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

$A(1, 2, 3)$, $B(2, 3, 1)$, and $C(3, 1, 2)$ are three points. If the point $P$ divides $AB$ in the ratio $1:2$ and the point $Q$ divides $BC$ in the ratio $-2:3$, then the distance between $P$ and $Q$ is

The point dividing the join of $(3, -2, 1)$ and $(-2, 3, 11)$ in the ratio $2:3$ is

Let $P(\alpha, 4, 7)$ and $Q(3, \beta, 8)$ be two points. If the $YZ$-plane divides the line segment joining $P$ and $Q$ in the ratio $2:3$ and the $ZX$-plane divides the line segment joining $P$ and $Q$ in the ratio $4:5$,then the length of the line segment $PQ$ is:

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.

The ratio in which $B\left(\frac{33}{5}, \frac{28}{5}, \frac{38}{5}\right)$ divides the line segment joining $A(3, 2, 4)$ and $C(9, 8, 10)$ is