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:

The equation of the plane,passing through the midpoint of the line segment joining the points $P(1, 2, 5)$ and $Q(3, 4, 3)$ and perpendicular to it,is

Find the coordinates of the foot of the perpendicular drawn from the origin to the plane $5y + 8 = 0$.

The Cartesian equation of the plane whose vector equation is $\vec{r}=(1+\lambda-\mu) \hat{i}+(2-\lambda) \hat{j}+(3-2 \lambda+2 \mu) \hat{k}$,where $\lambda, \mu$ are scalars,is:

If the foot of the perpendicular drawn from the point $(0, 0, 0)$ to the plane is $(4, -2, -5)$,then the equation of the plane is $......$

The distance from a point $(1, 1, 1)$ to a variable plane $\pi$ is $12$ units and the points of intersections of the plane $\pi$ and $X, Y, Z$-axes are $A, B, C$ respectively. If the point of intersection of the planes through the points $A, B, C$ and parallel to the coordinate planes is $P$,then the equation of the locus of $P$ is