How do you find the equation for the curve of intersection between a plane and a cone?

Answer 1

There are two different things you could mean.

If you're looking for the equation of a curve in 3-space -- f(x,y,z) = 0 -- it isn't going to happen. A one-dimensional figure takes two such equations, not one. One equation gives you a two-dimensional figure.

If you're in the x-y plane, minding your own business, and a cone comes by and intersects it -- well, that's how you get "conic sections" like ellipses, hyperbolae, and parabolas. I imagine you're pretty familiar with the equations for those. :)