Degenerate Conic Sections

  • #1
the set of points described by the quadratic equation

a y^2 + b xy + c x^2 + d y + e x + f = 0

can be 1) a parabola, an ellipse, an hyperbola or 2) an empty set, a line, two intersecting lines, two parallel lines, a circle, a point, and pherhaps something else...

I want two know which of these will I get.

I know the rule b^2-4ac. but the degenerate cases deceive me too often. Is there a method to decide which set the quadratic equation describes? Of course you can try completing squares in several ways, but that takes lots of trials and thought, doesn't it? is there a better way?

thanks!
 

Answers and Replies

  • #2
AlephZero
Science Advisor
Homework Helper
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Start by writing the equation in matrix form as

[tex] \begin{bmatrix}x & y & 1 \end{bmatrix}
\begin{bmatrix}a & h & g \cr h & b & f \cr g & f & c \end{bmatrix}
\begin{bmatrix}x \cr y \cr 1\end{bmatrix}
= 0[/tex]

(When you multipliy it out, the coefficients are in a different order from your notation, but this is the "standard" form).

Then consider the properties of the matrix of coefficients.

(It's more fun to work out the details for yourself than just be told the answer!)

BTW, when I was a kid we were taught to remember the matrix entries by "all hairy gorillas have big feet, good for climbing" :smile:
 
  • #3
I don't get it yet... some further hint?
 
  • #4
I learned that you put the coefficients in a matrix of
[a b/2 d/2]
[b/2 c e/2]
[d/2 e/2 f]

if the determinant of the matrix is 0, it is degenerate

if b^2>4ac it is a ellipse or a point
if b^2=4ac it is a parabola or a line
if b^2<4ac it is a hyperbola or two lines
if b=0 and a=c, then it is a circle or a point (special case of ellipse)

the determinant is calculated from a general matrix

[a b c]
[d e f]
[g h i]

using the formula aei + bfg + cdh - ceg - bdi - afh
 
Last edited:

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