Question

which degenerate conic is formed when a double cone is sliced at the apex by a plane perpendicular to the base of the cone?

Answer 1

A plane perpendicular to the base of a double cone that intersects at the apex forms a point, which is a degenerate conic section.

Step-by-step explanation:

When a double cone is sliced at the apex by a plane that is perpendicular to the base of the cone, the degenerate conic formed is a point. In the context of conic sections, a point is considered a degenerate case of a conic because it does not form the usual open or closed curve like a circle, ellipse, parabola, or hyperbola. Instead, it represents the case where the plane's intersection with the cone reduces to a single point - the apex itself.

Answer 2

When a double cone is sliced at the apex by a plane perpendicular to the base of the cone, the resulting conic section is hyperbola

What is hyperbola?

A double cone can be made to "pinch" its two halves together at their top point by slicing it through the apex with a plane perpendicular to the base. In this instance, the pinching results in a degenerate conic, which produces a hyperbola with infinitely near branches.

Imagine the hyperbola's two branches coming out of the apex point and expanding eternally to either side yet never getting too far apart. The essence of the "collapsed" cone shape at the apex is captured by this degenerate hyperbola.

Consequently, even though it may not make sense, the degenerate conic that forms in this situation is indeed a hyperbola—albeit one with infinitely near branches.