How do I find the size of a non-proportional rectangle inside a given ellipse?

I understand how to calculate the largest possible rectangle that can be drawn inside an ellipse, but my problem is: I a have a rectangle of given proportions and an ellipse of a given size (not the same proportions) and I need to know how big that rectangle will be when centered inside the ellipse and sized with all four corners intersecting.


If the rectangle is centered inside the ellipse, and for all 4 corners to lie on the ellipse, the 4 corners must be the solution to the equation system of:

[1] implicit equation of the ellipse: x^2/a^2 + y^2/b^2 = 1

[2] The proportion of the rectangle (aspect ratio) x / y = c.

Just substitute in and solve the equation for x and y. 2 * abs(x) will be the width and 2 * abs(y) will be the height of the rectangle.

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