Given: Equation of line is

_{To find: image of point (5, 9, 3)}

Formula Used: Equation of a line is

Vector form:

Cartesian form:

where is a point on the line and with b₁ : b₂ : b₃ being the direction ratios of the line.

If 2 lines of direction ratios a₁:a₂:a₃ and b₁:b₂:b₃ are perpendicular, then a₁b₁+a₂b₂+a₃b₃ = 0

_{Mid-point of line segment joining (}x₁, y₁, z₁) and (x₂, y₂, z₂) is

_Explanation:

Let

So the foot of the perpendicular is (2λ + 1, 3λ + 2, 4λ + 3)

The direction ratios of the perpendicular is

(2λ + 1 - 5) : (3λ + 2 - 9) : (4λ + 3 - 3)

⇒ (2λ – 4) : (3λ – 7) : (4λ)

Direction ratio of the line is 2 : 3 : 4

From the direction ratio of the line and the direction ratio of its perpendicular, we have

2(2λ – 4) + 3(3λ – 7) + 4(4λ) = 0

⇒ 4λ – 8 + 9λ – 21 + 16λ = 0

⇒ 29λ = 29

⇒ λ = 1

So, the foot of the perpendicular is (3, 5, 7)

The foot of the perpendicular is the mid-point of the line joining (5, 9, 3) and (α, β, γ)

So, we have

So, the image is (1, 1, 11)