Let be the mirror image of the point (2, 3, 5) in the line x – 1 2 = y – 2 3 = z – 3 4 . Then 2 + 3 + 4 is equal to [2024]

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Solution

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

1 34

2 32

3 31

4 33

Ans.
(4)

Given equation of the line is $\frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4}$

Let A(2, 3, 5) be the given point and let B be the foot of the perpendicular drawn from the point A on the line

$\frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4} = λ, λ \in R$

$\Rightarrow x = 2 λ + 1, y = 3 λ + 2, z = 4 λ + 3$

$∴ B \equiv (2 λ + 1, 3 λ + 2, 4 λ + 3)$

Direction ratios of AB are $2 λ - 1, 3 λ - 1, 4 λ - 2$ .

Now, AB is perpendicular to the given line.

$∴ 2 (2 λ - 1) + 3 (3 λ - 1) + 4 (4 λ - 2) = 0$

$\Rightarrow 4 λ - 2 + 9 λ - 3 + 16 λ - 8 = 0$

$\Rightarrow 29 λ - 13 = 0 \Rightarrow λ = \frac{13}{29}$

$∴$ Coordinates of B are $(\frac{55}{29}, \frac{97}{29}, \frac{139}{29})$

Now, using mid point formula,

$\frac{55}{29} = \frac{2 + α}{2}; \frac{97}{29} = \frac{3 + β}{2}; \frac{139}{29} = \frac{5 + γ}{2}$

$\Rightarrow α = \frac{52}{29}, β = \frac{107}{29}, γ = \frac{133}{29}$

$∴ 2 α + 3 β + 4 γ = 2 \times \frac{52}{29} + \frac{3 \times 107}{29} + \frac{4 \times 133}{29} = \frac{957}{29} = 33$ .

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