Classification of figure from the general equation of conics

[sadhu]

during my study on conics , I found a formula in the book regarding the classification of figure from the general equation of conics

ax²+2hxy+by²+2gx+2fy+c=0

it was given that
$\Delta=abc+2fgh-af^{2}-{bg}^{2}-{ch}^{2}$

$if \Delta \neq 0$
then if
$h^{2}=ab...parabola$
$h^{2}<ab...ellipse$
$h^{2}>ab...hyperbola$

if
$\Delta <0...circle ,h=0,a=b\neq 0,g^{2}+f^{2}-ac>0$

if
$\Delta = 0$
if
$$h^{2}>=ab...line$$
$$h^{2}<ab...unique point$$



No explanation regarding the derivation of result was given
neither i could find it on net

hope someone knows it...

thanks in advance

 

[QuantumQuest]

A full analysis and explanation would require a long document so I'll give an online resource at projecteuclid.org which investigates by elementary algebra the values of $y$ as $x$ increases from $-\infty$ to $+\infty$ and the values of $x$ as $y$ increases from $-\infty$ to $+\infty$ and classifies the loci by their principal graphical properties so found.