Largest Square That Can Be Inscribed in an Ellipse

If a square is inscribed in an ellipse, the distance from the centre of the square to any of its corners will be equal to the distance between the origin and the point  
  in the diagram below. where  

the equation of the ellipse is  
\[\frac{x^2}{a^2} + \frac{y^2}{b^2} =1\]
\[\frac{x^2}{a^2} + \frac{x^2}{b^2} =x^2 ( ( \frac{a^2+b^2}{a^2 b^2} =1 \rightarrow x = \frac{\sqrt{a^2 +b^2}}{ab}\]
\[y=\frac{\sqrt{a^2 +b^2}}{ab}\]
  and the vertices of the square are at  
\[(\pm \frac{\sqrt{a^2 +b^2}}{ab}, \pm \frac{\sqrt{a^2 +b^2}}{ab} ) \]