Degenerated Conic

pankaj · #1 $Old$ October 22nd, 2009, 11:11 AM

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.

Thanks in advance.

Opalg · #2 $Old$ October 22nd, 2009, 01:33 PM

Quote:

Originally Posted by pankaj $View Post$

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 $x^2+y^2=-1$ .

pankaj · #3 $Old$ October 22nd, 2009, 06:51 PM

Quote:

Originally Posted by Opalg $View Post$

$h^2-ab>0$ : 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