Given an equation, determine what conic section the equation represents

Question

I have the general equation $$Ax^2+Bxy+Cy^2+Dx+Ey+F=0$$ and I want to determine what conic section it represents according to the value of its coefficients or relations between them, is there a theorem where that is explained?

Thank you so much.

Answer 1

This equation can be written in matrix notation, in terms of a symmetric matrix to simplify some subsequent formulae, as \begin{equation}Q=(x,y)\begin{pmatrix}A & B/2\\ B/2&C\end{pmatrix}(x,y)^{T}+(D\quad E)(x,y)^{T}+F=0. \end{equation} the first matrix X is called the matrix of the quadratic form.
Q is a hyperbola if and only if det A<0,
Q is a parabola if and only if det A=0, and
Q is an ellipse if and only if det A>0.