For a right cone, it is a hyperbola which becomes and isosceles triangle when the section passes through the apex.
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If passing through the tip, you get two straight lines that cross one another. Otherwise you get a hyperbola.
No. Some of the classic curves studied by mathematicians: ellipses, hyperbola are cross sections of a cone taken at an angle.
If it a right cone then it is a circle, otherwise an ellipse.
Circle
It is a rectangle.
By definition, the circular cross-section of a cone changes linearly in width as you go along its axis. By definition, the cross-section of a prism is constant along its axis. So, by definition, a cone prism is an impossible shape.