Question

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 $R$.

Answer 1

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

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

Hence by section formula, the coordinates of point $R$ are given by,

$(\frac{k\left(8\right)+2}{k+1},\frac{k\left(0\right)-3}{k+1},\frac{k\left(10\right)+4}{k+1})=(\frac{8k+2}{k+1},\frac{-3}{k+1},\frac{10k+4}{k+1})$

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

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

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

$\Rightarrow 4k=2$

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

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