$A$ plane meets the coordinate axes at $A, B,$ and $C$ such that the centroid of triangle $ABC$ is $(1, 2, 3)$. Find the equation of the plane.

The distance from a point $(1, 1, 1)$ to a variable plane $\pi$ is $12$ units and the points of intersections of the plane $\pi$ and $X, Y, Z$-axes are $A, B, C$ respectively. If the point of intersection of the planes through the points $A, B, C$ and parallel to the coordinate planes is $P$,then the equation of the locus of $P$ is

The Cartesian equation of the plane passing through the point $A(7, 8, 6)$ and parallel to the $XY$-plane is:

$A$ variable plane is at a distance of $6$ units from the origin. If it meets the coordinate axes in $A, B$, and $C$, then the equation of the locus of the centroid of the $\triangle ABC$ is

The equation of the plane passing through the point $(1, 2, 1)$ and perpendicular to the planes $x + 2y + 2z - 7 = 0$ and $3x + 3y + 2z - 5 = 0$ is

$A$ plane $x$ passes through the point $(1, 1, 1)$. If $b, c, a$ are the direction ratios of a normal to the plane,where $a, b, c$ $(a < b < c)$ are the factors of $2001$,then the equation of the plane is