Ellipse or hyperbola?

Question

$C$ is the equation $$-2x^2+6xy+6y^2 = 1.$$

How can you see whether it is an ellipse or a hyperbola?

I've calculated the eigenvalues and eigenvectors but I don't know how to continue.

Thanks!

Answer 1

If the eigenvalues have the same sign, you'll get an ellipse. If they're opposite sign, you'll get a hyperbola. You may find this article helpful.

https://en.wikipedia.org/wiki/Matrix_representation_of_conic_sections