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

The ratio in which the point $P$,whose abscissa is $3$,divides the line segment joining $A(6, 5)$ and $B(-1, 4)$ is equal to .........

If $P$ divides the line segment joining the points $A(1, 2, -1)$ and $B(-1, 0, 1)$ externally in the ratio $1: 2$ and $Q = (1, 3, -1)$,then $PQ =$

$A$ point $R$ with $x$-coordinate $4$ lies on the line segment joining the points $P(2, -3, 4)$ and $Q(8, 0, 10)$. Find the coordinates of the point $R$.

$A(1, 2, 3)$,$B(2, 3, 1)$,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

If the point dividing internally the line segment joining the points $(a, b)$ and $(5, 7)$ in the ratio $2 : 1$ is $(4, 6)$,then