Need help on understanding cone geometry

[chyo]

Hi all,

Homework Statement

Given a right circular cone with origin at the centre of the base, the positive z-axis pointing towards the apex, and the height is h and radius of base is r. What is the cartesian equation of the cone?

Homework Equations

The Attempt at a Solution

The equation that I get is (h-z)^2 = (h/r)^2 (x^2+y^2). Can anyone confirm this?

Assuming that my above equation is correct, how is it that the general equation of a cone is instead x^2 + y^2 = z^2? Where did the extra terms from the first equation go to?

Also, what significance does it bring when the equation of a cone becomes ax^2 + by^2 = (h-cz)^2? If I compare it with the equation i obtained, I suppose that this should mean that the height of the cone is equal to its base radius? What about the constants a, b, c; what do they represent in the physical sense?

Thanks much!

 

[Dick]

Your equation looks fine to me. x^2 + y^2 = z^2 is the equation a 45 degree cone with apex at the origin. Not very general. And for this one, ax^2 + by^2 = (h-cz)^2, if a!=b, that's a vertical cone (axis along z) with an elliptical cross-section.