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.

Question

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

(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 \(\frac{x + 3}{5}=\frac{y - 1}{2}=\frac{z + 4}{3}\) = λ ....(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 – 3, 2λ – 1, 3λ – 7 

Since, PM is perpendicular to the line whose direction ratios are 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).