Question

Why exactly is \(31x^2 +12xy+y^2 −50x−125=0\) the equation of a degenerate conic section? I have tested it using the determinant of the matrix, and it does not equal 0, so I don't understand why it is degenerate. Could someone give me an explanation?

Answer 1

http://wolframescherbach.wordpress.com/2013/08/25/degenerate-conic-sections/

Answer 2

it looks like one to me

Answer 3

It seems like a sound way of figuring it out, but maybe if I gave you the solution that the textbook gave me, it would help understand why?

Answer 4

This is what my book says:
If one takes the discriminant in terms of the variable x to solve for y, we find that the expres- sion under the radical is never positive (it is of the form √−expression, where expression is a parabola that lies entirely under the x axis – e.g. always negative), implying that it is not the graph of a conic at all but a degenerate pair of lines (the asymptotes of what the classic b^2 − 4ac technique would have said is a hyperbola).

Answer 5

I tried graphing it and this is what it looked like http://www.wolframalpha.com/input/?i=31x2%2B12xy%2By2 −50x−125%3D0+

and then I went on to read some more about degenerate and found this: http://users.ipfw.edu/CoffmanA/pov/lsoc.html

Answer 6

none of the parameters for either non-degenerate and degenerate is satisfied, and that is why WolframAlpha defined it as pair of intersecting lines, which is true if we investigated further values under proper mathematical definitions

Answer 7

I have to say, though I can't quite say that I found an exact answer, the resources that you've linked here are very helpful, and I can probably figure it out from them. I couldn't find anything on this topic other than the matrix representation, so this actually is very helpful. Thank you. For context, I am going ahead of the pace, so I'm venturing through this material by myself.

Answer 8

ya the matrix helps, but it doesn't tell us quite easily

Answer 9

@Mimi_x3 she's a math goddess, she might know