4.21. Determine that the plane $z-2=0$ intersects the ellipsoid $x^{2} / 12+y^{2} / 4+z^{2} / 16=1$ along an ellipse. Formulate its equation and find the semi-axes and vertices.
Solution. Substitute the number 4 for $z^{2}$ in the surface equation,

$$
\begin{aligned}
& x^{2} / 12+y^{2} / 4+4 / 16=1 \\
& x^{2} / 12+y^{2} / 4=3 / 4 \\
& x^{2} / 3^{2}+y^{2} / 3=1
\end{aligned}
$$

from which the ellipse has semi-axes $a=3, b=\sqrt{3}$.