If the plane contains the given line then it must pass through $\left(3,b,3\right)$ and should be parallel to $2\widehat{i}+4\widehat{j}+3\widehat{k}$
Also, the plane passes through the points $\left(a,1,2\right)\left(2,1,4\right)\left(2,3,5\right)$.
Equation of the plane is
$\left|\begin{array}{ccc}x-2& y-1& z-4\\ 0& 2& 1\\ 2& 4& 3\end{array}\right|=0$
$\Rightarrow \left(x-2\right)\left(2\right)-\left(y-1\right)\left(-2\right)+\left(z-4\right)\left(-4\right)=0$
$\Rightarrow x-2+y-1-2z+8=0$
$\Rightarrow x+y-2z+5=0$
$\therefore$ points $\left(3,b,3\right)\left(a,1,2\right)$ satisfy the equation of the plane, if
$3+b-6+5=0\Rightarrow b=-2$
$a+1-4+5=0\Rightarrow a=-2$