Answer
Ellipsoid; the trace is an ellipse.
Work Step by Step
The equation
$$
x^2+ \left(\frac{y}{4}\right)^{2} +z^2=1
$$
is an ellipsoid.
(See equations on page 691.)
To find the trace with the plane $ z=1/4$, we have
$$ x^2+ \left(\frac{y}{4}\right)^{2} +1/16=1\Longrightarrow x^2+ \left(\frac{y}{4}\right)^{2} =15/16$$
which is an ellipse in the $ xy $-plane.
