Q5. For a cone with half-angle α, how does the section change as β goes from α up to 90° (plane not through the vertex)?
At β = α it is a parabola. For α < β < 90° it is an ellipse whose eccentricity cos β/cos α falls from 1 towards 0, so the ellipse gets rounder. At β = 90° it is a circle.
Q6. A cone has α = 30°. At what angle β does a plane cut an ellipse of eccentricity 1/2?
cos β = e cos α = (1/2)(√3/2) = √3/4 ≈ 0.433, so β ≈ 64.3°. This lies between 30° and 90°, as an ellipse needs.
Q7. Can a plane parallel to the axis of a cone cut an ellipse?
No. Such a plane has β = 0°, which is less than α. If it misses the vertex it cuts a hyperbola; if it passes through the vertex, so that it contains the axis, it cuts a pair of intersecting lines.
Q8. Find the eccentricity of the section made by a plane missing the vertex and parallel to the axis of a cone with α = 30°.
β = 0°, so e = cos 0°/cos 30° = 1/(√3/2) = 2/√3 ≈ 1.155. It is greater than 1, as a hyperbola needs.
Q9. Why does a parabola not close up, while an ellipse does?
For an ellipse the plane cuts right across one nappe, meeting every generator of that nappe, so the curve goes all the way round. For a parabola the plane is parallel to one generator and never meets it, so the curve runs off without end along that direction.