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.

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).

So 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.

By using 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:

\(\left ( \frac{mx_{2}+nx_{1}}{m+n},\frac{my_{2}+ny_{1}}{m+n},\frac{mz_{2}+nz_{1}}{m+n} \right )\)

So, the coordinates of the point R are given by

\(\left ( \frac{8k+2}{k+1}, \frac{-3}{k+1},\frac{10k+4}{k+1}\right )\)

Hence, we have,

\(\left ( \frac{8k+2}{k+1}, \frac{-3}{k+1},\frac{10k+4}{k+1}\right ) = (4, y, z)\)

⇒ (8k+2)/(k+1) = 4

8k + 2 = 4(k + 1)

8k + 2 = 4k + 4

8k – 4k = 4 – 2

4k = 2

k = 2/4

k = 1/2

On substituting the values, we get,

⇒ y = -3/(1/2) + 1 = -3/(3/2) = (-3 x 2)/3 = -2

z = {10(1/2) + 4}/{(1/2) + 1} = (5 + 4)/(3/2) = (9 x 2)/3 = 3 x 2 = 6

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

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