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

(D) Let the point $P$ divide the line segment $QR$ in the ratio $\lambda : 1$.
Using the section formula,the coordinates of $P$ are given by:
$P = \left( \frac{5\lambda + 2}{\lambda + 1}, \frac{\lambda + 2}{\lambda + 1}, \frac{-2\lambda + 1}{\lambda + 1} \right)$.
Given that the $x-$coordinate of $P$ is $4$,we have:
$\frac{5\lambda + 2}{\lambda + 1} = 4$.
Solving for $\lambda$:
$5\lambda + 2 = 4(\lambda + 1) \Rightarrow 5\lambda + 2 = 4\lambda + 4 \Rightarrow \lambda = 2$.
Now,substitute $\lambda = 2$ into the expression for the $z-$coordinate:
$z = \frac{-2(2) + 1}{2 + 1} = \frac{-4 + 1}{3} = \frac{-3}{3} = -1$.
Thus,the $z-$coordinate of the point is $-1$.