4.18. What surface is defined by the equation

$$
x^{2}-2 x-3 y^{2}+12 y+2 z^{2}+12 z-11=0 ?
$$
Solution. Let's complete the squares for $x, y, z$:

$$
\begin{aligned}
& x^{2}-2 x+1^{2}-1^{2}-3\left(y^{2}-4 y+2^{2}-2^{2}\right)+ \\
& \quad+2\left(z^{2}+6 z+3^{2}-3^{2}\right)-11=0 \\
& (x-1)^{2}-3(y-2)^{2}+2(z+3)^{2}=18 \\
& (x-1)^{2} / 18-(y-2)^{2} / 6+(z+3)^{2} / 9=1
\end{aligned}
$$

- The equation represents a one-sheet hyperboloid with the center at point $O^{\prime}(1,2,-3)$.