An right circular cone with vertex at the origin, and with it axis pointing along the $z$ axis, is scaled (stretched) along the $y$ axis direction by a factor of $2$. The angle between the surface of the unstretched cone and its axis is $20^\circ$.
A plane with equation $\sqrt{2} x + \sqrt{2} y + 2 \sqrt{3} z = 100 \sqrt{3} $ cuts through this stretched cone. The intersection between the plane and the stretched cone is an ellipse. Find its semi-major and semi-minor axes lengths.
My approach:
I took the following steps for the solution:
- Found the equation of the (stretched) elliptical cone.
- Parameterized the plane, using two unit perpendicular axes that span the plane.
- Used the parametrization in the equation of the cone, to find the equation governing the parameters. The equation is that of an ellipse.
- Extracted the semi-axes lengths from the equation of the ellipse.
I got the semi-major axis length as $44.029848$ and the semi-minor axis length as $20.70276$.
I would like to verify these values. I appreciate if someone with access to a CAD software would create a model for the elliptical cone and the cutting plane, and verify the lengths of semi-axes, or the area of the cut.
Your input is highly appreciated.