Answer
False
Work Step by Step
The equation of a hyperboloid is $ \left(\frac{x}{a}\right)^{2}+\left(\frac{y}{b}\right)^{2}=\left(\frac{z}{c}\right)^{2}+1$ and to find the trace we freeze one of the three variables, for example $ z=z_0$. Then the equation of the hyperboloid becomes
$$ \left(\frac{x}{a}\right)^{2}+\left(\frac{y}{b}\right)^{2}=\left(\frac{z_0}{c}\right)^{2}+1$$
which represent an ellipse. So the statement is false.
