jmc

The standard form for the equation of a vertically oriented hyperbola centered at $(h,k)$ is $$\frac{(y-k{)}^2}{a^2}-\frac{(x-h{)}^2}{b^2}=1.$$But the given equation is not in standard form, because the terms $2y$ and $3x$ appear instead of $y$ and $x.$ So we factor out $2^2$ and $3^2$ from the two terms on the left-hand side, giving $$\frac{2^2(y-1{)}^2}{5^2}-\frac{3^2(x-\frac{4}{3}{)}^2}{4^2}=1,$$or $$\frac{(y-1{)}^2}{{\left(\frac{5}{2}\right)}^2}-\frac{(x-\frac{4}{3}{)}^2}{{\left(\frac{4}{3}\right)}^2}=1.$$This equation is in standard form, so we can read off the center of the hyperbola as $\boxed{\left(\frac{4}{3},1\right)}.$