When learning conic section I learnt about $\Delta$.
For any conic in general form : $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$
Here $\Delta=abc +2fgh - af^2 - bg^2 -ch^2$
The conic is said to be pair of straight line if $\Delta =0$
a circle if $\Delta \neq 0$ , $a = b$ , $h =0$
a parabola if $\Delta \neq 0$ , $h^2 = ab$
an ellipse if $\Delta \neq 0$ , $h^2 < ab$
an hyberbola if $\Delta \neq 0$ , $h^2 >ab$
a rectangular hyperbola if it follows the above condition and $a + b = 0$
Here how was $\Delta$ derived from the above conic equation and how is it used to find out which type of conic is this? (In other words I want proof of derivation of $\Delta$ and its applications too. )
Also when I read the word $\Delta$ first thing that comes to my mind is determinants. Is $\Delta$ related to the determinant of some sort of above general conic equation?