Dimension of Fibres

Question

I want to check the dimensions of fibres of projections of $V=Z(y_0(x_0^2-x_1^2)+y_1x_1x_2) \subseteq \mathbb{P}^2 \times \mathbb{P}^1$, $p : V \rightarrow \mathbb{P}^2, q : V \rightarrow \mathbb{P}^1$. Is it true that dimension of all fibres of $q$ is $1$ since it is given by one polynomial equation (a hypersurface of $\mathbb{P}^2$)? What about dimension of fibres of $p$? My guess for $p$ the dimension is $0$ except for $[0:0:1]$. Thank you.