If the line makes a right angle with the line , then is equal to : [2024]

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Solution

Q.
If the line $\frac{2 - x}{3} = \frac{3 y - 2}{4 λ + 1} = 4 - z$ makes a right angle with the line $\frac{x + 3}{3 μ} = \frac{1 - 2 y}{6} = \frac{5 - z}{7}$ , then $4 λ + 9 μ$ is equal to : [2024]

1 13

2 6

3 5

4 4

Ans.
(2)

We have lines $\frac{2 - x}{3} = \frac{3 y - 2}{4 λ + 1} = 4 - z$ and $\frac{x + 3}{3 μ} = \frac{1 - 2 y}{6} = \frac{5 - z}{7}$ are perpendicular.

i.e., $\frac{x - 2}{- 3} = \frac{y - 2 / 3}{\frac{4 λ + 1}{3}} = \frac{z - 4}{- 1}$ and $\frac{x - (- 3)}{3 μ} = \frac{y - 1 / 2}{\frac{- 6}{2}} = \frac{z - 5}{- 7}$ are perpendicular.

$\Rightarrow - 3 (3 μ) + \frac{4 λ + 1}{3} (- 3) + (- 1) (- 7) = 0$

$\Rightarrow - 9 μ - 4 λ - 1 + 7 = 0$

$\Rightarrow - 9 μ - 4 λ + 6 = 0 \Rightarrow 4 λ + 9 μ = 6$ .

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