Find the equation of the plane passing through the origin and parallel to the plane $3x - 4y + 5z - 6 = 0$.

The length of the perpendicular drawn from the point $(2, 1, 4)$ to the plane containing the lines $\vec r = (\hat i + \hat j) + \lambda (\hat i + 2\hat j - \hat k)$ and $\vec r = (\hat i + \hat j) + \mu (-\hat i + \hat j - 2\hat k)$ is

Equations of planes parallel to the plane $x-2y+2z+4=0$ which are at a distance of $1$ unit from the point $(1, 2, 3)$ are $.....$

Let $P$ be the plane passing through the points $(5,3,0), (13,3,-2)$ and $(1,6,2)$. For $\alpha \in N$,if the distances of the points $A(3,4,\alpha)$ and $B(2,\alpha,a)$ from the plane $P$ are $2$ and $3$ respectively,then the positive value of $a$ is:

If $a, b, c$ are the intercepts made by the plane passing through the point $(1, 2, 3)$ parallel to the plane $3x + 4y - 5z = 0$ on the $X, Y, Z$-axes respectively,then $3a + b + 5c =$

The equation of a plane passing through $(-1, 2, 3)$ and whose normal makes equal angles with the coordinate axes is