Ok. Here is what i do know:

in an equation: Ax² + Bxy + Cy² + Dx + Ey + F = 0

i need to eliminate the Bxy part, and find an angle of rotation.

cot(2Ɵ) =(a-c)/b

from there i can find Ɵ and create a new x',y' equation without a Bxy part.

Here is my question:

when there are 2 possible values for Ɵ, which do i use?

this has come up on my homework a lot.

Here's an example:

xy+1 = 0

cot(2Ɵ) = 0

2Ɵ could either be (pi/2) or (3pi/2)

Ɵ could either be (pi/4) or (3pi/4)

another example:

2x² - 3xy - 2y² + 10 = 0

cot(2Ɵ) = (-4/3)

by constructing a right triangle i can figure that cos(2Ɵ) can either be (-4/5) or (4/5)

from what i can tell, sometimes either value of Ɵ will result in me making the same graph, but the x'y' equation still differs. Are both answers always equally correct? If so, then which way is the general way expected on a math test, because some answers wont ask for a graph, just the x'y' equation.

what do i do and why?

My class is a very high level honors class, and i am not going to be able to get a chance to ask the teacher before the test on monday. Thanks so much in advance.