Question

If the shortest distance between the lines `vecr_1 = αhati + 2hatj + 2hatk + λ(hati - 2hatj + 2hatk)`, λ∈R, α > 0 `vecr_2 = - 4hati - hatk + μ(3hati - 2hatj - 2hatk)`, μ∈R is 9, then α is equal to ______.

पर्याय

• 3

• 4

• 5

• 6

Answer 1

If the shortest distance between the lines `vecr_1 = αhati + 2hatj + 2hatk + λ(hati - 2hatj + 2hatk)`, λ∈R, α > 0 `vecr_2 = - 4hati - hatk + μ(3hati - 2hatj - 2hatk)`, μ∈R is 9, then α is equal to 6.

Explanation:

Given the equation of lines

`vecr_1 = αhati + 2hatj + 2hatk + λ(hati - 2hatj + 2hatk)`

`vecr_2 = - 4hati - hatk + μ(3hati - 2hatj - 2hatk)`

Shortest distance = `|((veca_1 xx veca_2).(vecb_1 xx vecb_2))/|vecb_1 xx vecb_2||`

∴ 9 = `|(((α + 4)hati + 2hatj + 3hatk).(8hati + 8hatj + 4hatk))/sqrt(64 + 64 + 16)|`

⇒ `|(8(α + 4) + 16 + 12)/12|` = 9

∴ α = 6, as α > 0