The equation [tex]x^2-9y^2=900[/tex] defines a hyperbolic cylinder.
1. Divide the equation [tex]x^2-9y^2=900[/tex] by 900:
[tex] \dfrac{x^2}{900} - \dfrac{y^2}{100} =1[/tex].
This is the equation of conic section when z=0 (or z=const) and that is hyperbola equation.
2. To find the domain and the range you should express y:
[tex]\dfrac{y^2}{100} =\dfrac{x^2}{900} - 1 \\ \\ y^2= \dfrac{x^2}{9} -100 \\ \\ y=\pm \sqrt{\dfrac{x^2}{9} -100} [/tex].
Since you have square root,
[tex]\dfrac{x^2}{9} -100\ge 0 \\ x^2-900\ge 0 \\ (x-30)(x+30)\ge 0 \\ x\in(-\infty,-30)\cup (30,\infty)[/tex]. This is the domain, the range is [tex]y\in (-\infty,\infty)[/tex].
