Question

In the following cases, find the coordinates of the foot of the perpendicular drawn from the origin. 

(a) 2x + 3y + 4z – 12 = 0

(b) (1, 1, 0), (1, 2, 1), (- 2, 2, – 1)

(c) x + y + z = 1

(d) 5y + 8 = 0

Answer 1

(a) Direction ratio of any line normal to 

2x + 3y + 4z – 12 = 0 is (2,3,4) 

∴ Equation of line through (0, 0, 0) and is ‘⊥’ to given lines is

any point on the lines is (2λ, 3λ, 4λ) but this lines on the given plane

(b) The given equation passes through (1,1,0) equation of plane is (0, 3 , 4) 

Equation of a lines through the orgin and is ‘⊥’ to given lines is

and any points on the line is (0, 3λ, 4λ) but this point lines on the given plane.

(c) Direction of any line normal to the given plane is (1,1,1) 

Equation of a lines through (0,0,0) and is ‘⊥’ to the given lines is

any point on this line (λ,λ,λ) but this point lies on the plane x +y + 2 = 1

(d) Directions of normal to the plane is (0,5,0) equation of a line through (0,0,0) and ‘⊥’ to the plane is

any point on this line is (0,5λ,0) but this point lines on the plane