If the shortest distance between the lines x - γ / 2 = y - 4 / 3 = z - 3 / 4 and x - 2 / 4 = y - 4 / 6 = z - 7 / 8

Question

If the shortest distance between the lines \(\frac{\mathrm{x}-\lambda}{2}=\frac{\mathrm{y}-4}{3}=\frac{\mathrm{z}-3}{4}\) and

\(\frac{x-2}{4}=\frac{y-4}{6}=\frac{z-7}{8}\) is \(\frac{13}{\sqrt{29}}\), then a value of \(\lambda\) is : 

(1) \(-\frac{13}{25}\)

(2) \(\frac{13}{25}\)

(3) 1

(4) -1

Answer 1

Correct option is : (3) 1 

\(\left.\begin{array}{l}\overline{\mathrm{r}}_{1}=(\lambda \hat{i}+4 \hat{j}+3 \hat{k})+\alpha(2 \hat{i}+3 \hat{j}+4 \hat{k}) \\ \bar{r}_{2}=(2 \hat{i}+4 \hat{j}+7 \hat{k})+\beta(2 \hat{i}+3 \hat{j}+4 \hat{k})\end{array}\right\} \begin{aligned} & \bar{b}=2 \hat{i}+3 \hat{j}+4 \hat{k} \\ & \bar{a}_{1}+\lambda \hat{i}+4 \hat{j}+3 \hat{k} \\ & \bar{a}_{2}=2 \hat{i}+4 \hat{j}+7 \hat{k}\end{aligned}\)

Shortest dist. \(=\frac{\left|\overline{\mathrm{b}} \times\left(\overline{\mathrm{a}}_{2}-\overline{\mathrm{a}}_{1}\right)\right|}{|\mathrm{b}|}=\frac{13}{\sqrt{29}}\)

\(\frac{|(2 \hat{i}+3 \hat{j}+4 \hat{k}) \times((2-\lambda) \hat{i}+4 \hat{k})|}{\sqrt{29}}=\frac{13}{\sqrt{29}}\)

\(|-8 \hat{\mathrm{j}}-3(2-\lambda) \hat{\mathrm{k}}+12 \hat{\mathrm{i}}+4(2-\lambda) \hat{\mathrm{j}}|=13\)

\(|12 \hat{\mathrm{i}}-4 \lambda \hat{\mathrm{j}}+(3 \lambda-6) \hat{\mathrm{k}}|=13\)

\(144+16 \ \lambda^{2}+(3 \lambda-6)^{2}=169\)

\(16 \lambda^{2}+(3 \lambda-6)^{2}=25\)

\(\Rightarrow\lambda =1\)