Answer
Hyperbola,
Center:$ \quad (1,2)$
Vertices: $ \quad (1,0),(1,4)$
Foci: $ \quad (1,2-\sqrt{5}),(1,2+\sqrt{5})$
Asymptotes: $ \quad y-2=2(x-1), \quad y-2=-2(x-1)$
Work Step by Step
Complete the squares
$(y^{2}-4y+2^{2})-4(x^{2}+2x+1)=4+2^{2}-4$
$(y-2)^{2}-4(x-1)^{2}=4$
$\displaystyle \frac{(y-2)^{2}}{4}-\frac{(x-1)^{2}}{1}=1$
Hyperbola$, (h,k)=(1,2),a=2,b=1$
Table 4:$\begin{array}{cccc}
{\text { Foci }}&{\text { Vertices }}&{\text{Equation}}&{\text{asymptotes}}\\\hline
(h,k\displaystyle \pm c)&{(h,k\displaystyle \pm a)}&{\displaystyle \frac{(y-k)^{2}}{a^{2}}-\frac{(x-h)^{2}}{b^{2}}=1,}&{y-k=\displaystyle \pm\frac{a}{b}(x-h)}\end{array}$
$b^{2}=c^{2}-a^{2}\quad $
Find $c:$
$c^{2}=a^{2}+b^{2}=4+1=5$
$c=\sqrt{5}$
Center: $ \quad (1,2)$
Vertices: $ \quad (h,k\pm a)=(1,2\pm 2)$
Foci:$ \quad (h,k\pm c)=(1,2\pm\sqrt{5})$
Asymptotes: $y-k=\displaystyle \pm\frac{a}{b}(x-h)$
$y-2=\pm 2(x-1)$
