Answer
Hyperboloid of One sheet.
Work Step by Step
Here, we have: $y^2+z^21+x^2$
Comparing the above form with the equation $\displaystyle \frac{z^{2}}{c^{2}}=\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}$
we find that we have a Hyperboloid of One sheet.
In this case, we see that when $z=k$, the traces are hyperbolas.
When $y=k$, then the traces are hyperbolas. That is, $z^2-x^2=1+k^2$.
When $x=k$ (axis is the x-axis), the traces are ellipses.
Because the traces are hyperbolas and ellipses and have no gap at the center, we have a Hyperboloid of One sheet.
