Let $S$ be the set of all real values of $\lambda$ such that a plane passing through the points $(-\lambda^2, 1, 1), (1, -\lambda^2, 1)$ and $(1, 1, -\lambda^2)$ also passes through the point $(-1, -1, 1)$. Then $S$ is equal to

The equation of the plane mid-parallel to the planes $2x - 3y + 6z + 21 = 0$ and $2x - 3y + 6z - 14 = 0$ is given by

The coordinates of the foot of the perpendicular drawn from the origin to the plane $2x + y - 2z = 18$ are

Let $R^3$ denote the three-dimensional space. Take two points $P=(1, 2, 3)$ and $Q=(4, 2, 7)$. Let $\operatorname{dist}(X, Y)$ denote the distance between two points $X$ and $Y$ in $R^3$. Let
$S=\{X \in R^3: (\operatorname{dist}(X, P))^2 - (\operatorname{dist}(X, Q))^2 = 50\}$
$T=\{Y \in R^3: (\operatorname{dist}(Y, Q))^2 - (\operatorname{dist}(Y, P))^2 = 50\}$
Then which of the following statements is (are) $TRUE$?
$(A)$ There is a triangle whose area is $1$ and all of whose vertices are from $S$.
$(B)$ There are two distinct points $L$ and $M$ in $T$ such that each point on the line segment $LM$ is also in $T$.
$(C)$ There are infinitely many rectangles of perimeter $48$,two of whose vertices are from $S$ and the other two vertices are from $T$.
$(D)$ There is a square of perimeter $48$,two of whose vertices are from $S$ and the other two vertices are from $T$.

If the plane $56x + 4y + 9z = 2016$ meets the coordinate axes at points $A$,$B$,and $C$,then the centroid of the $\triangle ABC$ is

The plane which bisects the line segment joining the points $(-3, -3, 4)$ and $(3, 7, 6)$ at right angles,passes through which one of the following points?