Let be the image of the point (8, 5, 7) in the line . Then is equal to : [2024]

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Solution

Q.
Let $(α, β, γ)$ be the image of the point (8, 5, 7) in the line $\frac{x - 1}{2} = \frac{y + 1}{3} = \frac{z - 2}{5}$ . Then $α + β + γ$ is equal to : [2024]

1 16

2 14

3 18

4 20

Ans.
(2)

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

$\Rightarrow x = 2 λ + 1, y = 3 λ - 1 and z = 5 λ + 2$

Clearly, $\vec{P Q} \cdot (2 \hat{i} + 3 \hat{j} + 5 \hat{k}) = 0$

$\Rightarrow (2 λ - 7) \cdot 2 + (3 λ - 6) \cdot 3 + (5 λ - 5) \cdot 5 = 0$

$\Rightarrow 38 λ = 57$

$\Rightarrow λ = \frac{57}{38} = \frac{3}{2}$

$∴$ $(4, \frac{7}{2}, \frac{19}{2})$ are the coordinates of Q.

Now, Q is the midpoint of PP'.

$∴ \frac{8 + α}{2} = 4, \frac{5 + β}{2} = \frac{7}{2}, \frac{7 + γ}{2} = \frac{19}{2}$

$\Rightarrow (α, β, γ) = (0, 2, 12)$

$∴ α + β + γ = 14$ .

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