Solution
First line $\mathrm{L}_{1}$ passes through point $\mathrm{A}(-1,2,4)$ with drs $\alpha,-1,-\alpha$
First line $\mathrm{L}_{1}$ passes through point $\mathrm{B}(0,1,1)$ with drs $\alpha, 2,2 \alpha$
shortest distance $\mathrm{b} / \mathrm{w}$ skew lines $=\frac{\left|\begin{array}{ccc}\mathrm{x}_{2}-\mathrm{x}_{1} & \mathrm{y}_{2}-\mathrm{y}_{1} & \mathrm{z}_{2}-\mathrm{z}_{1} \\ \mathrm{a}_{1} & \mathrm{~b}_{1} & \mathrm{c}_{1} \\ \mathrm{a}_{2} & \mathrm{~b}_{2} & \mathrm{c}_{2}\end{array}\right|}{\left\|\begin{array}{ccc}\hat{\mathrm{i}} & \hat{\mathrm{j}} & \hat{\mathrm{k}} \\ \mathrm{a}_{1} & \mathrm{~b}_{1} & \mathrm{c}_{1} \\ \mathrm{a}_{2} & \mathrm{~b}_{2} & \mathrm{c}_{2}\end{array}\right\|}$
\[
=\frac{\left|\begin{array}{ccc}
1 & -1 & -3 \\
\alpha & -1 & -\alpha \\
\alpha & 2 & 2 \alpha
\end{array}\right|}{\left|\begin{array}{ccc}
\hat{\mathrm{i}} & \hat{\mathrm{j}} & \hat{\mathrm{k}} \\
\alpha & -1 & -\alpha \\
\alpha & 2 & 2 \alpha
\end{array}\right|}=\sqrt{2}
\]
\[
\begin{aligned}
& \Rightarrow \frac{\left|(-2 \alpha+2 \alpha)+1\left(2 \alpha^{2}+\alpha^{2}\right)-3(3 \alpha)\right|}{\left|0-3 \alpha^{2} \hat{j}+3 \alpha \hat{k}\right|}=\sqrt{2} \\
& \Rightarrow \frac{\left|3 \alpha^{2}-9 \alpha\right|}{\sqrt{9 \alpha^{4}+9 \alpha^{2} \mid}}=\sqrt{2} \Rightarrow \frac{3|\alpha-3 \| \alpha|}{3 \alpha \sqrt{\alpha^{2}+1}}=\sqrt{2} \\
& \Rightarrow|\alpha-3|=\sqrt{2} \sqrt{\alpha^{2}+1} \Rightarrow \alpha^{2}+9-6 \alpha=2 \alpha^{2}+2 \\
& \Rightarrow \alpha^{2}+6 \alpha-7=0 \Rightarrow(\alpha+7)(\alpha-1)=0 \\
& \alpha=-7,1 \text { sum of } \alpha=-6 \text { Answer(2) }
\end{aligned}
\]
.
