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?

For any $x$, you can solve for exactly two solutions in $y$. One solution set has $y \geq 1$, the other $y \leq \frac{1}{2}$. This will give you the decomposition that witnesses that the set is disconnected.
