# Classification of figure from the general equation of conics

1. Jan 18, 2008

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........