$A(3, 2, -1), B(4, 1, 0), C(2, 1, 4)$ are the vertices of a triangle $ABC$. If the bisector of $\angle BAC$ intersects the side $BC$ at $D(p, q, r)$, then $\sqrt{2p + q + r} =$

Point $A$ divides the line segment joining points $(-5, 1)$ and $(3, 5)$ in the ratio $k:1$. If the coordinates of points $B$ and $C$ are $(1, 5)$ and $(7, -2)$ respectively,and the area of $\Delta ABC$ is $2$ square units,what is the value of $k$?

The point dividing the line segment joining the points $(1, 2, 3)$ and $(3, -5, 6)$ in the ratio $3: -5$ is

In what ratio does the $x$-axis divide the line segment joining the points $(3, 4)$ and $(5, 6)$?

If $A(1,2,3), B(2,-3,1), C(3,2,-1)$ are three vertices of a tetrahedron $ABCD$ and $G\left(\frac{5}{2}, \frac{3}{2}, \frac{9}{4}\right)$ is its centroid, then the point which divides $GD$ in the ratio $1:2$ is

If the point $(a, 8, -2)$ divides the line segment joining the points $(1, 4, 6)$ and $(5, 2, 10)$ in the ratio $m: n$,then $\frac{2m}{n} - \frac{a}{3} =$