If the mirror image of the point P(3, 4, 9) in the line is , then is : [2024]

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Solution

Q.
If the mirror image of the point P(3, 4, 9) in the line $\frac{x - 1}{3} = \frac{y + 1}{2} = \frac{z - 2}{1}$ is $(α, β, γ)$ , then $14 (α + β + γ)$ is : [2024]

1 102

2 138

3 132

4 108

Ans.
(4)

Given equation of line is

$\frac{x - 1}{3} = \frac{y + 1}{2} = \frac{z - 2}{1} = λ$ (say)

$∴$ Any point on line, say $m = (3 λ + 1, 2 λ - 1, λ + 2)$

Now, d.r.'s of $P M = < 3 λ - 2, 2 λ - 5, λ - 7 >$ and d.r.'s of given line = $< 3, 2, 1 > ∵ P M ⊥ line$

$∴ 3 (3 λ - 2) + 2 (2 λ - 5) + 1 (λ - 7) = 0$

$\Rightarrow 14 λ = 23 \Rightarrow λ = \frac{23}{14}$

Now, M is the mid point of P(3, 4, 9) and $Q (α, β, γ)$

$∴ \frac{83}{14} = \frac{3 + α}{2} \Rightarrow α = \frac{62}{7} \Rightarrow \frac{16}{7} = \frac{4 + β}{2} \Rightarrow β = \frac{4}{7}$

and $\frac{51}{14} = \frac{9 + γ}{2} \Rightarrow γ = \frac{- 12}{7}$

So, $14 (α + β + γ) = 14 \times \frac{54}{7} = 108$ .

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