Suppose that the point  R lies on the line segment joining the points P(2, -3, 4) and 0(8, 0, 10) with $x-$co ordinate 6. Then distance of, R from origin is

Suppose R divides PQ in the ratio k : 1. 

$\therefore \mathrm{R}\equiv \left(\frac{8\mathrm{k}+2}{\mathrm{k}+1},\frac{-3}{\mathrm{k}+1},\frac{10\mathrm{k}+4}{\mathrm{k}+1}\right)$

It is given that the x--coordinate of point R is 6. 

$\begin{array}{l}\therefore \frac{8\mathrm{k}+2}{\mathrm{k}+1}=6\Rightarrow 8\mathrm{k}+2=6\mathrm{k}+6\\ \Rightarrow 2\mathrm{k}=4\Rightarrow \mathrm{k}=2\end{array}$

Therefore, the coordinates of point R are given by
$\left(6,\frac{-3}{2+1},\frac{10\times 2+4}{2+4}\right)\equiv (6,-1,4)$