Question

If the shortest distance between the lines `(x - 1)/2 = (y - 2)/3 = (z - 3)/λ` and `(x - 2)/1 = (y - 4)/4 = (z - 5)/5` is `1/sqrt(3)`, then the sum of all possible values of λ is ______.

Options

• 16

• 6

• 12

• 15

Answer 1

If the shortest distance between the lines `(x - 1)/2 = (y - 2)/3 = (z - 3)/λ` and `(x - 2)/1 = (y - 4)/4 = (z - 5)/5` is `1/sqrt(3)`, then the sum of all possible values of λ is 16.

Explanation:

Given points and direction ratios are shown below,

a₁ = (1, 2, 3), a₂ = (2, 4, 5)

`vecb_1 = 2hati + 3hatj + λhatk`, `vecb_2 = hati + 4hatj + 5hatk`

Apply the shortest distance formula,

Shortest distance = `|(a_2 - a_1).(b_1 xx b_2)|/|b_1 xx b_2|`

S.D. = `|((2 - 1)hati + (4 - 2)hatj + (5 - 3)hatk).(vecb_1 xx vecb_2)|/|b_1 xx b_2|`  ...(i)

Take, `vecb_1 xx vecb_2 = |(hati, hatj, hatk),(2, 3, λ),(1, 4, 5)|`

= `hati(15 - 4λ) + hatj(λ - 10) + hatk(5)`

= `(15 - 4λ)hati + (λ - 10)hatj + 5hatk`

`|vecb_1 xx vecb_2| = sqrt((15 - 4λ)^2 + (λ - 10)^2 + 25)`

From equation (i),

S.D. = `|(hati + 2hatj + 2hatk).[(15 - 4λ)hati + (λ - 10)hatj + 5k]|/sqrt((15 - 4λ)^2 + (λ - 10)^2 + 25)`

`|15 - 4λ + 2λ - 20 + 10|/sqrt((15 - 4λ)^2 + (λ - 10)^2 + 25) = 1/sqrt(3)`

Take square both sides,

3 (5 – 2λ)² = 225 + 16λ² – 120 λ + λ² + 100 - 20λ + 25

12λ² + 75 – 60λ = 17λ² – 140 λ + 350

5λ² – 80λ + 275 = 0 `\implies` λ² – 16λ + 55 = 0

(λ – 5) (λ – 11) = 0 `\implies` λ = 5, 11

Sum of values of λ = 5 + 11 = 16