If the plane $2x - y + 2z + 3 = 0$ has distances of $\frac{1}{3}$ and $\frac{2}{3}$ units from the planes $4x - 2y + 4z + \lambda = 0$ and $2x - y + 2z + \mu = 0$ respectively,then the maximum value of $\lambda + \mu$ is equal to:

Let $\pi_1$ be the plane passing through the point $2\hat{i}-\hat{j}+\hat{k}$ and perpendicular to the vector $a\hat{i}+2\hat{j}-3\hat{k}$, and $\pi_2$ be the plane passing through the point $\hat{i}+2\hat{j}-\hat{k}$ and perpendicular to the vector $\hat{i}-2\hat{j}+\hat{k}$. If $\theta$ is the angle between the planes $\pi_1$ and $\pi_2$ and $\cos \theta = -\sqrt{\frac{3}{7}}$, then the integral value of $a$ is:

Let the foot of the perpendicular drawn from the point $(1, 2, 3)$ to a plane be $(-1, 3, -2)$. Then the perpendicular distance from the origin to the plane is

$A$ variable plane passes through a fixed point $(\alpha, \beta, \gamma)$ and meets the coordinate axes in $A, B$ and $C$. Let $P_1, P_2$ and $P_3$ be the planes passing through $A, B, C$ and parallel to the coordinate planes $YZ, ZX, XY$ respectively. Then,the locus of the point of intersection of the planes $P_1, P_2$ and $P_3$ is

The plane,passing through the points $(0,-1,2)$ and $(-1,2,1)$ and parallel to the line passing through $(5,1,-7)$ and $(1,-1,-1)$,also passes through the point.

$A$ 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.