Determine whether the intersection of surfaces is a parabola Let $C$ be the curve of intersection between the cone $z=\sqrt{x^2 + y^2}$ and the plane $z=1-x$. Is $C$ a parabola?
I can see that letting $x=t$ we have parametric equations
$x=t\\y=\pm\sqrt{1-2t}\\z=1-t$
and so the projection onto the $xy$ plane is a parabola. But I don't know how to see whether $C$ itself is a parabola. 

If you want to go back to the classical definition of conic sections, all of them are what you get by intersecting a right regular (circular) cone with a plane that doesn’t contain the vertex of the cone. The generatrices of the cone are the straight lines lying in the cone, all passing throught the vertex. Since it’s a right cone, the generatrices all make the same acute angle $\theta$ with the axis. In your example, $\theta=45^\circ$.
Now, in general, if your plane is perpendicular to the axis, you get a circle; if the plane makes an angle to the axis that’s strictly between $90^\circ$ and $\theta$, you get an ellipse; if less than t$\theta$, you get a hyperbola. But if the plane is parallel to one of the generatrices, i.e. if the angle between the plane and the axis is $\theta$, you get a parabola.
In your case, you have a generatrix given by the equations $\{z=-x, y=0\}$, and your plane is certainly parallel to this line. So your intersection is a parabola.
