Answer
See graph
Work Step by Step
We are given the hyperbola:
$(x-1)^2-(y-2)^2=3$
Bring the equation to the standard form:
$\dfrac{(x-h)^2}{a^2}-\dfrac{(y-k)^2}{b^2}=1$
$\dfrac{(x-1)^2}{3}-\dfrac{(y-2)^2}{3}=1$
The transverse axis is parallel to the $x$-axis.
Determine $h,k,a,b,c$:
$h=1$
$k=2$
$a^2=3\Rightarrow a=\sqrt 3$
$b^2=3\Rightarrow b=\sqrt 3$
$c^2=a^2+b^2$
$c^2=3+3$
$c^2=6$
$c=\sqrt 6$
The centre of the hyperbola is:
$(h,k)=(1,2)$
Determine the coordinates of the vertices:
$(h-a,k)=(1-\sqrt 3,2)$
$(h+a,k)=(1+\sqrt 3,2)$
Determine the coordinates of the foci:
$(h-c,k)=(1-\sqrt{6},2)$
$(h+c,k)=(1+\sqrt{6},2)$
Determine the asymptotes:
$y-k=\pm\dfrac{b}{a}(x-h)$
$y-2=\pm\dfrac{\sqrt 3}{\sqrt 3}(x-1)$
$y-2=\pm(x-1)$
Graph the hyperbola: