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