If a variable plane, at a distance of 3 units from the origin, intersects the coordinate axes at A, B, and C, then the locus of the centroid of $Δ A B C$ is

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$\frac{1}{x^{2}} + \frac{1}{y^{2}} + \frac{1}{z^{2}} = 1$

$\frac{1}{x^{2}} + \frac{1}{y^{2}} + \frac{1}{z^{2}} = 3$

$\frac{1}{x^{2}} + \frac{1}{y^{2}} + \frac{1}{z^{2}} = 9$

$\frac{1}{x^{2}} + \frac{1}{y^{2}} + \frac{1}{z^{2}} = \frac{1}{9}$

answer is A.

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Detailed Solution

Equation of the plane is $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1 \Rightarrow \frac{x}{3 x_{1}} + \frac{y}{3 y_{1}} + \frac{z}{3 z_{1}} = 1$
The distance from origin to the plane = 3
$\Rightarrow \frac{1}{\sqrt{\frac{1}{9 x_{1}^{2}} + \frac{1}{9 y_{1}^{2}} + \frac{1}{9 z_{1}^{2}}}} = 3 \Rightarrow \frac{1}{9 x_{1}^{2}} + \frac{1}{9 y_{1}^{2}} + \frac{1}{9 z_{1}^{2}} = \frac{1}{9} \Rightarrow \frac{1}{x_{1}^{2}} + \frac{1}{y_{1}^{2}} + \frac{1}{z_{1}^{2}} = 1.$
The locus of P is $\frac{1}{x^{2}} + \frac{1}{y^{2}} + \frac{1}{z^{2}} = 1.$