Let . Let and be two lines. if the line passes through the point of intersection of and , and is parallel to , then passes through the point: [2025]

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Solution

Q.
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 $L_{3}$ 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: [2025]

1 (2, 8, 5)

2 (–1, –1, 1)

3 (5, 17, 4)

4 (8, 26, 12)

Ans.
(4)

We have, $L_{1} : - \hat{i} + 2 \hat{j} + \hat{k} + λ (\hat{i} + 2 \hat{j} + \hat{k}) and L_{2} : \hat{j} + \hat{k} + μ (2 \hat{i} + 7 \hat{j} + 3 \hat{k})$

Point of intersection of $L_{1}$ & $L_{2}$ is given by

$(- 1 + λ) \hat{i} + (2 + 2 λ) \hat{j} + (1 + λ) \hat{k} = 2 μ \hat{i} + (1 + 7 μ) \hat{j} + (1 + 3 μ) \hat{k}$

$\Rightarrow - 1 + λ = 2 μ, 2 + 2 λ = 1 + 7 μ, 1 + λ = 1 + 3 μ$

$\Rightarrow μ = 1 and λ = 3$

$∴$ Position vector of point of intersection of $L_{1}$ and $L_{2}$ is $2 \hat{i} + 8 \hat{j} + 4 \hat{k}$ .

Hence, $L_{3}$ is given by

$2 \hat{i} + 8 \hat{j} + 4 \hat{k} + γ (\vec{a} + \vec{b}) = 2 \hat{i} + 8 \hat{j} + 4 \hat{k} + γ (3 \hat{i} + 9 \hat{j} + 4 \hat{k})$

For $γ = 2,$ line $L_{3}$ passes through point (8, 26, 12).

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