Determining degernerate conics

Question

How can I tell the type of this conic, and determine if it is degenerate or not?

x² + 4xy + 2x + 2y² - 1 = 0

Answer 1

First you gotta combine "like terms" and look at how the terms cooperate with the squares of the variables like (x^2) or (x+2)^2 for example

Remember complete the square method.. Well that is how you put this equation in the write form which makes a certain shape!

There is a squared term for both the x and the y

Doing the perfect square method I put everything on the left side and I get these 2 squares

However both squares are definitely not perfect squares and the only way you can make a proper shape on a graph either circle, ellipse, parabola, etc. you need perfect squares to exist in the given equation!!

Therefore this is definitely degenerative because it cannot generate a proper conic "shape"

Answer 2

Dear John D.

Obviously this equation is not a degenerate case.

In any Trigonometry book you can find a theorem in the chapter of Rotation of Axes.

The Theorem postulates that

THEOREM: Except for degenerate cases, the equation

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

1. Defines a parabola if B^2 -4AC =0
2. Defines an ellipse ( or a circle ) if B^2 -4AC < 0
3. Defines a hyperbola If B^2 -4AC >0

In your case B^2-4AC =16-8 >0.

Hence the conic section represents a Hyperbola

Adam B.