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Question
Find m, if the lines `(1 - x)/3 =(7y - 14)/(2"m") = (z - 3)/2` and `(7 - 7x)/(3"m") = (y - 5)/1 = (6 - z)/5` are at right angles
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Solution
The equation of the lines are
`(1 - x)/3 =(7y - 14)/(2"m") = (z - 3)/2`
i.e., `(x - 1)/(3) = (y - 2)/(2/7 "m") = (z - 3)/2`
and `(7 - 7x)/(3"m") = (y - 5)/1 = (6 - z)/5`
i.e. `(x - 1)/((-3)/7 "m") = (y - 5)/1 = (z - 6)/(-5)`
∴ Direction ratios of two lines are
`-3, 2/7"m", 2` annd `(-3)/7"m", 1, -5`
Since the lines are at right angles ......(perpendicular)
∴ `(-3)((-3)/7"m") + (2/7"m")(1) + (2)(-5)` = 0
∴ `9/7"m" + 2/7"m" - 10` = 0
∴ 11m − 70 = 0
∴ m = `70/11`
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