If the points $A(1, 3, 5)$, $B(2, 4, 6)$, and $C(4, 5, k)$ form a right-angled triangle, then the number of possible values of $k$ is:

If $a>0, b>0$, then the maximum area of the parallelogram whose three vertices are $O(0,0)$, $A(a \cos \theta, b \sin \theta)$, and $B(a \cos \theta, -b \sin \theta)$ is

If the mid-points of the sides $AB, BC, CA$ of a triangle are $(1, 5, -1), (0, 4, -2), (2, 3, 4)$ respectively,then the length of the median drawn from $C$ to $AB$ is

Let $A(2, 2, -3)$,$B(5, 6, 9)$,and $C(2, 7, 9)$ be the vertices of a triangle. The angle bisector of $\angle A$ meets $BC$ at the point $D$. Find the coordinates of $D$.

If $P(0, 7, 10)$,$Q(-1, 6, 6)$,and $R(-4, 9, 6)$ are three points in space,then $\triangle PQR$ is:

Let $ABCD$ be a tetrahedron in which the coordinates of each of its vertices are in arithmetic progression. If the centroid $G$ of the tetrahedron is $(2, 3, k)$,then the distance of $G$ from the origin is