# A point R with x-coordinate 4 lies on the line segment joining the pointsP

Question:

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

[Hint suppose $\mathrm{R}$ divides $\mathrm{PQ}$ in the ratio $k$. 1. The coordinates of the point $\mathrm{R}$ are given by $\left(\frac{8 k+2}{k+1}, \frac{-3}{k+1}, \frac{10 k+4}{k+1}\right)$ ]

Solution:

The coordinates of points P and Q are given as P (2, –3, 4) and Q (8, 0, 10).

Let R divide line segment PQ in the ratio k:1.

Hence, by section formula, the coordinates of point $\mathrm{R}$ are given by $\left(\frac{k(8)+2}{k+1}, \frac{k(0)-3}{k+1}, \frac{k(10)+4}{k+1}\right)=\left(\frac{8 k+2}{k+1}, \frac{-3}{k+1}, \frac{10 k+4}{k+1}\right)$

It is given that the x-coordinate of point R is 4.

$\therefore \frac{8 k+2}{k+1}=4$

$\Rightarrow 8 k+2=4 k+4$

$\Rightarrow 4 k=2$

$\Rightarrow k=\frac{1}{2}$

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