How to prove $\frac{y^2-x^2}{x+y+1}=\pm1$ is a hyperbola? How to prove $\frac{y^2-x^2}{x+y+1}=\pm1$ is a hyperbola, knowing the canonical form is $\frac{y^2}{a^2}-\frac{x^2}{b^2}=\pm1$ where $a$ and $b$ are constants? Thanks !

Let 
$$
\frac{y^2-x^2}{x+y+1}=1\\
\Rightarrow y^2-x^2=x+y+1\\
\Rightarrow y^2-x^2-x-y=1
$$
Complete the squares for x and y . You will get  rectangular hyperbola. Similar will be the case if
$$
\frac{y^2-x^2}{x+y+1}=-1$$
