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Given the equation of an ellipse, say $$ x^{2} - xy + y^2 = 2, $$ I would like to find the equation of the line which passes through the major axis.

I tried solving for x and y and applying a rotation matrix. Doing so, I found that

$$x' = \cos \theta (-\sqrt{8-3y^2} - y) + \sin \theta (-\sqrt{8 - 3x^2} - x) $$

$$y' = \sin \theta (\sqrt{8-3y^2} + y) + \cos \theta (-\sqrt{8 - 3x^2} - x). $$

Setting the second equation to equal zero and solving in terms of θ I ended up with the result: $$ \tan \theta = \frac{\sqrt{8-3x^2} + x}{\sqrt{8-3y^2 + y}}. $$

This result doesn't seem to help me as it is ultimately telling me something very trivial that I already know about the relationship between x and y coordinates and the tan function.

I know that my choice of coordinates might not be particularly strategic. Are there any other mistakes I'm making in how I'm approaching this?

Thank you!