimedia.org/wikipedia/commons/3/35/Phase_plane_nodes.svg

I tried this, but, for some reason, don't works... Can you help me!?

 

[Dr. Courtney]

https://en.wikipedia.org/wiki/Quadric

http://mathworld.wolfram.com/QuadraticSurface.html

 

[Bruno Tolentino]

I want to express the quadric equation in this way: $$\begin{bmatrix}<br /> A & B \\<br /> C & D<br /> \end{bmatrix}<br /> :<br /> \begin{bmatrix}<br /> x^2 & xy \\<br /> yx & y^2<br /> \end{bmatrix}<br /> +<br /> \begin{bmatrix}<br /> E\\<br /> F<br /> \end{bmatrix}<br /> \cdot<br /> \begin{bmatrix}<br /> x\\<br /> y<br /> \end{bmatrix}<br /> +<br /> G=0$$

And I want to analyze the square matrix, the first matrix, the matrix of the coefficients.