Solution:

Given:

The coordinates of the points P \[\left( \mathbf{2},\text{ }\mathbf{3},\text{ }\mathbf{4} \right)\] and Q \[\left( \mathbf{8},\text{ }\mathbf{0},\text{ }\mathbf{10} \right)\].

Let the coordinates of the required point be \[\left( 4,\text{ }y,\text{ }z \right)\].

So now, let the point R \[\left( 4,\text{ }y,\text{ }z \right)\]divides the line segment joining the points P \[\left( \mathbf{2},\text{ }\mathbf{3},\text{ }\mathbf{4} \right)\] and Q \[\left( \mathbf{8},\text{ }\mathbf{0},\text{ }\mathbf{10} \right)\] in the ratio \[\mathbf{k}:\text{ }\mathbf{1}\].

By using Section Formula,

We know that the coordinates of the point R which divides the line segment joining two points \[P~({{x}_{1}},\text{ }{{y}_{1}},\text{ }{{z}_{1}})\]and \[Q\text{ }({{x}_{2}},\text{ }{{y}_{2}},\text{ }{{z}_{2}})\]internally in the ratio m: n is given by:

So, the coordinates of the point R are given by

So, we have

\[\begin{array}{*{35}{l}}

8k\text{ }+\text{ }2\text{ }=\text{ }4\text{ }\left( k\text{ }+\text{ }1 \right)  \\

8k\text{ }+\text{ }2\text{ }=\text{ }4k\text{ }+\text{ }4  \\

8k\text{ }\text{ }4k\text{ }=\text{ }4\text{ }\text{ }2  \\

4k\text{ }=\text{ }2  \\

k\text{ }=\text{ }2/4  \\

=\text{ }1/2  \\

\end{array}\]

Now let us substitute the values, we get

= \[6\]

Therefore, The coordinates of the required point are \[\left( 4,\text{ }-2,\text{ }6 \right)\].