Area of a conic section

This problem was proposed by Shafiqur Rahman on Facebook: Find the area and length of the semi axes of the section of the paraboloid $2x^2+y^2=z$ by the plane $x + 2 y + z = 4$.

We find a rigid motion that maps the given plane to the plane $z = 0$. We apply the same transformation to the given paraboloid and make $z=0$ then we get a equation of the conic on the $z=0$ plane. Next we get the reduced equation of the conic and its semiaxes.

Read the details here: Area of a conic section