Finding Angles for Conic Section

[splac6996]

I have a hyperbola with the following equation and I am trying to find my angles to eventually rotate this (conic section), but I don't know what to do because when I follow the standard form of (A-C)/B, I get zero. Is there some trick that I don't know about.

x^2+4xy+y^2=12

 

[photis]

First, you can write the equation in this way
$$\left[ \begin{array}{cc} x & y \end{array} \right] \cdot \left[ \begin{array}{cc} 1 & 2 \\ 2 & 1 \end{array} \right] \cdot \left[ \begin{array}{c} x \\ y \end{array} \right] = 12$$

Now, you have to diagonalize the symmetric matrix
$$A = \left[ \begin{array}{cc} 1 & 2 \\ 2 & 1 \end{array} \right]$$

Then, see this!

 

[splac6996]

I have never seen this technique before is that the only way to do this?