I sent the following question to another forum more than a week ago but haven't got any responses. The moderator of that forum suggested that I pose the following question here:
Suppose we have an ellipse $x^2/a^2 + y^2/b^2 = 1$ (centered at the origin). Let $n>4$. There are $n$ rays going out of the origin, at angles $0, 2\pi/n, 4\pi/n, 6\pi/n,...,2\pi(n-1)/n$. Let $(x_1,y_1),...,(x_n,y_n)$ be the points of intersection of the rays and the ellipse. The product from $k=0$ to $n-1$ of $(x_k)^2 + (y_k)^2$ is equal to one. Can $a$ and $b$ be rational? Note that this is obviously possible if $a=b=1$, since then the ellipse becomes a circle of radius 1. But what about if $a$ is not equal to $b$? Can $a$ and $b$ still be rational?