The perpendicular distance from the origin to the plane containing the points having position vectors $\hat{i}+2\hat{j}+3\hat{k}$, $2\hat{i}+3\hat{j}-4\hat{k}$, and $3\hat{i}-4\hat{j}+5\hat{k}$ is

Find the coordinates of the foot of the perpendicular drawn from the origin to the plane $2x + 3y + 4z - 12 = 0$.

$A$ plane meets the coordinate axes at $P, Q, R$ respectively. If the centroid of $\triangle P Q R$ is $\left(1, \frac{1}{2}, \frac{1}{3}\right)$, then the equation of the plane is

The coordinates of the foot of the perpendicular drawn from the origin to the plane $2x - y + 5z - 3 = 0$ are $ . . . . . . $.

If for some $\alpha$ and $\beta$ in $\mathbb{R},$ the intersection of the following three planes $x+4y-2z=1$,$x+7y-5z=\beta$,and $x+5y+\alpha z=5$ is a line in $\mathbb{R}^{3},$ then $\alpha+\beta$ is equal to

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 $T$.