If the mirror image of the point $(1, 3, 5)$ with respect to the plane $4 x - 5 y + 2 z = 8$ is $(α, β, γ)$ , then $5 (α + β + γ)$ equals:

47

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39

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43

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41

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Solution

The correct option is A $47$
Image of $(1, 3, 5)$ in the plane $4 x - 5 y + 2 z = 8$ is $(α, β, γ)$
$\Rightarrow \frac{α - 1}{4} = \frac{β - 3}{- 5} = \frac{γ - 5}{2} = - 2 [\frac{4 (1) - 5 (3) + 2 (5) - 8}{4^{2} + 5^{2} + 2^{2}}] = \frac{2}{5}$
$∴ α = 1 + 4 (\frac{2}{5}) = \frac{13}{5}$
$β = 3 - 5 (\frac{2}{5}) = 1 = \frac{5}{5}$
$γ = 5 + 2 (\frac{2}{5}) = \frac{29}{5}$
Thus, $5 (α + β + γ) = 5 (\frac{13}{5} + \frac{5}{5} + \frac{29}{5}) = 47$

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