Let $C$ be an open ended cylinder shell running along the $z$-axis, with radius $1$ - i.e, $x^2+y^2=1$.
Let $S$ be a flat plane defined by $z=x+3$.
The cylinder $C$ and flat plane $S$ intersect to form an ellipse, $E$.
What is the equation of the cone that starts at the origin, opens up towards positive $z$, and intersects this ellipse, $E$?
I would prefer the equation be in terms of $z$, and not a parametrization.
So far, seeing as how naïve I am, I tried to use a regular cone, with $z=3\sqrt{x^2+y^2}$ coming pretty close to intersecting the ellipse $E$. I think the whole cone needs to be somehow sheared so that it goes through/intersects the ellipse $E$ - however, I have no clue whatsoever how to do this.
I tried searching for this on Google to no avail - I have no clue what such a cone would be called - I called it a sheared cone, but I'm probably wrong.