Fibers of $V(ax_0^2+bx_1x_2)$ in $\mathbb{P}^1\times \mathbb{P}^2$

Question

What are the fibers of $Z=V(ax_0^2+bx_1x_2)$ for $(a:b)\in \mathbb{P}^1, (x_0:x_1,x_2)\in \mathbb{P}^2$?

If we fix $(a:b)$ or $(x_0:x_1,x_2)$ and dehomogenize, we can see that $Z$ is a family of curves or lines respectively. Do these describe the fibers of $Z$?