The equation of the plane which is bisecting the line segment joining the points $A(2,3,4)$ and $B(-4,1,-2)$ and is perpendicular to it,is

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

If ${P_1}$ and ${P_2}$ are the lengths of the perpendiculars from the points $(2, 3, 4)$ and $(1, 1, 4)$ respectively to the plane $3x - 6y + 2z + 11 = 0$,then ${P_1}$ and ${P_2}$ are the roots of the equation:

The Cartesian equation of the plane passing through the point $(1, -2, 3)$ and perpendicular to the vector $-\hat{i} + 2\hat{j} - 3\hat{k}$ is:

$A$ plane meets the coordinate axes at $A, B, C$ such that the centroid of the $\Delta ABC$ is the point $(\alpha, \beta, \gamma)$. Show that the equation of the plane is $\frac{x}{\alpha} + \frac{y}{\beta} + \frac{z}{\gamma} = 3$.

The equation of the plane passing through the point $(2, 3, 4)$ and parallel to the plane $x + 2y + 4z = 5$ is: