Answer
$x^{2}-y^{2}-z^{2}=k$
When k=0$, \quad x^{2}=y^{2}+z^{2}$,
a right circular cone with x as the axis, vertex at the origin (image 1.)
For positive k,
a family of 2-sheet hyperboloids with x as their axis (image 2.)
For negative k,
family of single sheet hyperboloids with x as their axis (image 3.)
Work Step by Step
(Graphed with geogebra.)