Question

The length of the perpendicular from (0, 2, 3) to the line `(x + 3)/5 = (y - 1)/2 = (z + 4)/3` is

विकल्प

• 4

• `sqrt21`

• `sqrt41`

• 7

Answer 1

`sqrt21`

Explanation:

Let M be the foot of the perpendicular drawn from the point P(0, 2, 3) to the line 

`(x + 3)/5 = (y - 1)/2 = (z + 4)/3`

Let `(x + 3)/5 = (y - 1)/2 = (z + 4)/3 = lambda`

∴ The co-ordinates of any point on the line are

M ≡ (5λ - 3, 2λ + 1, 3λ - 4)    ...(i)

∴ The direction ratios of PM are

5λ - 3, 2λ + 1, 3λ - 4

Direction ratios of given line are 5, 2, 3

Since PM is perpendicular to the given line,

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

⇒ 25λ - 15 + 4λ - 2 + 9λ - 21 = 0

⇒ λ = 1

∴ M ≡ (2, 3, -1)      ....[From (i)]

∴ PM = `sqrt((2 - 0)^2 + (3 - 2)^2 + (- 1 - 3)^2) = sqrt21`