Let a = i +2 j + k and b =2 i +7 j +3 k . Let L_ 1 : r =(- i +2 j… - Vidyadip

MCQ

Let $\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{b}}=2 \hat{\mathrm{i}}+7 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}$. 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:

A
$(8,26,12)$
B
$(2,8,5)$
C
$(-1,-1,1)$
D
$(5,17,4)$

✓

Answer

(A)
Sol. $L_{1}: \overrightarrow{\mathrm{r}}=(-\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}})+\lambda(\hat{i}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}})$
$\Rightarrow \overrightarrow{\mathrm{r}}=(\lambda-1) \hat{\mathrm{i}}+2(\lambda+1) \hat{\mathrm{j}}+(\lambda+1) \hat{\mathrm{k}}$
$L_{2}: \overrightarrow{\mathrm{r}}=(\hat{\mathrm{j}}+\hat{\mathrm{k}})+\mu(2 \hat{\mathrm{i}}+7 \hat{\mathrm{j}}+3 \hat{\mathrm{k}})$
$\Rightarrow \overrightarrow{\mathrm{r}}=2 \mu \hat{\mathrm{i}}+(1+7 \mu) \hat{\mathrm{j}}+(1+3 \mu) \hat{\mathrm{k}}$
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 $\mu=1$
$\Rightarrow \overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}=3 \hat{\mathrm{i}}+9 \hat{\mathrm{j}}+4 \hat{\mathrm{k}}$
$L_{3}: \overrightarrow{\mathrm{r}}=2 \hat{\mathrm{i}}+8 \hat{\mathrm{j}}+4 \hat{\mathrm{k}}+\alpha(3 \hat{\mathrm{i}}+9 \hat{\mathrm{j}}+4 \hat{\mathrm{k}})$
For $\alpha=2, \overrightarrow{\mathrm{r}}=8 \hat{\mathbf{i}}+26 \hat{\mathrm{j}}+12 \hat{\mathrm{k}}$

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