# A variable plane is at a distance $k$ from origin and meets the coordinate axes at the points $A,B,C$ respectively. then the locus of centroid of the $\Delta\:ABC$ is ?

$(a)\:\large\frac{1}{k^2}=\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}\:\:\qquad\:(b)\:\large\frac{3}{k^2}=\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}\:\:\qquad\:(c)\:\large\frac{4}{k^2}=\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}\:\:\qquad\:(d)\:\large\frac{9}{k^2}=\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}$

Toolbox:
• Centroid of triangle with vertices $A(x_1,y_1,z_1),\:B(x_2,y_2,z_2)\:\:and\:\:C(x_3,y_3,z_3)$ is given by $\bigg(\large\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3},\frac{z_1+z_2+z_3}{3}\bigg)$
• Distance of a plane $ax+by+cz+d=0$ from origin is $\bigg|\large\frac{d}{\sqrt {a^2+b^2+c^2}}\bigg|$
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 $\large\frac{9}{k^2}=\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}$