Answer
See graph
Work Step by Step
We are given the hyperbola:
$\dfrac{(y-2)^2}{36}-\dfrac{(x+1)^2}{49}=1$
The equation is in the standard form:
$\dfrac{(y-k)^2}{a^2}-\dfrac{(x-h)^2}{b^2}=1$
The transverse axis is parallel to the $y$-axis.
Determine $h,k,a,b,c$:
$h=-1$
$k=2$
$a^2=36\Rightarrow a=\sqrt{36}=6$
$b^2=49\Rightarrow b=\sqrt {49}=7$
$c^2=a^2+b^2$
$c^2=36+49$
$c^2=85$
$c=\sqrt{85}$
The centre of the hyperbola is:
$(h,k)=(-1,2)$
Determine the coordinates of the vertices:
$(h,k-a)=(-1,2-6)=(-1,-4)$
$(h,k+a)=(-1,2+6)=(-1,8)$
Determine the coordinates of the foci:
$(h,k-c)=(-1,2-\sqrt{85})$
$(h,k+c)=(-1,2+\sqrt{85})$
Determine the asymptotes:
$y-k=\pm\dfrac{a}{b}(x-h)$
$y-2=\pm\dfrac{6}{7}(x+1)$
$y+2=\pm\dfrac{1}{2}(x-1)$
Graph the hyperbola:
