Does a circular cone with half-angle $x$ always fit within an elliptical cone of minor half-angle $x$?

Question

I have two right cones. One is a circular cone with height $h$ and half-angle $x$. EDIT: Both share the same apex and axis.

The other is an elliptical cone, with height $h$, minor half-angle $x$ and major half-angle $y$ where $y > x$.

Will the entire volume of the first (circular) cone always fit within the second?

This is for some software, and I need to know if I can make simplifying assumptions about the angles involved. If the circular always fits within the elliptical, than I can do a (much easier) circular cone check in some cases.

Maybe this just simplifies down to whether a circle of radius $r$ will always fit within an ellipse of minor axis $r$? (If so, then it's true, right?)

Please forgive any mistakes in terminology, this is a bit outside my knowledge base.

Answer 1

The answer is yes. One way to see this is to consider the common axis for the two cones, and a point $p$ on this axis. Say this point is a distance $d$ from the common apex of the cones, and consider the plane $P$ orthogonal to the axis at the point $p$. This will intersect the two cones in a circle and an ellipse respectively. The circle will have radius $r=d\tan(x)$ and will be centred at $p$, while the ellipse will have major and minor axes $d\tan(x)$ and $d\tan(y)$ respectively. Since $y>x$, $d\tan(y)>d\tan(x)$. This ellipse will also be centred at $p$, and so must contain the circle. As this is the case for every point $p$ on the axis, the circular cone s contained in the elliptical cone.