Let \(\vec{\mathrm{a}}=\overset{^}{\mathrm{i}}+2\overset{^}{\mathrm{j}}+\overset{^}{\mathrm{k}}\text{ and }\vec{\mathrm{…

Given

\({L}_{1}:\vec{\mathrm{r}}=(-\overset{^}{\mathrm{i}}+2\overset{^}{\mathrm{j}}+\overset{^}{\mathrm{k}})+\lambda (\overset{^}{\mathrm{i}}+2\overset{^}{\mathrm{j}}+\overset{^}{\mathrm{k}})\\ \Rightarrow \vec{\mathrm{r}}=(\lambda -1)\overset{^}{\mathrm{i}}+2(\lambda +1)\overset{^}{\mathrm{j}}+(\lambda +1)\overset{^}{\mathrm{k}}\)

Next,

\({L}_{2}:\vec{\mathrm{r}}=(\overset{^}{\mathrm{j}}+\overset{^}{\mathrm{k}})+\mu (2\overset{^}{\mathrm{i}}+7\overset{^}{\mathrm{j}}+3\overset{^}{\mathrm{k}})\\ \Rightarrow \vec{\mathrm{r}}=2\mu \overset{^}{\mathrm{i}}+(1+7\mu )\overset{^}{\mathrm{j}}+(1+3\mu )\overset{^}{\mathrm{k}}\)

for point of intersection of \({L}_{1}\text{and}{L}_{2}\)

\(\lambda -1=2\mu \text{and}2(\lambda +1)=1+7\mu \\ \text{On solving, }\lambda =3\text{ and }\mu =1\\ \Rightarrow \vec{\mathrm{a}}+\vec{\mathrm{b}}=3\overset{^}{\mathrm{i}}+9\overset{^}{\mathrm{j}}+4\overset{^}{\mathrm{k}}\\ \text{hence, }\\ {\mathrm{L}}_{3}:\vec{\mathrm{r}}=2\overset{^}{\mathrm{i}}+8\overset{^}{\mathrm{j}}+4\overset{^}{\mathrm{k}}+\alpha (3\overset{^}{\mathrm{i}}+9\overset{^}{\mathrm{j}}+4\overset{^}{\mathrm{k}})\\ \text{from options, we can see}{\mathrm{L}}_{3}\mathrm{passes}\\ \mathrm{through}(8,26,12)\)