When a solid right circular cone is intersected by a plane that cuts through all of its generators (i.e. passes through both lateral slopes without intersecting the base and is not perpendicular to the central axis), what geometric curve is formed at the planar cross-section?
Options:
(A) Circle
(B) Ellipse
(C) Parabola
(D) Hyperbola
Answer Key:
B
Step-by-Step Solution & Explanation
**Method 1: Classical Conic Section Definition**
- When the cutting plane is perpendicular to the cone's axis of rotation, the cross-section is a **circle**.
- When the cutting plane cuts obliquely across all generators (at an angle to the axis greater than the semi-vertical angle), the section is an **ellipse**.
- When the cutting plane is parallel to one generator, the section is a **parabola**.
- When the plane is parallel to the axis (or cuts at an angle less than the semi-vertical angle, intersecting both nappes), the section is a **hyperbola**.
Option **B** is correct.