A variable plane which remains at a constant distance 3p from the origin cut the coordinate axes at A,B and C. the locus of the centroid of triangle ABC is

Let equation of the variable plane be _{$\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$}

This meets the coordinate axes at A(a,0,0),B(0,b,0) and C(0,0,c)

Let _{$P\left(\alpha ,\beta ,\gamma \right)$} be the centroid of the_{$\Delta ABC$} .

Then _{$\alpha =\frac{a+0+0}{3},\beta =\frac{0+b+0}{3},\gamma =\frac{0+0+c}{3}$}

_{$\therefore a=3\alpha ,b=3\beta ,c=3\gamma \to \left(2\right)$}

Plane (1) is at constant distance 3p from the origin ,

 so _{$3p=\frac{\left|\frac{0}{a}+\frac{0}{b}+\frac{0}{c}-1\right|}{\sqrt{{\left(\frac{1}{a}\right)}^2+{\left(\frac{1}{b}\right)}^2+{\left(\frac{1}{c}\right)}^2}}$}

_{$\Rightarrow \frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}=\frac{1}{9p^2}\mathrm{.............}\left(3\right)$}

From (2) and (3) , we get _{$\frac{1}{9{\alpha }^2}+\frac{1}{9{\beta }^2}+\frac{1}{9{\gamma }^2}=\frac{1}{9p^2}\Rightarrow {\alpha }^{-2}+{\beta }^{-2}+{\gamma }^{-2}=p^{-2}$}

Generalizing, _{$\alpha ,\beta ,\gamma$} .

Locus of centroid _{$P\left(\alpha ,\beta ,\gamma \right)$} is _{$x^{-2}+y^{-2}+z^{-2}=p^{-2}$}