Degenerate Conics

Date: 03/04/98 at 16:35:26
From: Anonymous
Subject: Conic Sections

I am aware that there are a number of degenerate cases for the graphs 
of equations in conic form. As I have been told, one of these cases is 
two parallel lines. I have sliced a cone with a plane mentally in 
every way that I can think of, and cannot figure out how parallel 
lines may be achieved. I recently discovered that it had something to 
do with lines that become parallel at infinity, but could still not 
understand how the case could be generated. I was hoping that I could 
find some help here.

Date: 03/05/98 at 17:47:12
From: Doctor Sam
Subject: Re: Conic Sections

Hi,

As far as I know, there are only three degenerate conics: a point, a 
line, and a pair of intersecting lines.

Geometrically, you can get the conic sections by slicing a pair of 
cones that touch at their vertices. By slicing appropriately, you can 
get a circle, an ellipse, a parabola, or a hyperbola. The degenerate 
cases arise if you slice 

    (a) through the vertex parallel to the bases...this gives a point;  
    (b) through the vertex tangent to the cones...this gives a line;
    (c) through the vertex but intersecting both cones...this gives 
        two intersecting lines.

There is no way to get two parallel lines.

You can see this algebraically as well. The general conic has an 
equation of the form

   Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0.

If A = B = C = 0, then the cone degenerates to a line.

If all the coefficients are zero, this degenerates to a point.

If all but B are zero, the equation degenerates into Bxy = 0, which 
has two linear solutions -- x = 0 and y = 0 -- two intersecting lines.

And that's it. You cannot get two parallel lines out of that quadratic 
equation, I believe.

I hope that helps.

-Doctor Sam, The Math Forum
Check out our web site! http://mathforum.org/dr.math/   
    

Date: 07/10/98 at 17:47:12
From: Doctor Peterson
Subject: Re: Conic Sections

You should also read a conversation from our archives about this topic.  
It not only gives an example of parallel lines but explains (if you read
to the end) why it can be justly called a degenerate conic, aside from 
the fact that the equation fits the general form:

  Conic Sections and Parallel Lines
    http://mathforum.org/library/drmath/view/54756.html

-Doctor Peterson, The Math Forum
Check out our web site! http://mathforum.org/dr.math/