If the distance between the planes $4x - 2y - 4z + 1 = 0$ and $4x - 2y - 4z + d = 0$ is $7$,then the value of $d$ is:

$A$ point $P(a, a, a)$ lies on a line passing through the origin and making equal angles with the axes,where $a$ is a constant. $A$ plane passes through $P$ and is perpendicular to $OP$. If this plane makes intercepts on the axes,what is the sum of the reciprocals of these intercepts?

Find the coordinates of the foot of the perpendicular drawn from the origin to the plane $2x - 3y + 4z - 6 = 0$.

If the plane $\vec{r} = (\lambda + \mu)\hat{i} + (2 + \mu)\hat{j} + (3\lambda + 2\mu)\hat{k}$, where $\lambda$ and $\mu$ are parameters, intersects coordinate axes at points $(a, 0, 0), (0, b, 0), (0, 0, c)$, then $a + b + c = $

$A$ 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 equation of a plane passing through $(-1, 2, 3)$ and whose normal makes equal angles with the coordinate axes is