Question

Find the co-ordinates of the foot of the perpendicular drawn from the point (0, 2, 3) to the line `(x + 3)/(5) = (y - 1)/(2) = (z + 4)/(3)`.

Answer 1

Let P = (0, 2, 3)

Let M be the foot of the perpendicular drawn from P to the line

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

The coordinates of any point on the line are given by 

x = 5λ – 3, y = 2λ + 1, z = 3λ – 4

Let M = (5λ – 3, 2λ + 1, 3λ – 4)                ...(1)

The direction ratios of PM are

5λ – 3 – 0, 2λ + 1 – 2, 3λ – 4 – 3

i.e. 5λ – 3λ, 2λ – 1, 3λ – 7

Since, PM is perpendicular to the line whose direction ratios atr 5, 2, 3,

5(5λ – 3) + 2(2λ – 1) + 3(3λ – 7) = 0

∴ 25λ – 15 + 4λ – 2 + 9λ – 21 = 0

∴ 38λ – 38 = 0

∴ λ = 1

Substituting λ = 1 in (1), we get

M = (5 – 3, 2 + 1, 3 – 4) = (2, 3, –1).

Hence, the coordinates of the foot of perpendicular are (2, 3, –1).