Question

If the lines `(x + 1)/"k" = ("y + 3")/(- 1) = ("z" - 4)/1` and `(x + 10)/(-1) = ("y + 1")/(-3) = ("z - 1")/4` intersect each other, then the value of k is ______.

Options

• 7

• -8

• 9

• -10

Answer 1

If the lines `(x + 1)/"k" = ("y + 3")/(- 1) = ("z" - 4)/1` and `(x + 10)/(-1) = ("y + 1")/(-3) = ("z - 1")/4` intersect each other, then the value of k is -10.

Explanation:

Here, 

x₁ = - 1, y₁ = - 3, z₁ = 4, a₁ = k, b₁ = - 1, c₁ = 1,

x₂ = - 10, y₂ = - 1, z₂ = 1, a₂ = - 1, b₂ = - 3, c₂ = 4

Two lines are intersecting, if shortest distance is zero

i.e. if `|(x_2 - x_1, "y"_2 - "y"_1, "z"_2 - "z"_1),("a"_1, "b"_1, "c"_1),("a"_2, "b"_2, "c"_2)|` = 0

`=> |(-9,2,-3),("k",-1,1),(-1,-3,4)|` = 0

⇒ - 9(- 4 + 3) - 2(4k + 1) - 3(- 3k - 1) = 0

⇒ k = - 10