TITLE: Equation of intersection of two cones
QUESTION [6 upvotes]: The equations of two cones are given;
$(x-x_{0})^2+(y-y_{0})^2=\frac {(z-z_{0})^2}{m^2}$ 
and 
$(x-x_{1})^2+(y-y_{1})^2=\frac {(z-z_{1})^2}{m^2}$ 
How to find the equations of intersections
1) When the intersection is an ellipse? 
2) When the intersection is a hyperbola?

REPLY [2 votes]: As the two cones have the same aperture and parallel axes, their intersection is a hyperbola if both vertices are external to the other cone, and is an ellipse if a vertex is inside the other cone. Finally, if a vertex is on the surface of the other cone, then the cones are tangent and their intersection is a line.

In any case, by subtracting one equation from the other one you will find the equation of the plane containing the intersection.