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.