Let the line be parallel to the vector and pass through the point (2,6,7), and the line be parallel to the vector and pass through the point (4,3,5). If the line is parallel to the vector and intersects the lines at the points C and D, respe

Question

Let the line $L_{1}$ be parallel to the vector $- 3 \hat{i} + 2 \hat{j} + 4 \hat{k}$ and pass through the point (2,6,7), and the line $L_{2}$ be parallel to the vector $2 \hat{i} + \hat{j} + 3 \hat{k}$ and pass through the point (4,3,5). If the line $L_{3}$ is parallel to the vector $- 3 \hat{i} + 5 \hat{j} + 16 \hat{k}$ and intersects the lines $L_{1} and L_{2}$ at the points C and D, respectively, then $| \vec{C D} |^{2}$ is equal to: [2026]

Smart Achievers · Accepted Answer

(3)

$L_{1} : \frac{x - 2}{- 3} = \frac{y - 6}{2} = \frac{z - 7}{4}$

Point $C$ on $L_{1}$ : $(- 3 λ_{1} + 2, 2 λ_{1} + 6, 4 λ_{1} + 7)$

$L_{2} : \frac{x - 4}{2} = \frac{y - 3}{1} = \frac{z - 5}{3}$

Point $D$ on $L_{2}$ : $(2 λ_{2} + 4, λ_{2} + 3, 3 λ_{2} + 5)$

D.R's of line $L_{3}$ :

$L_{3} : \frac{2 λ_{2} + 3 λ_{1} + 2}{- 3} = \frac{λ_{2} - 2 λ_{1} - 3}{5} = \frac{3 λ_{2} - 4 λ_{1} - 2}{16}$

$λ_{1} = - 3, λ_{2} = 2$

$C (11, 0, - 5)$

$D (8, 5, 11)$

$| \vec{C D} |^{2} = 3^{2} + 5^{2} + 16^{2} = 290$

Solution

1 171

2 89

3 290

4 312

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