Answer
$\dfrac{y^2}{9}-\dfrac{x^2}{4}=1$
Work Step by Step
The standard form of the equation of the hyperbola with a horizontal transverse axis can be expressed as: $\dfrac{(x-h)^2}{a^2}-\dfrac{(y-k)^2}{b^2}=1$
The vertices and foci have the form $(\pm a, 0) $ and $(\pm c,0)$.
The standard form of the equation of the hyperbola with a vertical
transverse axis can be expressed as: $\dfrac{(y-k)^2}{a^2}-\dfrac{(x-h)^2}{b^2}=1$
The vertices and foci have the form $(0, \pm, a) $ and $(0, \pm c)$.
The center is the midpoint of the vertices: $(4,2)$
We have: $ a=3$
Now, $\dfrac{a}{b}=\dfrac{3}{2}$
$\implies b=2$
$\dfrac{(y-0)^2}{3^2}-\dfrac{(x-0)^2}{2^2}=1$
or, $\dfrac{y^2}{9}-\dfrac{x^2}{4}=1$
