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Question
If the lines `(x-1)/(-3) = (y -2)/(2k) = (z-3)/2 and (x-1)/(3k) = (y-1)/1 = (z -6)/(-5)` are perpendicular, find the value of k.
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Solution 1
The direction ratios of the given lines are – 3, 2k, 2 and 3k, 1, −5.
∵ If the lines are perpendicular then a1a2 + b1b2 + c1c2 = 0
∴ – 3.3k + 2k.1 + 2.( –5) = 0
⇒ 9k + 2k – 10 = 0
⇒ −7k = 10
k = `(-10)/7`
Solution 2
The given lines are,
`(x - 1)/-3 = (y - 2)/(2k) = (z - 3)/2` ....(i)
and `(x - 1)/(3k) = (y - 1)/1 = (z - 6)/-5` ....(ii)
The direction ratio of the line (i) are < - 3, 2k, 2 >
The direction ratio of the line (ii) are < 3k, 1, -5 >
if -9k + 2k - 10 = 0 if 7k = -10 if `k = -10/7`
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