How to find the sides of a rectangle inscribed in an ellipse?

Question

JavaScript

I can’t find a formula for how to find the width and height of a rectangle inscribed in an ellipse, knowing the vertical and horizontal radii of the ellipse and the aspect ratio of the rectangle.
Example:
horizontal radius = 10
vertical radius = 5
width = 2 height
you need to find the height and width values ​​if the rectangle is inscribed in an ellipse
If there is a ready-made npm library with these formulas or an example in JS, please tell me.
I ask mathematicians to suggest the formulas themselves, google offers me only circles with one radius.
Thanks to all!

Answer 1

An infinite number of rectangles can be inscribed in an ellipse.
The question does not make sense, or there are not enough additional conditions

Answer 2

Ellipse formula:
x ² / a ² + y ² / b ² \u003d 1
where a is the major semi-axis, b is the minor semi-axis of the ellipse. From ^the rectangle condition
x = 2y
4y ² / a ^{2 +} y ² / b ² = ^{1 _} _ x = 7.0710678 So the rectangle will have a width of 14.1421356 and a height of 7.0710678