Answer
Hyperbola
Center:$ \quad (0,0)$
Vertices: $ \quad (-\sqrt{2},0),(\sqrt{10},0)$
Foci: $ \quad (-\sqrt{10},0), \quad (\sqrt{10},0)$
Asymptotes: $ \quad y=2x, \quad y=-2x$
Work Step by Step
Both variables squared, a minus between terms$\Rightarrow$Hyperbola
Divide by 8,
$\displaystyle \frac{x^{2}}{2}-\frac{y^{2}}{8}=1$
Table 4:
$\begin{array}{cccc}
{\text{Foci}}&{\text { Vertices }}&{\text{Equation}}&{\text{asymptotes}}\\\hline
(h\pm c,k)&{(h \pm a, k)}&{ \displaystyle \frac{(x-h)^{2}}{a^{2}}-\frac{(y-k)^{2}}{b^{2}}=1,}&{y-k=\displaystyle \pm\frac{b}{a}(x-h)}\end{array}$
$b^{2}=c^{2}-a^{2},\quad a=\sqrt{2}, \quad b=\sqrt{8}=2\sqrt{2}$
Find c$:$
$c^{2}=a^{2}+b^{2}=2+8=10$
$c=\sqrt{10}$
Center:$ \quad (0,0)$
Vertices: $ \quad (h \pm a, k)=(\pm\sqrt{2},0)$
Foci: $ \quad (h\pm c,k)=(\pm\sqrt{10},0)$
Asymptotes:$ \quad y-k=\displaystyle \pm\frac{b}{a}(x-h)$
$y=\pm 2x$
