Question

If line `(2x - 4)/lambda = ("y" - 1)/2 = ("z" - 3)/1` and `(x - 1)/1 = (3"y" - 1)/lambda = ("z" - 2)/1` are perpendicular to each other then λ = ______.

Options

• 7

• `- 7/6`

• 6

• `- 6/7`

Answer 1

If line `(2x - 4)/lambda = ("y" - 1)/2 = ("z" - 3)/1` and `(x - 1)/1 = (3"y" - 1)/lambda = ("z" - 2)/1` are perpendicular to each other then λ = `underline(- 6/7)`.

Explanation:

If lines `(x - x_1)/"a"_1 = ("y" - "y"_1)/"b"_1 = ("z" - "z"_1)/"c"_1 and (x - x_2)/"a"_2 = ("y" - "y"_2)/"b"_2 = ("z" - "z"_2)/"c"_2`

are perpendicular, then a₁a₂ + b₁b₂ + c₁c₂ = 0

Given lines are `(2x - 4)/lambda = ("y" - 1)/2 = ("z" - 3)/1`

`=> (x - 2)/(lambda/2) = ("y - 1")/2 = ("z" - 3)/1`   ....(i)

and `(x - 1)/1 = (3"y" - 1)/lambda = ("z" - 2)/1`

`=> (x - 1)/1 = ("y" - 1/3)/(lambda/3) = ("z - 2")/1`    ...(ii)

Since, lines (i) and (ii) are perpendicular.

`therefore lambda/2 xx 1 + 2 xx lambda/3 + 1 xx 1 = 0`

`=> lambda/2 + (2lambda)/3 + 1 = 0`

⇒ 3λ + 4λ + 6 = 0

⇒ 7λ = - 6

⇒ λ = `-6/7`