Let vector a=i+2j+k and vector b=2i+7j+3k. Let L1: vector r=(-i+2j+k)+λvector a, λ ∈ R

Question

Let \(\vec{a}=\hat{i}+2 \hat{j}+\hat{k}\) and \(\vec{b}=2 \hat{i}+7 \hat{j}+3 \hat{k}.\) Let

\(L_{1}: \overrightarrow{\mathrm{r}}=(-\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}})+\lambda \overrightarrow{\mathrm{a}}, \lambda \in \mathrm{R}\) and

\(L_{2}: \vec{r}=(\hat{j}+\hat{k})+\mu \vec{b}, \mu \in R\) be two lines. If the line \(L_{3}\) passes through the point of intersection of \(L_{1}\) and \(L_{2}\), and is parallel to \(\vec{a}+\vec{b},\) then \(L_{3}\) passes through the point:

(1) (8, 26, 12)

(2) (2, 8, 5)

(3) (-1, -1, 1)

(4) (5, 17, 4)

Answer 1

Correct option is (1) (8, 26, 12) 

\(L_{1}: \overrightarrow{\mathrm{r}}=(-\hat{i}+2 \hat{j}+\hat{k})+\lambda(\hat{i}+2 \hat{j}+\hat{k})\)

\(\Rightarrow \overrightarrow{\mathrm{r}}=(\lambda-1) \hat{\mathrm{i}}+2(\lambda+1) \hat{\mathrm{j}}+(\lambda+1) \hat{\mathrm{k}}\)

\(L_{2}: \overrightarrow{\mathrm{r}}=(\hat{\mathrm{j}}+\hat{\mathrm{k}})+\mu(2 \hat{\mathrm{i}}+7 \hat{\mathrm{j}}+3 \hat{\mathrm{k}})\)

\(\Rightarrow \overrightarrow{\mathrm{r}}=2 \mu \hat{\mathrm{i}}+(1+7 \mu) \hat{\mathrm{j}}+(1+3 \mu) \hat{\mathrm{k}}\)

For point of intersection equating respective components

\(\Rightarrow \lambda-1=2 \mu\)   .....(1)

\(2(\lambda+1)=1+7 \mu\)     .....(2)

\(\lambda+1=1+3 \mu\)      .....(3)

We get

\(\Rightarrow \lambda=3\) and \(\mu=1\)

\(\Rightarrow \overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}=3 \hat{\mathrm{i}}+9 \hat{\mathrm{j}}+4 \hat{\mathrm{k}}\)

\(L_{3}: \vec{r}=2 \hat{i}+8 \hat{j}+4 \hat{k}+\alpha(3 \hat{i}+9 \hat{j}+4 \hat{k})\)

For \(\alpha=2, \overrightarrow{\mathrm{r}}=8 \hat{\mathbf{i}}+26 \hat{\mathrm{j}}+12 \hat{\mathrm{k}}\)