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The reflection (image) of the line (x - 1)/9 = (y - 2)/(-1) = (z + 3)/(-3) in the plane 3x - 3y + 10z = 26 is:

1. (x + 2)/9 = (y - 3)/(-1) = (z + 7)/(-3)
2. (x - 4)/9 = (y + 1)/(-1) = (z - 7)/(-3)
3. (x - 4)/9 = (y - 2)/(-1) = (z + 3)/(-3)
4. (x - 4)/9 = (y - 2)/(-1) = (z + 1)/(-3)

Correct answer: (x - 4)/9 = (y + 1)/(-1) = (z - 7)/(-3)

Solution

The direction vector d = (9, -1, -3). Normal to plane n = (3, -3, 10). d. n = 27 + 3 - 30 = 0, so the line is parallel to the plane. Reflect the point (1, 2, -3) in the plane: foot of perpendicular F = (1,2,-3) + t(3,-3,10) where 3(1+3t) - 3(2-3t) + 10(-3+10t) = 26. This gives 3+9t-6+9t-30+100t=26, so 118t=59, t=1/2. F=(5/2, 1/2, 2). Image point P' = 2F - (1,2,-3) = (4,-1,7). Image line passes through (4,-1,7) with direction (9,-1,-3): (x-4)/9 = (y+1)/(-1) = (z-7)/(-3).

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