Formation of a hyperbola using a plane cutting a doubole con

[barryj]

Homework Statement

My question. When studying conics, the parabola circle, and ellipse can be easily see by passing a plane through a double cone. The hyperbola is generated when the plane is passed through the double cone where it passes through the top and the bottom cone. My question is, does the plane make a hyperbola if the plane is not parallel to the vertical axis of the cone? It seems that most illustrations show the plane parallel to the vertical axis. If the plane is not parallel to the vertical, is the intersection a symmetrical hyperbola?

Homework Equations

irrelevant

The Attempt at a Solution

 

[gneill]

If the plane is not parallel to the vertical, is the intersection a symmetrical hyperbola?

Nope. It'll be one of the other conic sections (circle, ellipse, or parabola).

 

[Dick]

If the plane intersects the cone in two separate curves then it's a hyperbola. And, yes, it's symmetrical (even if it might seem like it wouldn't be).

 

[Ray Vickson]

Homework Statement

My question. When studying conics, the parabola circle, and ellipse can be easily see by passing a plane through a double cone. The hyperbola is generated when the plane is passed through the double cone where it passes through the top and the bottom cone. My question is, does the plane make a hyperbola if the plane is not parallel to the vertical axis of the cone? It seems that most illustrations show the plane parallel to the vertical axis. If the plane is not parallel to the vertical, is the intersection a symmetrical hyperbola?

Homework Equations

irrelevant

The Attempt at a Solution

See, eg.,
http://lh6.ggpht.com/-QlPR2LJwMIo/SwG06HzEfcI/AAAAAAAAAEA/Pp7RaURjYXg/s1280/conic%20sections.jpg

 

[gneill]

Funny how they never seem to show the case where the plane is vertical and contains the centerline of the cone. That yields the so-called straight-line or "degenerate" orbits where the body on orbit moves in a straight line either directly into or away from the central body.

 

[barryj]

Yes, and I have never seen where the plan is tilted a bit. A previous answer says it is still a hyperbola. The equation above shows a tilted plane.