The coordinates of the foot of the perpendicular drawn from the origin to the plane $2x + y - 2z = 18$ are

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

The equation $xy = 0$ in three-dimensional space represents

Find the intercepts made by the plane $\bar{r} \cdot (2, -3, 4) = 12$ on the coordinate axes.

Let $S$ be the set of all real values of $\lambda$ such that a plane passing through the points $(-\lambda^2, 1, 1)$,$(1, -\lambda^2, 1)$,and $(1, 1, -\lambda^2)$ also passes through the point $(-1, -1, 1)$. Then $S$ is equal to

If for some $\alpha$ and $\beta$ in $\mathbb{R},$ the intersection of the following three planes $x+4y-2z=1$,$x+7y-5z=\beta$,and $x+5y+\alpha z=5$ is a line in $\mathbb{R}^{3},$ then $\alpha+\beta$ is equal to