I know the categories of non-degenerate conics in the Euclidean plane are circle, ellipse, parabola, hyperbola. The general equation of a conic is $ax^2 + bxy + cy^2 + dx + ey + f$. None of the standard forms of the above conics contain an $xy$ term. My question is, in what kind of conics does the $xy$ appear?

  • $\begingroup$ The conic $xy=1$ is a hyperbola. In fact, it is the hyperbola $x^2-y^2=1$ rotated 45 degrees clockwise. $\endgroup$ – Baby Dragon Sep 15 '13 at 2:42

You might be interested in the Wikipedia article "Rotation of axes."

To summarize: the $xy$ term appears in conics whose axes do not lie along the $x$- and $y$-axes. A conic with non-zero $bxy$ term is rotated by $\arctan(\frac{b}{a-c})$. To "unrotate" a conic, you need to substitute new expressions for $x$ and $y$ -- you can't just remove the $bxy$ term and expect to get the same conic, rotated a bit.

The quantity $b^2 - 4ac$ does not change under rotation, so you can use it to classify conics as ellipse, parabola, and hyperbola.


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