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I have the an ellipse with its semi-minor axis length $x$, and semi major axis $4x$. However, it is oriented $45$ degrees from the axis (but is still centred at the origin). I want to do some work with such a shape, but don't know how to express it algebraically. What is the equation for this ellipse?

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    http://en.wikipedia.org/wiki/Matrix_representation_of_conic_sections2012-02-11
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    $x$ is not particularly helpful as a parameter if you also intend to use it as an equation variable2012-02-11
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    In general, compute what you already know (say, equation of the ellipse aligned with the $x$-axis) and then apply rotation by the desired angle. Look here: http://en.wikipedia.org/wiki/Rotation_matrix.2012-02-11

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