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

Let $R(4,y,z)$ be any point which divides the join $P(2,-3,4)$ and $Q(8,0,10)$ in the ratio k : 1 internally.

$\therefore$ Coordinates of R is

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

But x coordinates of R is 4

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

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

$\therefore y=\frac{-3}{\frac{1}{2}+1}=\frac{-3}{\frac{3}{2}}=-2$

$z=\frac{\frac{10\times 1}{2}+4}{\frac{1}{2}+1}=\frac{9}{\frac{3}{2}}=6$

Thus coordinates of R is $(4,-2,6).$