$A$ line with direction ratios $2, 1, 2$ intersects the lines $x = y + a = z$ and $x + a = 2y = 2z$. Find the coordinates of the points of intersection.

The transformed equation of $2x^2 + 3y^2 - z^2 - 8x + 18y + 2z + 9 = 0$ when the axes are translated to the point $(2, -3, 1)$ 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 $\sqrt{6}+\sqrt{13}+\sqrt{11}$

The locus of a point $P$ such that $PA + PB = 4$ where $A(2, 3, 4)$ and $B(-2, 3, 4)$ 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:

Assertion $(A):$ If $(-1,3,2)$ and $(5,3,2)$ are respectively the orthocentre and circumcentre of a triangle, then $(3,3,2)$ is its centroid.
Reason $(R):$ Centroid of the triangle divides the line segment joining the orthocentre and the circumcentre in the ratio $1: 2$.
Which one of the following is true?