Question

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Answer 1

The equation $x^2-9y^2=900$ defines a hyperbolic cylinder.

1. Divide the equation $x^2-9y^2=900$  by 900:
$\dfrac{x^2}{900} - \dfrac{y^2}{100} =1$.
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:
$\dfrac{y^2}{100} =\dfrac{x^2}{900} - 1 \\ \\ y^2= \dfrac{x^2}{9} -100 \\ \\ y=\pm \sqrt{\dfrac{x^2}{9} -100}$.
Since you have square root, 
$\dfrac{x^2}{9} -100\ge 0 \\ x^2-900\ge 0 \\ (x-30)(x+30)\ge 0 \\ x\in(-\infty,-30)\cup (30,\infty)$. This is the domain, the range is $y\in (-\infty,\infty)$.