Question

Find the value of λ so that the lines `(1 - x)/(3) = (7y - 14)/(λ) = (z - 3)/(2) and (7 - 7x)/(3λ) = (y - 5)/(1) = (6 - z)/(5)` are at right angles.

Answer 1

The equations of the given lines are

`(1 -  x)/(3) = (7y - 14)/(2λ) = (z - 3)/(2)` ...(1)
and
`(7 - 7x)/(3λ) = (y - 5)/(1) = (6 - z)/(5)`   ...(2)

Equation (1) can be written as:

`(-(x - 1))/(3) = (7(y - 2))/(2λ) = (z - 3)/(2)`

i.e. `(x - 1)/(-3) = (y - 2)/((2λ)/7) = (z - 3)/(2)`

The direction ratios of this line are

`"a"_1 = -3, b_1 = (2λ)/(7), c_1 = 2`

Equation (2) can be written as :

`(-7(x - 1))/(3λ) = (y - 5)/(1) = (-(z - 6))/(5)`

i.e.`(x - 1)/(-(3λ)/7) = (y - 5)/(1) = (z - 6)/(-5)`

The direction ratios of this line are

`a_2 = (-3λ)/(7), b_2 =1, c_2 = -5`

Since the lines (1) and (2) are at right angles,

a₁a₂ + b₁b₂ + c₁c₂ = 0

∴ `(-3)((-3λ)/7) + ((2λ)/7)(1) + 2(-5) = 0`

∴ `((9λ)/7) + ((2λ)/7) - 10 = 0`

∴ `(11λ)/(7)` = 10

∴ `λ = (70)/(11)`.