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I have only worked with ellipses aligned with the x or y axis. However, how can I approach the following:

Suppose we have an ellipse centered at the origin of the following form

$ax^2 + b xy +c y^2 + d = 0$

How would I go about finding the axes on which it lies. As clearly this will be a rotated ellipse.

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    Are you familiar with "completing the square"? Can you see how to apply it to your problem?2012-09-12

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Using Derivation of the rotation formula, find $\theta$ to remove $xy$ from the equation.

Here $x=x'\cos \theta-y'\sin \theta$ and $y=x'\sin\theta +y'\cos\theta$

So, $x'=x\cos \theta+y\sin \theta,y'=y\cos \theta-x\sin \theta$.

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    @ Money.SE we have a push to avoid "answers in comments" and would have pushed Brian to resubmit his comment as a full answer. Removing yours would just leave this with no 'answers' which would be silly, the linked articles are great. So long as the links don't break.2016-11-08