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$C$ 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!

Michael Galuza
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Robin Hart
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  • Why is this getting close votes and downvotes? The user wants to see if a quadratic form is an ellipse or a hyperbola and has demonstrated work in this direction. The question has a concrete answer in terms of the determinant of the matrix of the form, which can be computed in terms of the eigenvalues (the work the user has already done). – hunter Aug 13 '15 at 10:24
  • @hunter "...has demonstrated work...". really? and where is this work? OP doesn't understand what is ellipse/hyperbola or what is quadratic form and eigenvalues, and I can't help with it. – Michael Galuza Aug 13 '15 at 10:28
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    @MichaelGaluza: Do you expect the OP to type all his computations leading to the eigenvalues? Come on! He is supposed to show some context, not necessarily his own work (as discussed in this thread on Meta MSE), in order for us to evaluate his knowledge of the subject and know how to answer. Since the OP claims he has already computed the eigenvalues, this clearly shows at least a basic understanding of the issue. He just doesn't know what to do with those eigenvalues. – Alex M. Aug 13 '15 at 10:58
  • @AlexM. It's really bad way (IMHO, of course). I see it so: "Folks, $C$ is . How can I determine <something trivial/obvious/very basic>. I've calculated (actually no, I googled and there are some words from first page), but I don't know how to continue." There is no sense in close votes in such case: write something like "My attempt: need help" and good luck! – Michael Galuza Aug 13 '15 at 11:07

1 Answers1

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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

hunter
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