Let L1 and L2 be the following straight lines.L1:x−11=y−1=z−13 and L2:x−1−3=y−1=z−11. Suppose the straight line L:x−αl=y−1m=z−γ−2 lies in the plane containing L1 and L2 and passes through the point of intersection of L1 and L2. If the line L bisects the acute angle between the lines L1 and L2 , then the value of (2α+γ+l+m) is_______

Question

Let  L1  and L2  be the following straight lines.L1:x−11=y−1=z−13 and  L2:x−1−3=y−1=z−11. Suppose the straight line L:x−αl=y−1m=z−γ−2  lies in the plane containing L1  and L2 and passes through the point of intersection of L1 and  L2. If the line L bisects the acute angle between the lines L1  and L2 , then the value of  (2α+γ+l+m)  is_______

Infinity Learn · Accepted Answer

Vector equation of the given straight lines are
$r = (\hat{i} + \hat{k}) + λ (\hat{i} - \hat{j} + 3 \hat{k})$ and $r = (\hat{i} + \hat{k}) + v (- 3 \hat{i} - \hat{j} + \hat{k})$
$∵ (\hat{i} - \hat{j} + 3 \hat{k}) . (- 3 \hat{i} - \hat{j} + \hat{k}) = - 3 + 1 + 3 = 1$ is positive,
$∴$ Angle between supporting line vectors of lines $L_{1}$ and $L_{2}$ is acute, and point of intersection of given lies $L_{1}$ and $L_{2}$ is (1, 0, 1).
vector along the acute angle bisector of vectors
$(\hat{i} - \hat{j} + 3 \hat{k})$ and $(- 3 \hat{i} - \hat{j} + \hat{k})$ is $(\hat{i} - \hat{j} + 2 \hat{k})$ or $(\hat{i} + \hat{j} - 2 \hat{k})$ .
It is given that line $L : \frac{x - α}{l} = \frac{y - 1}{m} = \frac{z - γ}{- 2}$ is the bisector of the acute angle between the lines $L_{1}$ and $L_{2}$ , so $l = 1$ and $m = 1$ and $\frac{1 - α}{1} = \frac{0 - 1}{1} = \frac{1 - γ}{- 2}$
$\Rightarrow α = 2, γ = - 1 ∴ α - γ = 3, l + m = 2$

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