Everything is over $\mathbb C$. Let $f:X \to \mathbb P^2$ be a flat projective morphism and $x_0\in \mathbb P^2$ be a fixed point in $\mathbb P^2$, and all the lines ($\mathbb P^1$) through it are denoted by $\{L_\alpha\}_{\alpha \in \mathbb P^1}$. We assume that $X$ is smooth, and all the fibers $f^{-1}(x)$ except for $x=x_0$ are smooth. I would like to know if the following is true:
There exists at least one $L_\alpha$ such that $f^{-1}(L_\alpha)$ is smooth.
I tried to use some sort of Bertini theorem, which says if a linear system has no base point, then a general divisor would be smooth. However, here $\{f^{-1}(L_\alpha)\}$ certainly contains base locus $f^{-1}(x_0)$. So I guess the statement is not true. But I want an example to see how this fails.
Thanks in advance.