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?

Your set $S$ contains the points $(0,{1\over2})$ and $(0,1)$, but does not contain any points on the line $y={3\over4}$. Therefore $$S\cap\{(x,y)|y<{3\over4}\},\quad S\cap\{(x,y)|y>{3\over4}\}$$
is a partition of $S$ into disjoint (relatively) open subsets of ${\mathbb R}^2$. This shows that $S$ is not connected.
