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.
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.