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For a > 0, the feet of perpendiculars from points A(a, -2a, 3) and B(0, 4, 5) onto the plane lx + my + nz = 0 are C(0, -a, -1) and D respectively. Find the length of segment CD.

1. sqrt(31)
2. sqrt(41)
3. sqrt(55)
4. sqrt(66)

Correct answer: sqrt(66)

Solution

The direction AC = (-a, a, -4) gives the normal direction. Substituting C into the plane equation gives a² = 4, so a = 2. The plane is x - y + 2z = 0. The foot D from B(0,4,5) is found by the foot-of-perpendicular formula to be (-1, 5, 3). With C = (0, -2, -1), |CD|² = 1 + 49 + 16 = 66.

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