Let the points on the coordinated axes at which the variable plane meets be

$A(a,0,0),\:\:B(0,b,0),\:\:C(0,0,c)$ respectively.

Then the equation of the variable plane is given by $\large\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$

Centroid of the $\Delta\:ABC$, $ G(x_1,y_1,z_1)$ is given by $(\large\frac{a}{3},\frac{b}{3},\frac{c}{3})$

$\Rightarrow\:x_1=\large\frac{a}{3},$ $y_1=\large\frac{b}{3},$ $z_1=\large\frac{c}{3}$

$\Rightarrow\:a=3x_1,\:\:b=3y_1\:\:and\:\:c=3z_1$

Given that the plane is at a distanced of $k$ from origin.

$\therefore\:\bigg|\large\frac{-1}{\sqrt {\large\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}}}\bigg|=k$

$\Rightarrow\:\large\frac{1}{k^2}=\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}$

$\Rightarrow\:\large\frac{9}{k^2}=\frac{1}{x_1^2}+\frac{1}{y_1^2}+\frac{1}{z_1^2}$

$\therefore$ The locus of centroid $G$ is $\large\frac{9}{k^2}=\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}$