If the image of the point (4, 4, 3) in the line is , then is equal to [2025]

Question

If the image of the point (4, 4, 3) in the line $\frac{x - 1}{2} = \frac{y - 2}{1} = \frac{z - 1}{3}$ is $(α, β, γ)$ , then $α + β + γ$ is equal to [2025]

Smart Achievers · Accepted Answer

(1)

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

$\Rightarrow M \equiv (2 λ + 1, λ + 2, 3 λ + 1)$

$\vec{P M} = (2 λ - 3) \hat{i} + (λ - 2) \hat{j} + (3 λ - 2) \hat{k}$

Since, $\vec{P M}$ is perpendicular to the given line

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

$\Rightarrow λ = 1$

$∴$ The coordinates of point M is (3, 3, 4).

Let Q be the image of the point P. Then, M be the mid-point of PQ.

$∴ (\frac{4 + α}{2}, \frac{4 + β}{2}, \frac{3 + γ}{2}) \equiv (3, 3, 4)$

$\Rightarrow$ $α = 2, β = 2, γ = 5$ $∴$ $α + β + γ = 9$

Solution

Q.
If the image of the point (4, 4, 3) in the line $\frac{x - 1}{2} = \frac{y - 2}{1} = \frac{z - 1}{3}$ is $(α, β, γ)$ , then $α + β + γ$ is equal to [2025]

1 9

2 7

3 12

4 8

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Solution

Q. If the image of the point (4, 4, 3) in the line x–12=y–21=z–13 is (α,β,γ), then α+β+γ is equal to [2025]

1 9

2 7

3 12

4 8

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Q.
If the image of the point (4, 4, 3) in the line $\frac{x - 1}{2} = \frac{y - 2}{1} = \frac{z - 1}{3}$ is $(α, β, γ)$ , then $α + β + γ$ is equal to [2025]