Let $L_{1}$ and $L_{2}$ denotes the lines $\bar{r} = \bar{i} + λ (- \bar{i} + 2 \bar{j} + 2 \bar{k}), λ \in R$ and $\bar{r} = μ (2 \bar{i} - \bar{j} + 2 \bar{k}), μ \in R$ respectively. If $L_{3}$ is a line which is perpendicular to both $L_{1}$ and $L_{2}$ and cuts both of them, which of the following options discribes $L_{3}$ ?

Moderate

A

$\bar{r} = \frac{1}{3} (2 \bar{i} + \bar{k}) + t (2 \bar{i} + 2 \bar{j} - \bar{k}), t \in R$
B

$\bar{r} = \frac{2}{9} (2 \bar{i} + \bar{j} + 2 \bar{k}) + t (2 \bar{i} + 2 \bar{j} - \bar{k}), t \in R$
C

$\bar{r} = t (2 \bar{i} + 2 \bar{j} - \bar{k}), t \in R$
D

$\bar{r} = \frac{2}{9} (4 \bar{i} + \bar{j} + \bar{k}) + t (2 \bar{i} + 2 \bar{j} - \bar{k}), t \in R$

Solution

Point on $L_{1}$ and $L_{2}$ are respectively
$A (1 - λ, 2 λ, 2 λ)$ and $B (2 μ, - μ, 2 μ)$
$\bar{A B} = (2 μ + λ - 1) \hat{i} + (- μ - 2 λ) \hat{j} + (2 μ - 2 λ) \hat{k}$
vector along their shortest distance $= 2 \hat{i} + 2 \hat{j} - \hat{k}$
Hence $\frac{2 μ + λ - 1}{2} = \frac{- μ - 2 λ}{2} = \frac{2 μ - 2 λ}{- 1}$
$\Rightarrow λ = \frac{1}{9}$ and $μ = \frac{2}{9}$

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