Why $xy+yz+xz=1$ is two-sheeted hyperboloid? I can't see why $xy+yz+xz=1$ is two-sheeted hyperboloid.
I know that the equation for two-sheeted hyperboloid is: $\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=-1$. 

You need to perform Gauß' reduction to see it:
\begin{align*}
xy+yz+zx&=(x+z)(y+z)-z^2=\frac14\Bigl((x+z+y+z)^2-(x+z-y-z)^2\Bigr)-z^2\\
&=\Bigl(\frac{x+y+2z}2\Bigr)^2-\Bigl(\frac{x-y}2\Bigr)^2-z^2.
\end{align*}
Hence the equation of the quadric can be written as 
$$\Bigl(\frac{x-y}2\Bigr)^2+z^2-\Bigl(\frac{x+y+2z}2\Bigr)^2=-1.$$
