The required plane must contain $z$-axis and parallel to normal vector of the given plane
A parallel vector to $z$-axis is $<\mathrm{0,0},1>$
Normal vector to the given plane is $<3,-\mathrm{4,1}>$
A normal vector to the required plane $=\left|\begin{array}{ccc}\widehat{i}& \widehat{j}& \widehat{k}\\ 0& 0& 1\\ 3& -4& 1\end{array}\right|$
$=4\widehat{i}+3\widehat{j}$
$\because$ Required plane contain $z$ axis, so it must pass through $\left(\mathrm{0,0},0\right)$
Equation of the required plane is $4\left(x-0\right)+3\left(y-0\right)=0\Rightarrow 4x+3y=0$
So, the perpendicular distance from $\left(\mathrm{1,2},3\right)$ to $4x+3y=0$ is $\left|\frac{4+6}{5}\right|=2$ units