I have the following question here...
Find an equation of a plane through the origin such that the intersection between the plane and the elliptical cylinder $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ is a circle.
Wouldn't this simply just be $ax+by=0$??? The plane has to pass the origin, so $d=0$ and since the plane has to be perpendicular to the cylinder to get circular cross section, we get that $z=0$. This results in $ax+by=0$. I feel like this is too simple and I am overlooking something.
If for instance $a\ge b$, take a plane $$z=ky.$$ The intersection with your cylinder will be an ellipse, whose semi-axes $a$ and $b\sqrt{1+k^2}$ are equal iff $$k=\pm \frac{\sqrt{a^2-b^2}}b.$$