# Ellipse Question

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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Read this article or this PDF. If they enable you to solve the problem, you can post your solution and (after a few hours) accept it. If not, you can ask for further help. – Brian M. Scott Sep 12 '12 at 5:29
Are you familiar with "completing the square"? Can you see how to apply it to your problem? – Gerry Myerson Sep 12 '12 at 5:40

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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Yes I figured out how to do this problem thanks to Brian's link. – katari Sep 12 '12 at 5:55
Sorry, I have observed the link just now, thanks Brian for your PDF link. – lab bhattacharjee Sep 12 '12 at 6:02
@katari, could you please un-accept the answer as I want to delete it as it's redundant. – lab bhattacharjee Sep 12 '12 at 6:41