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Question
Find the coordinates of the foot of the perpendicular drawn from point (5, 7, 3) to the line `(x - 15)/3 = (y - 29)/8 = (z - 5)/-5`.
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Solution
Given point is P(5, 7, 3) and line is
`(x - 15)/3 = (y - 29)/8 = (z - 5)/-5` = k
Let any point Q on this line with coordinates (3k + 15, 8k + 29, – 5k + 5).
Now direction ratio of line PQ is
(3k + 15 – 5), (8k + 29 – 7), (– 5k + 5 – 3)
= 3k + 10, 8k + 22, – 5k + 2
and direction ratio of given line l are (3, 8, – 5)
∵ PQ ⊥ l
∴ 3(3k + 10) + 8(8k + 22) – 5(– 5k + 2) = 0
9k + 30 + 64k + 176 + 25k – 10 = 0
98k + 196 = 0
k = `(-196)/98` = – 2
Hence foot of perpendicular drawn on the given line is [3 × (– 2) + 15, 8 × (– 2) + 29, – 5 × (– 2) + 5] = (9, 13, 15).
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