For a general conic $Q(x,y)=ax^2+2hxy+by^2+2gx+2fy+c$ we define a matrix $A$ as follows:
$A=\left( \begin{matrix} a& h& g\\ h& b& f\\ g& f& c\end{matrix} \right)$.
Then we define $\tau=a+b$, $\delta=\left| \begin{matrix} a& h\\ h& b\end{matrix} \right| $ ve $\Delta=\left| \begin{matrix} a& h& g\\ h& b& f\\ g& f& c\end{matrix} \right|$ With these arguments, we say that $Q(x,y)$ gives us
an ellipse if $\Delta \neq 0$ and $\delta >0$ ,
a parabola if $\Delta \neq 0$ and $\delta =0$ ,
a hyperbola if $\Delta \neq 0$ and $\delta <0$ .
Now, my questions: Where does this $A$ matrix and all $\tau, \delta, \Delta$ definitions come from?Why do we need $\Delta \neq 0$?
$Q(x,y)=ax^2+2hxy+by^2+2gx+2fy+c$ must be equal (with some corrections maybe) to $$(x,y,1)^TA(x,y,1)$$ For all $(x,y)$ vectors in the plane. If $\delta$ and $\Delta$ are simultaneously positive, this exactly means that $A$ is a positive definite matrix. (Notice that $A$ is symmetrical.) Then, it is obvious why the set of $(x,y)$ satisfying $$(x,y,1)^TA(x,y,1)=C, \ C >0 \ constant $$ is an ellipsis.
Other cases?... Play with them in similar ways.