For the equation $z=\sqrt{x^2 + y^2} - x - y$. At $z=100$ what is the equation of the shape on the plane of $z=100$? Use the variables $x$ and $y$.

Question

For the 3D graph $z=\sqrt{x^2 + y^2} - x - y$ for the plane $z=100$ what is the equation for $y$ in terms of $x$.

I see a shape that looks a lot like a parabola or a hyperbola but do not know what it is or what the equation of the shape is

Answer 1

Note that\begin{align}\require{cancel}\sqrt{x^2+y^2}-x-y=100&\iff\sqrt{x^2+y^2}=x+y+100\\&\implies\cancel{x^2}+\cancel{y^2}=\cancel{x^2}+\cancel{y^2}+10\,000+200x+200y+2xy\\&\iff 100x+100y+xy=-5\,000,\end{align}which is the equation of a hyperbola.