Question

A plane intersects a double-napped cone only at the cone's vertex.

Which terms describe the degenerate conic section that is formed?

Select each correct answer.

degenerate parabola

degenerate ellipse

line

degenerate hyperbola

point

pair of intersecting lines

Answer 1

The answer is:

Degenerate ellipse.

Point.

Explanation:

We know that when a plane intersects a double-napped cone only at the cone's vertex then the cross-section that is obtained is the point.

Also we know that a point is also considered as a circle with radius 0.

and a circle is also a kind of a ellipse when the length of both the transverse and conjugate axis are equal.

Hence, the term that best describe the degenerate conic section that is formed is:

degenerate ellipse

point

Answer 2

In addition to ...

degenerate ellipse

point

it is also a degenerate circle (special case of ellipse).

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The vertex of a double-napped cone is a point. So, the intersection of only the vertex with a plane is a point, not a degenerate parabola (a line) or degenerate hyperbola (pair of crossing lines).

Essentially, the plane must intersect at an angle greater than the angle of the edge of the cone with the axis of the cone. If the plane is not at the vertex, such an intersection angle will result in a circle or ellipse. As the plane gets closer to the vertex, the ellipse gets smaller, degenerating to a single point.