# Finding the curve of intersection of a cylinder and cone

I have a cone $$x^2 + y^2 -z^2 =0$$ and a cylinder $$x^2 +y^2 -2ax =0$$. Together they look like this.

If one were to project the intersection onto the xy plane, the curve given by this intersection would be an ellipse.

To attain the equation for this ellipse I have attempted to solve the equations of these quadrics simultaneosly when $$z=0$$ but this has been leading me nowhere.

Any suggestions?

The context for this work is to find the surface area of the cone inside the cylinder. To do this I consider a surface integral whose region will be defined by this intersection curve.

• The projection of the curve onto the $x$-$y$ plane isn’t just any old ellipse: it’s a circle. Setting $z=0$ isn’t the right approach—you’re then trying to compute the intersection of three surfaces, which in this case consists only of the origin. – amd Apr 22 at 22:06

HINT: You need to start by parametrizing your cone. I suggest you work just with half of it, say $$z=\sqrt{x^2+y^2}$$. Presumably polar coordinates would be the best coordinates to use for this. Now find in those coordinates where the region you're interested in stops, i.e., the intersection with the cylinder. (The circle $$x^2+y^2-2ax=0$$ in the $$xy$$-plane has a very nice equation in polar coordinates.)