$A$ plane $\pi$ makes intercepts $3$ and $4$ respectively on $Z$-axis and $X$-axis. If $\pi$ is parallel to $Y$-axis, then its equation is:

Let $P$ be the plane passing through the points $(5,3,0), (13,3,-2)$ and $(1,6,2)$. For $\alpha \in N$,if the distances of the points $A(3,4,\alpha)$ and $B(2,\alpha,a)$ from the plane $P$ are $2$ and $3$ respectively,then the positive value of $a$ is:

The foot of the perpendicular drawn from the origin to the plane $x+y+3z-4=0$ is

Find the equation of the plane that passes through the three points $(1,1,0), (1,2,1),$ and $(-2,2,-1)$.

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 a plane is at a distance of $6$ units from the origin and the vector $2 \hat{i} + 6 \hat{j} - 3 \hat{k}$ is its normal, then the equation of the plane in Cartesian form is