Direction of $L$ is the cross product of the two given vectors:

$\left|\begin{array}{c}\hat{i}\hat{j}\hat{k}\\ 122\\ 221\end{array}\right|=\hat{i}(2-4)-\hat{j}(1-4)+\hat{k}(2-4)=-2\hat{i}+3\hat{j}-2\hat{k}$

Parameterise $L$ as $(1,1,1)+\lambda (-2,3,-2)$. A general point is $M=(1-2\lambda ,1+3\lambda ,1-2\lambda )$ (rewriting with direction $(2,-3,2)$ as in the source gives $M=(2\lambda +1,-3\lambda +1,2\lambda +1)$).

For $M$ to be the foot of perpendicular from origin, $\overrightarrow{OM}$ must be orthogonal to the direction:

$2(2\lambda +1)-3(-3\lambda +1)+2(2\lambda +1)=0\Rightarrow 4\lambda +2+9\lambda -3+4\lambda +2=0\Rightarrow 17\lambda +1=0\Rightarrow \lambda =-\frac{1}{17}$

Hence $(a,b,c)=\left(\frac{-2}{17}+1,\frac{3}{17}+1,\frac{-2}{17}+1\right)=\left(\frac{15}{17},\frac{20}{17},\frac{15}{17}\right)$.

$34(a+b+c)=34\cdot \frac{50}{17}=100$.