$A$ plane meets the coordinate axes at $A, B, C$ such that the centroid of the triangle $ABC$ is $(1, 2, 4)$. Then,the equation of the plane is

Let a plane $P$ contain the points $\hat{i}, \hat{j}$ and $\hat{i}+\hat{j}+\hat{k}$. Let $L$ be the line through the point $A(3, 0, -5)$ and parallel to the vector $\hat{i}-\hat{j}+\hat{k}$. The equation of the normal to the plane $P$ passing through point $A$ is:

The distance between the parallel planes $ax + by + cz + d = 0$ and $ax + by + cz + d' = 0$ is:

If the foot of the perpendicular drawn from the origin to the plane is $(3, 2, 1)$,then the equation of the plane is

Let $S$ be the mirror image of the point $Q(1,3,4)$ with respect to the plane $2x-y+z+3=0$ and let $R(3,5,\gamma)$ be a point on this plane. Then the square of the length of the line segment $SR$ is ..... .

Find the equation of a plane,given that the foot of the perpendicular drawn to the plane from the origin is $(2, 1, 2)$.