Answer
Center: $(1,2)$
Foci: $(1\pm \sqrt 2, 2)$
Vertices: $(0, 2)$ and $(2,2)$
Asymptote: $y=x+1 $ and $y=-x+3$
Work Step by Step
$x^2 -y^2-2x+4y=4$
$(x^2 -2x + \textbf{1}) -(y^2 -4y+\textbf{4})=4 \textbf {+1 - 4}$
$(x-1)^2 - (y-2)^2 = 1$
Hyperbola form: $\frac{(x-1)^2}{1^2} - \frac{(y-2)^2}{1^2} =1$
$\qquad \qquad \qquad \qquad \quad \enspace h=1, k=2, a=1, b=1$
Linear eccentricity (focal distance): $c=\sqrt {a^2 +b^2}$
$\qquad \qquad \qquad \qquad \qquad \qquad \quad c=\sqrt {1^2 +1^2}$
$\qquad \qquad \qquad \qquad \qquad \qquad \quad c=\sqrt {2}$
Center: $(h,k)$
$\qquad \quad (1,2)$
Foci: $(h \pm c, k)$
$\qquad (1\pm \sqrt 2, 2)$
Vertices: $(h \pm a, k)$
$\qquad \quad \enspace (1 \pm 1, 2)$
$\qquad \quad \enspace (0, 2)$ and $(2,2)$
Asymptote: $y=\pm\frac{b}{a} (x-h)+k$
$\qquad \qquad \enspace y=\pm \frac{1}{1} (x-1) +2 $
$\qquad \qquad \enspace y=x+1 $ and $y=-x+3$
