What is the angle between two diagonals of a cube?

For $a \neq 0$,if the sum of the distances of a point $P(x, y, z)$ from the points $F_1(a, 0, 0)$ and $F_2(-a, 0, 0)$ is a constant $2k$,then the locus of that point is

The number of values of $k$ for which the points $A(-4, 9, k)$, $B(-1, 6, k)$, and $C(0, 7, 10)$ form a right-angled isosceles triangle is:

If the ratio of the perpendicular distances of a variable point $P(x, y, z)$ from the $X$-axis and from the $YZ$-plane is $2:3$, then the equation of the locus of $P$ is

$A$ tetrahedron of volume $5$ has three of its vertices at the points $A(2,1,-1)$,$B(3,0,1)$,and $C(2,-1,3)$. If the fourth vertex $D$ lies on the $y$-axis,then the sum of the ordinates of all possible points $D$ is-

$A$ straight line drawn from the point $P(1,3,2)$,parallel to the line $\frac{x-2}{1}=\frac{y-4}{2}=\frac{z-6}{1}$,intersects the plane $L_1: x-y+3z=6$ at the point $Q$. Another straight line which passes through $Q$ and is perpendicular to the plane $L_1$ intersects the plane $L_2: 2x-y+z=-4$ at the point $R$. Then which of the following statements is(are) $TRUE$?
$(A)$ The length of the line segment $PQ$ is $\sqrt{6}$
$(B)$ The coordinates of $R$ are $(1,6,0)$
$(C)$ The centroid of the triangle $PQR$ is $\left(\frac{4}{3}, \frac{14}{3}, \frac{5}{3}\right)$
$(D)$ The perimeter of the triangle $PQR$ is $\sqrt{6}+\sqrt{13}+\sqrt{11}$