Imagine a plane and a cone intersecting to form a parabola. Explain why the plane has to intersect the axis of the cone.

Question

This a math homework problem and i don't understand it.

Answer 1

The plane must lie parallel to a the side of the cone to make a parabola.  That way it doesn't exit the other side of the cone, forming an ellipse, and it also doesn't intersect the other cone, forming a hyberbola.  The whole "other cone" thing might be confusing, so here's a picture.

 

http://www.andrews.edu/~calkins/math/webtexts/conicsl.jpg

Anyway, being parallel to the side of the cone means it can't be parallel to the cone's axis and therefore must cross it.  Hope that helps.

Answer 2

I have to respectfully disagree with Cynthia.

 

See this picture: http://en.wikipedia.org/wiki/File:Conic_sections_with_plane.svg

 

An ellipse will intersect the cone at an angle from the axis greater than the angle of the cone, thus forming a closed line.

 

A parabola will intersect the cone at an angle from the axis equal to the angle of the cone, essentially parallel to the opposite wall of the cone. Since the parabola is not parallel to the axis, it must, as some point, cross the axis.

 

A hyperbola will intersect the cone at an angle from the axis less than the angle of the cone, including possibly parallel to the axis, the only angle which will never cross the axis.

Answer 3

The question here is  wrong the plane must never intersect the axis in order to form a parabola.  If the plane intersects the axis then at some point it will form an ellipse. In order to make a parabola the plane would have to be parallel but not on the axis.  If the plane was on the axis exactly it would form an angle not a parabola.  So the question must be wrong. It is the plane must not intersect the axis of the cone.