Let λ be an integer. If the shortest distance between the lines x - λ = 2 y - 1 = - 2 z and x = y…

$\frac{x-\lambda }{1}=\frac{y-\frac{1}{2}}{\frac{1}{2}}=\frac{z-0}{-\frac{1}{2}}$

$\frac{x-0}{1}=\frac{y+2\lambda }{1}=\frac{z-\lambda }{1}$

Shortest distance$=\frac{\left(\overrightarrow{a_2}-\overrightarrow{a_1}\right)·\left(\overrightarrow{b_1}\times \overrightarrow{b_2}\right)}{\left|\overrightarrow{b_1}\times \overrightarrow{b_2}\right|}$

$\overrightarrow{b_1}\times \overrightarrow{b_2}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 1& \frac{1}{2}& -\frac{1}{2}\\ 1& 1& 1\end{array}\right|$

$=\hat{\mathrm{i}}\left(\frac{1}{2}+\frac{1}{2}\right)-\hat{\mathrm{j}}\left(1+\frac{1}{2}\right)+\hat{\mathrm{k}}\left(1-\frac{1}{2}\right)$

$=\hat{\mathrm{i}}-\frac{3}{2}\hat{\mathrm{j}}+\frac{\hat{\mathrm{k}}}{2}=\frac{2\hat{\mathrm{i}}-3\hat{\mathrm{j}}+\hat{\mathrm{k}}}{2}$

$\frac{\left(\overrightarrow{b_1}\times \overrightarrow{b_2}\right)}{\left|\overrightarrow{b_1}\times \overrightarrow{b_2}\right|}=\frac{2\hat{\mathrm{i}}-3\hat{\mathrm{j}}+\hat{\mathrm{k}}}{\sqrt{14}}$

$\frac{\left(\overrightarrow{a_2}-\overrightarrow{a_1}\right)·\left(\overrightarrow{b_1}\times \overrightarrow{b_2}\right)}{\left|\overrightarrow{b_1}\times \overrightarrow{b_2}\right|}=$

$\left(-\lambda \hat{\mathrm{i}}+\left(-2\lambda +\frac{1}{2}\right)\hat{\mathrm{j}}+\lambda \hat{\mathrm{k}}\right)·\left(\frac{2\hat{\mathrm{i}}-3\hat{\mathrm{j}}+\hat{\mathrm{k}}}{\sqrt{14}}\right)$

$=\left|\frac{-2\lambda +6\lambda -\frac{3}{2}+\lambda }{\sqrt{14}}\right|=\frac{\sqrt{7}}{2\sqrt{2}}$

$\left|5\lambda -\frac{3}{2}\right|=\frac{7}{2}$

$5\lambda =\frac{3}{2}\pm \frac{7}{2}$

$5\lambda =5,-2$

$\lambda =1,-\frac{2}{5}$