Is the set $C = \left\{ (x, y, z) \in \mathbb R^3 : y^2+z^2 = 1, xz =3 \right\}$ compact?

Question

Is the following set compact? How can I show it?

$$C = \left\{ (x, y, z) \in \mathbb R^3 : y^2+z^2 = 1, xz =3 \right\}$$

Clearly it is closed as it contains its boundary, but I can not show that it is bounded..

Answer 1

It is not compact, as it is not bounded.
Let $A:=\{z\in(-1,0)\cup(0,1)\vert (3/z,\sqrt{1-z^2},z)\}$ be an unbounded set contained in $C$.

Answer 2

It is not bounded, so not compact, by Heine-Borel. Look at $y^2+9/x^2=1$. Here $x$ is free to roam (away from $(-3,3)$).