Degenerate ConicsDate: 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/ |
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