Rule:

1. Get it in the general form

Ax² + Bxy + Cy² + Dx + Ey + F = 0,

2. Calculate the discriminant B²-4AC

3.  if discriminant = 0, the graph is a parabola
    if discriminant < 0, the graph is an ellipse
    if discriminant < 0 and A=C then it is a special ellipse,
                                          that is, a CIRCLE! 

    if discriminant > 0, hyperbola 



Multiply it out and get it in general order:

x² - 6x + 9 - y² = 1

x² - y² - 6x + 8 = 0

The general order for all conic sections is

Ax² + Bxy + Cy² + Dx + Ey + F = 0, so we write ours as:

1x² + 0xy - 1y² - 6x + 0y + 8 = 0

A=1, B=0, C=-1, D=-6, E=0, F=8

We calculate discriminant = B²-4AC
                            0²-4(1)(-1)
                            0+4
                            4

That's > 0 so the graph is a hyperbola.

Edwin