Let $A$ is $\left(1-3\mu , \mu -1, 2+5\mu \right)$

We know that the vector joining two points $\left(x_1, y_1, z_1\right)$ and $\left(x_2, y_2, z_2\right)$ is $\left(x_2-x_1\right)\hat{i}+\left(y_2-y_1\right)\hat{j}+\left(z_2-z_1\right)\hat{k}$

$\Rightarrow \overrightarrow{AB}=\left(3\mu +2\right)\widehat{i}+\left(3-\mu \right)\widehat{j}+\left(4-5\mu \right)\widehat{k}$ which is parallel to plane $x-4y+3z=1$

$\Rightarrow 1\left(3\mu +2\right)-4\left(3-\mu \right)+3\left(4-5\mu \right)=0$

$\Rightarrow 3\mu +2-12+4\mu +12-15\mu =0$

$\Rightarrow -8\mu +2=0$

$\Rightarrow \mu =\frac{1}{4}.$