Matrix representation of conic sections

The matrix representation of conic sections is one of the possible methods how to study conic sections and it's elements, namely axis, vertexes, focuses, tangents, relative position of a given point. We can also study conic sections whose directrix isn't paralel to our coordinate system.

Conic sections have the form of a polynom grade 2:

$Q\equiv Ax^{2}+By^{2}+Cx+Dy+Exy+F=0$

That can be easily transformed into a matrix:

$A_{Q}={\begin{pmatrix}F&C/2&D/2\\C/2&A&E/2\\S/2&D/2&B\end{pmatrix}}$

'Clasification of conic sections using matrixes'

We can distinct regular and degenerated conic sections with the determinant of A_Q.

Iff $|A_{Q}|=0$, the conic is degenerated.

If Q isn't degenerated, we can see what type of conic section it is by computing the subdeterminant resulting from removing the first row and the first column of A_Q.

$A_{00}={\begin{pmatrix}A&E/2\\D/2&B\end{pmatrix}}$

Failed to parse (syntax error): {\displaystyle Iff |A_{00}| < 0, it is hiperbole. \\ Iff |A_{00}| = 0, it is parabole. \\ Iff |A_{00}| > 0, it is elipse. }

In the case we obtained an elipse, we can make futher distinction between elipse and circumference by comparing the last two diagonal elements corresponding to x² and y².

Failed to parse (unknown function "\eq"): {\displaystyle If a_{11} \eq a_{22}, it is circumference. }

(This is still work in progress. I'm going to add the description how to calculate the elements very soon.)