Cylindrical equation to rectangular coordinates

Question

How would you convert $y^2=9+r^2$ and $x=3r$ from cylindrical to rectangular coordinates? And what would these equations look like on the $R^3$ plane?

Answer 1

If $r$ is a free parameter then $y^2=9+\frac {x^2}9\iff \left(\frac y3\right)^2-\left(\frac x9\right)^2=1$ is an hyperbola.

But if you are talking about the $r$ of polar coordinates then $r^2=x^2+y^2=9r^2+9+r^2\iff r^2+1=0$ which is impossible.