A point R with x-coordinate 4 lies on the line segment joining the points P(2, –3, 4) and Q (8, 0, 10). Find the coordinates of the point R.

[Hint Suppose R divides PQ in the ratio k : 1. The coordinates of the point R are given by

Given: The coordinates of the points P (2, -3, 4) and Q (8, 0, 10).

_{⇒ x1} = 2, y₁ = -3, z₁ = 4; x₂ = 8, y₂ = 0, z₂ = 10

Let the coordinates of the required point be (4, y, z).

Now, let the point R (4, y, z) divides the line segment joining the points P (2, -3, 4) and Q (8, 0, 10) in the ratio k : 1.

Also, by Section Formula,

We know that the coordinates of the point R which divides the line segment joining two points P (x₁, y₁, z₁) and Q (x₂, y₂, z₂) internally in the ratio m : n is given by:

So, the coordinates of the point R are given by

So, we have

⇒ 8k + 2 = 4 (k + 1)

⇒ 8k + 2 = 4k + 4

⇒ 8k – 4k = 4 – 2

⇒ 4k = 2

Hence, the coordinates of the required point are (4, -2, 6).