My task is this: 

Find the points on the surface $z^2 -xy = 1$ with the shortest distance to the origin.

My work so far:

Let $f(x,y,z) = x^2 + y^2 + z^2$ and $g(x,y,z)=z^2 -xy -1$ then we have to solve:

$\nabla f = \lambda\nabla g \to \begin{pmatrix}2x\\2y\\2z\end{pmatrix} = \lambda\begin{pmatrix}-y\\-x\\2z \end{pmatrix}$, if $z = 0$ then I get the points $(\pm \sqrt{2}, \pm \sqrt{2}, 0)$ which are not even on the graph. If $z \neq 0$ then $\lambda = 1 \to x= -y/2, y = -x/2$. Substituting this into our constraint we get: 

$z = \pm\sqrt{1- y^2/2}$ or $z = \pm \sqrt{1 -x^2/2}$. Im stuck here and need some help. Any hints or better approach would be appriciated!