How to prove that there's a plane with the required property?

Question

I'm finding this particularly difficult. Let's say a circular cone is given with its base on a plane $\pi$. Then, if we cut this cone with planes that are not parallel to $pi$ we will have an Ellipse (until a certain moment, when we will have another conic). But anyway, the task is to show that there's a place nor parallel to $\pi$ that determines a circumference. I will post a picture:

Any help is more than welcome!

Answer 1

As far as I think,(I may not be absolutely right), you are asking for the way to make a conic section (by a plane cut on a conic )different from an ellipse(or a circle ). If that is the query, then you can always make a hyperbola ; take a double conic (reflect your conic about its vertex, so that the axis of the double conic remains the same as the original conic (and all other proper) and cut it by a plane parallel to the axis. You get a hyperbola. and certainly ,a plane parallel to the axis is NOT parallel to $\pi$

Answer 2

The cone has an axis of symmetry. Pick any point on that axis and draw the plane $\pi'$, perpendicular to it. If the axis of symmetry is not perpendicular to $\pi$, then $\pi$ and $\pi'$ will not be parallel and $\pi'$ will intersect the cone in a circle.