Question

The co-ordinates of the foot of perpendicular drawn from origin to the plane 2x - y + 5z - 3 = 0 are ______.

Options

• `(2/sqrt30, (- 1)/sqrt30, 5/sqrt30)`

• (2, -1, 5)

• `(2/3, (-1)/3, 5/3)`

• `(1/5, (-1)/10, 1/2)`

Answer 1

The co-ordinates of the foot of perpendicular drawn from origin to the plane 2x - y + 5z - 3 = 0 are `underline((1/5; (-1)/10; 1/2))`.

Explanation:

Use foot of perpendicula (x, y, z) of a point (x₁, y₁, z₁) in a plane ax + by + cz + d = 0 is given by

`(x - x_1)/"a" = ("y" - "y"_1)/"b" = ("z" - "z"_1)/"c" = (-("a"x_1 + "b"y_1 + "c"z_1 + "d"))/("a"^2 + "b"^2 + "c"^2)`

Given equation of plane is 2x - y + 5z - 3 = 0

∴ Foot of perpendicular drawn from origin to the given plane

`=> (x - 0)/2 = ("y" - 0)/(- 1) = ("z" - 0)/5`

`= (- (0 + 0 + 0 - 3))/((2)^2 + (- 1)^2 + (5)^2)`

`=> x/2 = "y"/(-1) = "z"/5 = 3/30 = 1/10`

∴ `x/2 = 1/10, "y"/(-1) = 1/10, "z"/5 = 1/10`

`=> x = 1/5, "y" = - 1/10, "z" = 1/2`