If the equation of a plane passing through $A(1, p, 2)$ and $B(3, 2, 4)$ and parallel to the $z$-axis is $3x - 2y - q = 0$, then:

$A$ plane meets the coordinate axes at $A, B, C$ such that the centroid of the triangle $ABC$ is $(1, 2, 4)$. Then,the equation of the plane is

$A$ variable plane at a constant distance $p$ from the origin meets the coordinate axes at points $A, B, C$. Through these points,planes are drawn parallel to the coordinate planes. Find the locus of their point of intersection.

The length of the perpendicular from the origin to the plane $\bar{r} \cdot (\hat{i} - 2\hat{j} + 3\hat{k}) = 14$ is

In space,the equation $by + cz + d = 0$ represents a plane perpendicular to the

If $(x, y, z)$ is an arbitrary point lying on a plane $P$ which passes through the points $(42, 0, 0)$,$(0, 42, 0)$,and $(0, 0, 42)$,then the value of the expression $3 + \frac{x-11}{(y-19)^{2}(z-12)^{2}} + \frac{y-19}{(x-11)^{2}(z-12)^{2}} + \frac{z-12}{(x-11)^{2}(y-19)^{2}} - \frac{x+y+z}{14(x-11)(y-19)(z-12)}$ is: