Hyperbola in Cartesian Planes problem

[kasse]

Does the plane that intersects the cone need to be parallell to the axis of the cone to make the section a hyperbola, or is it enough that it is not parallell to a generator?

If the latter is correct, can one say that a parabola is a special case of a hyperbola?

 

[Gib Z]

The Latter is correct.

The hyperbola in Cartesian Planes is defined by Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0. As you Can see, a parabola is simply where B and C equals zero.

 

[kasse]

Thank you!

 

[Gib Z]

No problemo :)

 

[dextercioby]

If one takes one of the focal points of a hyperbola to infinity, then the remaining curve would be a parabola. Same is valid for an ellipse.

In other words let b tend to infinity in

$$\frac{x^{2}}{a^{2}}\pm \frac{y^{2}}{b^{2}} =1$$

and you'll get a parabola.

Daniel.