Given: Equation of plane is 2x – y + z + 3 = 0. P is at (1, 3, 4)

To find: image of P

Explanation:

From the figure, P’ is the image of P and B is the midpoint of PP’

If B is (a, b, c), then

B lies on the plane.

So, 2a – b + c + 3 = 0 … (1)

Also, equation of line PB is

So any point on the line PB will be of the form

x = 2λ +1, y = -λ + 3, z = λ + 4

Since (a, b, c) will also be of this form we can substitute these values in (1)

⇒ 2(2λ + 1) – (-λ + 3) + (λ + 4) + 3 = 0

⇒ 4λ + 2 + λ – 3 + λ + 4 + 3 = 0

⇒ 6λ = -6

⇒ λ = -1

So (a, b, c) = (-1, 4, 3)

Substituting these values in the equations of a, b and c,

Therefore,

x₁ = -3, y₁ = 5, z₁ = 2

Therefore, the image is (-3, 5, 2)