Inscribe a rectangle inside an ellipse

Question

A rectangle is to be inscribed inside a horizontal ellipse (whose major or minor axis is parallel to x axis). Is the horizontal orientation of the rectangle (two sides parallel to x axis) the only possibility to inscribe it?

Answer 1

There are two ways. You can place it horizontally or vertically.

Answer 2

Let the ellipse be $x^2+ay^2=b$. Let the centre of the rectangle be $(c,d)$, and the length of its diagonals be $2r$. The points on the rectangle are all distance $r$ from $(c,d)$, and two points are directly opposite the other two.

So there is an angle $\alpha$ that gives the coordinates of $P1$ below, and the opposite point $P2$ follows.

$$P1(c-r\cos\alpha,d-r\sin\alpha),\\P2(c+r\cos\alpha,d+r\sin\alpha),\\P3(c-r\cos\beta,d-r\sin\beta),\\P4(c+r\cos\beta,d+r\sin\beta)$$

Feed these points into the equation for the ellipse.

P1: $(c-r\cos\alpha)^2+a(d-r\sin\alpha)^2=b$

P2: $(c+r\cos\alpha)^2+a(d+r\sin\alpha)^2=b$

Subtract P1 from P2, divide by 4. The difference is $cr\cos\alpha+adr\sin\alpha=0$.

Repeat for $P3$ and $P4$ to get $cr\cos\beta+adr\sin\beta=0$

If $cd\ne0$, then $\tan\alpha=-ad/c=\tan\beta$, so it is not a rectangle.

If $c=0$ and $d\ne 0$ then $\sin\alpha=0=\sin\beta$. Similar if $d=0\ne c$.

If $c=d=0$ then $r^2(\cos^2\alpha+a\sin^2\alpha)=b=r^2(\cos^2\beta+a\sin^2\beta)$ so $\sin^2\alpha=\sin^2\beta$, and it is a rectangle with horizontal sides.

Answer 3

You don't need the condition two of its sides pass through the foci of the ellipse. You only need

Any conics is uniquely determined by 5 points on it.

Start with your inscribed rectangle $ABCD$, construct a line $\ell$ parallel to one of its sides, say $AB$, passing through the center of the rectangle $O$. Let $\ell$ intersect your ellipse at some point $E$. It is clear you can construct another ellipse having $O$ as center, passing through $A,B,C,D$ and $E$ and either the major or the minor axis of it is $\ell$. Because any conics is uniquely determined by any 5 points on it. The new ellipse constructed is the same as your old ellipse. This means $\ell$ is either the major or the minor axis of your original ellipse!