Chapter 8.6, Problem 53PS

Solution

(a)

To explain : In the given case, when the intersection is not a circle.

The point at which the upper cone ends and the lower cone starts will not be a circle.

Given information :

A plane perpendicular to the cone's axis passes through the cone, intersecting it in a circle whose radius decreases and then increases.

  

Explanation :

When a plane perpendicular to the cone's axis passes through the double napped cone and intersects the cone, it forms a conic section called circle.

When the plane is moved from up to down then the point at which the upper cone ends and the lower cone starts, the radius will be zero, and therefore that point will not be a circle.

(b)

To explain : In the given case, when the intersection is not a hyperbola.

The point at which the two lines intersect is not a hyperbola.

Given information :

A plane parallel to the cone's axis passes through the cone, intersecting it in a hyperbola whose vertices get closer together and then farther apart.

  

Explanation :

When a plane parallel to the cone's axis passes through the double napped cone and intersects the cone, it forms a conic section called hyperbola.

When the plane passes through the axis of the double-napped cone, the point at which the two lines intersect is not a hyperbola.

Therefore the point at which the two lines intersect is not a hyperbola.

(c)

To explain : In the given case, when the intersection is not a parabola.

The point at which the plane gives a line is not a parabola.

Given information :

a plane parallel to the cone's nappe passes through the cone, intersecting it in a parabola that first gets narrower, then flips and gets wider.

  

Explanation :

When a plane parallel to the cone's nappe passes through the cone and intersects the cone, it forms a conic section called parabola.

When the plane moves from down to up of the double-napped cone, at the line of napping of the cones then the plane gives a line and this point is not a parabola.

Therefore the point at which the plane gives a line is not a parabola.