Let be the foot of the perpendicular from the point (1, 2, 3) on the line . Then is equal to [2024]

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Solution

Q.
Let $(α, β, γ)$ be the foot of the perpendicular from the point (1, 2, 3) on the line $\frac{x + 3}{5} = \frac{y - 1}{2} = \frac{z + 4}{3}$ . Then $19 (α + β + γ)$ is equal to [2024]

1 100

2 99

3 101

4 102

Ans.
(3)

Let $\frac{x + 3}{5} = \frac{y - 1}{2} = \frac{z + 4}{3} = λ$

$\Rightarrow (α, β, γ) = (5 λ - 3, 2 λ + 1, 3 λ - 4)$

Let $B (α, β, γ)$ be the foot of perpendicular from A on the given line

Now, $5 \hat{i} + 2 \hat{j} + 3 \hat{k}$ is the direction vector of given line.

$\Rightarrow \vec{A B} \cdot (5 \hat{i} + 2 \hat{j} + 3 \hat{k}) = 0$

$\Rightarrow (5 λ - 4) 5 + (2 λ - 1) 2 + (3 λ - 7) 3 = 0$

$\Rightarrow 25 λ + 4 λ + 9 λ = 20 + 2 + 21$

$\Rightarrow 38 λ = 43 \Rightarrow λ = \frac{43}{38}$

$∴ 19 (α + β + γ) = 19 (5 λ - 3 + 2 λ + 1 + 3 λ - 4) = 19 (10 λ - 6)$

$= 19 (10 \times \frac{43}{38} - 6) = 215 - 114 = 101$ .

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