Answer
See graph
Work Step by Step
We are given the hyperbola:
$(y-2)^2-(x+3)^2=5$
Bring the equation to the standard form:
$\dfrac{(y-k)^2}{a^2}-\dfrac{(x-h)^2}{b^2}=1$
$\dfrac{(y-2)^2}{5}-\dfrac{(x+3)^2}{5}=1$
The transverse axis is parallel to the $y$-axis.
Determine $h,k,a,b,c$:
$h=-3$
$k=2$
$a^2=5\Rightarrow a=\sqrt {5}$
$b^2=5\Rightarrow b=\sqrt 5$
$c^2=a^2+b^2$
$c^2=5+5$
$c^2=10$
$c=\sqrt{10}$
Determine the coordinates of the vertices:
$(h,k-a)=(-3,2-\sqrt 5)$
$(h,k+a)=(-3,2+\sqrt 5)$
Determine the coordinates of the foci:
$(h,k-c)=(-3,2-\sqrt{10})$
$(h,k+c)=(-3,2+\sqrt{10})$
Determine the asymptotes:
$y-k=\pm\dfrac{a}{b}(x-h)$
$y-2=\pm\dfrac{\sqrt 5}{\sqrt 5}(x+3)$
$y-2=\pm (x+3)$
Graph the hyperbola:
