Is the set $\{(x, y) : 3x^2 − 2y^ 2 + 3y = 1\}$ connected? Is the set $\{(x, y)\in\mathbb{R}^2 : 3x^2 − 2y^ 2 + 3y = 1\}$ connected?
I have checked that it is an hyperbola, hence disconnected am i right?

Assuming, you mean 
$$S=\{(x,y)\in\mathbb R^2:3x^2-2y^2+3y=1\},  $$
observe that $f\colon(x,y)\mapsto y-\frac23$ is a continuos map. There is no point $(x,y)\in S$ with $f(x,y)=0$ because $3x^2-2\cdot \frac49+3\cdot \frac23=1$ (i.e. $3x^2=-\frac19$) has no real solution. On the other hand, $(0,1)\in S$ and $(0,\frac12)\in S$, so that $f$ does take both positive ans negative values. Therefore we can write $S=f^{-1}((-\infty,0))\cup f^{-1}((0,\infty))$ as disjoint union of nonempty open sets, i.e. $S$ is not connected.
