Question

The $x$-coordinate of a point on the line joining the points $P$$(2,2,1)$ and $Q$ $(5,1,-2)$ is $4$. Find its $z$-coordinate.

Answer 1

Let $R$ divides $PQ$ in the ratio $\lambda :1$
Coordinates of $R$= $(\frac{5\lambda +2}{\lambda +1},\frac{\lambda +2}{\lambda +1},\frac{-2\lambda +1}{\lambda +1})$


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

$\therefore \frac{5\lambda +2}{\lambda +1}=4\Rightarrow 5\lambda +2=4\lambda +4$

$\Rightarrow \lambda =2$

So, $z$- coordinate of $R$ = $\frac{-2\lambda +1}{\lambda +1}=\frac{-4+1}{2+1}=-1$