Question

question 3 of 25
if a right circular cone is intersected by a plane that goes through both
nappes of the cone but not through the vertex, as in the picture below, what
shape is produced?

a. a circle

b. a parabola

c. a hyperbola

d. a line

e. a point

f. an ellipse

Explanation:

To determine the shape formed when a plane intersects a right circular cone through both nappes (the two parts of the cone) and not through the vertex, we recall the definitions of conic sections:

• A circle is formed when the plane is perpendicular to the cone's axis and intersects only one nappe.
• A parabola is formed when the plane is parallel to the side of the cone (tangent to one nappe).
• A hyperbola is formed when the plane intersects both nappes of the cone and is not parallel to the cone's axis or through the vertex.
• A line, point, or ellipse do not match this intersection condition. The plane cutting through both nappes (as described) creates two separate open curves, which is the definition of a hyperbola.

Answer:

C. A hyperbola