Find the image of the point (5, 9, 3) in the line (x - 1)/2 = (y - 2)/3 = (z - 3)/4.

Question

Find the image of the point (5, 9, 3) in the line \(\cfrac{\text x-1}2=\cfrac{y - 2}3 = \cfrac{z-3}4\)

(x - 1)/2 = (y - 2)/3  = (z - 3)/4.

Answer 1

Given: Equation of line is

\(\cfrac{\text x-1}2=\cfrac{y - 2}3 = \cfrac{z-3}4\).

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

Formula Used: Equation of a line is

Vector form: \(\vec r=\vec a+\lambda\vec b\) 

Cartesian form:

where \(\vec a=\text x_1\hat i+y_1\hat j+z_1\hat k\) is a point on the line and \(\vec b=b_1\hat i+b_2\hat j+b_3\hat k\) 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

\(\cfrac{\text x-1}2=\cfrac{y - 2}3 = \cfrac{z-3}4\) = λ

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)