If a plane meets the coordinate axes at $A, B$,and $C$ such that the centroid of the triangle $ABC$ is $(1, 2, 4)$,then the equation of the plane is:

$A$ plane is parallel to two lines whose direction ratios are $2, 0, -2$ and $-2, 2, 0$ and it contains the point $(2, 2, 2)$. If it cuts the coordinate axes at $A, B, C$,then the volume of the tetrahedron $OABC$ (in cubic units) is

Let $P(1, -2, 5)$ be the foot of the perpendicular drawn from the origin to the plane $\pi_1$ and the same $P$ be the foot of the perpendicular from $(1, 2, -1)$ to the plane $\pi_2$. Then the acute angle between the planes $\pi_1$ and $\pi_2$ is

The plane passing through the point $(4, -1, 2)$ and parallel to the lines $\frac{x + 2}{3} = \frac{y - 2}{-1} = \frac{z + 1}{2}$ and $\frac{x - 2}{1} = \frac{y - 3}{2} = \frac{z - 4}{3}$ also passes through the point

$A$ variable plane is at a constant distance $h$ from the origin and meets the coordinate axes in $A, B, C$. The locus of the centroid of $\triangle ABC$ is

If the perpendicular distance from $(1, 2, 4)$ to the plane $2x + 2y - z + k = 0$ is $3$,then $k =$