If the $x$-coordinate of a point $P$ on the line joining the points $Q(2,2,1)$ and $R(5,1,-2)$ is $4$, then the $z$-coordinate of $P$ is

The harmonic conjugate of $(2,3,4)$ with respect to the points $(3,-2,2)$ and $(6,-17,-4)$ is

The ratio in which the $yz$-plane divides the line segment joining the points $(-3, 4, -2)$ and $(2, 1, 3)$ is:

If $P(-3, -2, 4)$,$Q(-9, -8, 10)$,and $R(-5, -4, 6)$ are collinear,then the ratio in which $R$ divides $PQ$ is

Let $A(3, 2, -4)$ and $B(9, 8, -10)$ be two points. Let $P_1$ divide $AB$ in the ratio $1:2$ and $P_2$ divide $AB$ in the ratio $2:1$. If the point $P(\alpha, \beta, \gamma)$ divides $P_1 P_2$ in the ratio $1:1$, then $\alpha + 2\beta + 2\gamma =$

Show that the points $A (1,-2,-8), B (5,0,-2)$ and $C (11,3,7)$ are collinear,and find the ratio in which $B$ divides $AC$.