Given

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

x_{1} = 2, y_{1} = -3, z_{1} = 4;

x_{2} = 8, y_{2} = 0, z_{2} = 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 (x_{1}, y_{1}, z_{1}) and Q (x_{2}, y_{2}, z_{2}) internally in the ratio m: n is given by:

\( \dfrac{mx_2+nx_1}{m+n}, \dfrac{my_2+ny_1}{m+n}, \dfrac{mz_2+nz_1}{m+n}\)

So, the coordinates of the point R are given by

\(\dfrac{8k+2}{k+1}, \dfrac{-3}{k+1}, \dfrac{10k+4}{k+1}\)

So, we have

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

\( \dfrac{8k+2}{k+1}=4\)

8k + 2 = 4 (k + 1)

8k + 2 = 4k + 4

8k – 4k = 4 – 2

4k = 2

k = \( \dfrac{2}{4}\)

= \( \dfrac{1}{2}\)

Now let us substitute the values, we get

So, the coordinates of the point R are,

\((4, \dfrac{-3}{\dfrac{1}{2}+1},\dfrac{10(\dfrac{1}{2})+4}{\dfrac{1}{2}+1})\)

= (4,-2,6)

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