# Thread: Degenerated Conic

1. ## Degenerated Conic

If the equation of the curve is

$
ax^2+2hxy+by^2+2gx+2fy+c=0
$

and this curve is such a conic that the focus lies on the directrix.

Can anybody describe the possible situations i.e what happens if $h^2-ab$ is $>,<$ or $=$ to 0(zero)

Or

Can anybody provide a link where such a situation has been described.

2. Originally Posted by pankaj
If the equation of the curve is

$
ax^2+2hxy+by^2+2gx+2fy+c=0
$

and this curve is such a conic that the focus lies on the directrix.

Can anybody describe the possible situations i.e what happens if $h^2-ab$ is $>,\ <$ or $=$ to 0(zero)
$h^2-ab>0$: degenerate hyperbola = two straight lines, e.g. $x^2-y^2=0$.
$h^2-ab=0$: degenerate parabola = single straight line, e.g. $x^2=0$.
$h^2-ab<0$: degenerate ellipse = single point or nothing at all, e.g. $x^2+y^2=0$ or $x^2+y^2=-1$.

3. Originally Posted by Opalg
$h^2-ab>0$: degenerate hyperbola = two straight lines, e.g. $x^2-y^2=0$.
$h^2-ab=0$: degenerate parabola = single straight line, e.g. $x^2=0$.
$h^2-ab<0$: degenerate ellipse = single point or nothing at all, e.g. $x^2+y^2=0$ or $x^2+y^2=-1$.
Is their proof available