Rotating a surface along a point does not changes the distance of the surface to that point.
Therefore, rotating $\displaystyle S$ with $\displaystyle \pi/4$ radians along $\displaystyle O$, we have $\displaystyle S(x,y,z)=\widehat{S}(t,s,z)=2t^{2}-z^{2}-1=0,$ by making the substitution $\displaystyle x(t,s)=\sqrt{2}t/2+\sqrt{2}s/2$ and $\displaystyle y(t,s)=-\sqrt{2}t/2+\sqrt{2}s/2$.
Clearly $\displaystyle \widehat{S}$ is a hyperbola of two parts (I made this term up, and if possible teach me the correct one).
Hence, the distance $\displaystyle \ell$ attains it smallest value at the peak points $\displaystyle (\pm\sqrt{2}/2,0,0)$, it is clear that $\displaystyle \ell=\sqrt{2}/2$.