Degenerate Conic Sections

[matiasmorant]

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!

 

[AlephZero]

Start by writing the equation in matrix form as

$$\begin{bmatrix}x & y & 1 \end{bmatrix} \begin{bmatrix}a & h & g \cr h & b & f \cr g & f & c \end{bmatrix} \begin{bmatrix}x \cr y \cr 1\end{bmatrix} = 0$$

(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"

 

[matiasmorant]

I don't get it yet... some further hint?

 

[Zombiedude347]

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

 

[CellCoree]

I would like to clarify that the set of points described by a quadratic equation is not limited to only conic sections. In fact, a quadratic equation can describe a wide range of curves and shapes, including parabolas, ellipses, hyperbolas, circles, and more. However, when certain conditions are met, the resulting curve may be considered a degenerate conic section.

To answer your question, there is no easy or foolproof method to determine which set a quadratic equation will describe. As you mentioned, the rule b^2-4ac can give you some information, but it is not always reliable. To accurately determine the type of curve, you may need to use methods such as completing the square or graphing the equation. It may also be helpful to familiarize yourself with the general shapes and characteristics of each type of conic section. With practice and experience, you may be able to identify the type of curve more quickly and accurately.