You have the correct eigenvalues, and you could continue to find the eigenvectors that way. But there is another method which seems to work better in this case.

When you rotate the axes through an angle , x becomes and y becomes . Make those substitutions in the equation , to get . If you choose so that the xy-term is zero then you will have found the directions of the principal axes. The coefficient of xy is This will be zero when So , or . That gives you the orientation of the conic, and you should then be able to complete the question.

The numbers that come in the eigenvalue/eigenvector equations are related to the trig functions of , which is what makes them messy.