Ellipse in a Rectangle What is the equation for an ellipse (or rather, family of ellipses) which has as its tangents the lines forming the rectangle $$x=\pm a, y=\pm b\;\; (a,b>0)$$?
This question is a modification/extension of this other question here posted recently.

By exploiting the affine map $(x,y)\mapsto\left(\frac{x}{a},\frac{y}{b}\right)$ the question boils down to finding the family of ellipses inscribed in a square with vertices at $(\pm 1,\pm 1)$. In a ellipse the line joining the midpoints of parallel chords always go through the center. Additionally, the orthoptic curve of an ellipse is the director circle. It follows that all the ellipses that are tangent to the sides of the previous square fulfill $a^2+b^2=2$, are centered at the origin and are symmetric with respect to the diagonals of the square.

Here it is a straightedge-and-compass construction.


*

*Take some point $P$ on the perimeter of the square and by reflecting it with respect to the center and the diagonals of the square construct the rectangle $PQRS$;

*Consider the tangents at $P,Q,R,S$ at the circumcircle of $PQRS$;

*Two intersections of such tangents are the vertices of the inscribed ellipse through $P$, that is simple to draw.

