Let a→=i^+2j^+k^ and b→=2i^+7j^+3k^. Let L1:r→=(-i^+2j^+k^)+λa→,λ∈R and L2:r→=(j^+k^)+μb→,μ∈R be two lines. If the line L3 passes through the point of intersection of L1 and L2, and is parallel to a→+b→, then L3 passes through the point:

L1:r→=(−i^+2j^+k)+λ(i^+2j^+k)⇒r→=(λ−1)i^+2(λ+1)j^+(λ+1)k^L2:r→=(j^+k^)+μ(2i^+7j^+3k^)⇒r→=2μi^+(1+7μ)j^+(1+3μ)k^For point of intersection equating respective components⇒λ−1=2μ2(λ+1)=1+7μλ+1=1+3μWe get⇒λ=3 and μ=1⇒a→+b→=3i^+9j^+4k^L3:r→=2i^+8j^+4k^+α(3i^+9j^+4k^)For α=2,r→=8i^+26j^+12k^

Let a→=i^+2j^+k^ and b→=2i^+7j^+3k^. Let L1:r→=(-i^+2j^+k^)+λa→,λ∈R and L2:r→=(j^+k^)+μb→,μ∈R be two lines. If the line L3 passes through the point of intersection of L1 and L2, and is parallel to a→+b→, then L3 passes through the point:

L1:r→=(−i^+2j^+k)+λ(i^+2j^+k)⇒r→=(λ−1)i^+2(λ+1)j^+(λ+1)k^L2:r→=(j^+k^)+μ(2i^+7j^+3k^)⇒r→=2μi^+(1+7μ)j^+(1+3μ)k^For point of intersection equating respective components⇒λ−1=2μ2(λ+1)=1+7μλ+1=1+3μWe get⇒λ=3 and μ=1⇒a→+b→=3i^+9j^+4k^L3:r→=2i^+8j^+4k^+α(3i^+9j^+4k^)For α=2,r→=8i^+26j^+12k^

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Let $\vec{a} = \hat{i} + 2 \hat{j} + \hat{k}$ and $\vec{b} = 2 \hat{i} + 7 \hat{j} + 3 \hat{k} .$ Let $L_{1} : \vec{r} = (- \hat{i} + 2 \hat{j} + \hat{k}) + λ \vec{a}, λ \in R$ and $L_{2} : \vec{r} = (\hat{j} + \hat{k}) + μ \vec{b}, μ \in R$ be two lines. If the line L₃ passes through the point of intersection of L₁ and L₂, and is parallel to $\vec{a} + \vec{b}$ , then L₃ passes through the point:

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(2,8, 5)

(–1, –1, 1)

(5, 17, 4)

(8, 26, 12)

answer is A.

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Detailed Solution

$\begin{array}{l} L_{1} : \vec{r} = (- \hat{i} + 2 \hat{j} + k) + λ (\hat{i} + 2 \hat{j} + k) \\ \Rightarrow \vec{r} = (λ - 1) \hat{i} + 2 (λ + 1) \hat{j} + (λ + 1) \hat{k} \\ L_{2} : \vec{r} = (\hat{j} + \hat{k}) + μ (2 \hat{i} + 7 \hat{j} + 3 \hat{k}) \\ \Rightarrow \vec{r} = 2 μ \hat{i} + (1 + 7 μ) \hat{j} + (1 + 3 μ) \hat{k} \end{array}$
For point of intersection equating respective components
$\begin{array}{l} \Rightarrow λ - 1 = 2 μ \\ 2 (λ + 1) = 1 + 7 μ \\ λ + 1 = 1 + 3 μ \end{array}$
We get
$\begin{array}{l} \Rightarrow λ = 3 and μ = 1 \\ \Rightarrow \vec{a} + \vec{b} = 3 \hat{i} + 9 \hat{j} + 4 \hat{k} \\ L_{3} : \vec{r} = 2 \hat{i} + 8 \hat{j} + 4 \hat{k} + α (3 \hat{i} + 9 \hat{j} + 4 \hat{k}) \\ For α = 2, \vec{r} = 8 \hat{i} + 26 \hat{j} + 12 \hat{k} \end{array}$