Question

The distance between the lines `x/2 = y/(-1) = z/2` and `(x - 1)/2 = (y - 1)/(-1) = (z - 1)/2` is ______.

पर्याय

• 1 unit

• `sqrt(3)` units

• `sqrt(2)` units

• 2 units

Answer 1

The distance between the lines `x/2 = y/(-1) = z/2` and `(x - 1)/2 = (y - 1)/(-1) = (z - 1)/2` is `underlinebb(sqrt(2)  units)`.

Explanation:

Given lines are

`x/2 = y/(-1) = z/2`

`\implies (x - 0)/2 = (y - 0)/(-1) = (z - 0)/2`

or r = `0hati + 0hatj + 0hatk + λ(2hati - hatj + 2hatk)`  ...(i)

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

`\implies` r = `(hati + hatj + hatk) + μ(2hati - hatj + 2hatk)`  ...(ii)

Now, distance between lines (i) and (ii) is given by

d = `|(b xx (a_2 - a_1))/|b||`  ...(iii)

Now, a₂ – a₁ = `hati + hatj + hatk`

So, b × (a₂ – a₁) = `|(hati, hatj, hatk),(2, -1, 2),(1, 1, 1)| = -3hati + 3hatk`

and | b | = `sqrt(4 + 1 + 4)` = 3

From equation (iii),

∴  d = `|-3hati + 3hatk|/3`

= `(3sqrt(2))/2`

= `sqrt(2)`