Answer
False.
Work Step by Step
The equation of an ellipsoid 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 ellipsoid becomes
$$ \left(\frac{x}{a}\right)^{2}+\left(\frac{y}{b}\right)^{2}=1-\left(\frac{z_0}{c}\right)^{2}$$
which represents an ellipse.
However, we can also trace a single point. For example:
($x=0,y=0,z=c$)
Thus, not all traces are ellipses.
