A vertically oriented 3D cone is sliced with a vertical 2D plane. What is the conic section that will form?

• A. Circle:
• B. Ellipse:
• C. Parabola:
• D. Hyperbola:

Answer: (D)

Verified step by step guidance

1. Understand the problem: A vertically oriented 3D cone is sliced with a vertical 2D plane. The goal is to determine the conic section formed by this intersection.
2. Recall the geometry of conic sections: Conic sections (circle, ellipse, parabola, and hyperbola) are formed by the intersection of a plane with a cone. The shape depends on the angle and orientation of the slicing plane relative to the cone.
3. Analyze the slicing plane: A vertical plane intersects the cone in such a way that it cuts through both nappes (the upper and lower parts) of the cone. This is a key characteristic of a hyperbola.
4. Compare with the definitions: A hyperbola is formed when the slicing plane is vertical and intersects both nappes of the cone. This is distinct from other conic sections like circles, ellipses, or parabolas, which involve different orientations or angles of the slicing plane.
5. Conclude: Based on the geometry of the slicing plane and its interaction with the cone, the conic section formed is a hyperbola.