$\text{ If a variable plane forms a tetrahedron of constant volume }64k^3\text{ with the coordinate }$$\text{ planes, then the locus of the centroid of the tetrahedron is }(k>0)$

$\text{ Let the variable plane intersects the coordinate axes at }A(a,0,0),B(0,b,0),C(0,0,c)$

$\begin{array}{l}\text{ Then, the equation of the plane will be }\\ \frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1\dots \dots .\left(1\right)\end{array}$

$\text{ Let }P(\alpha ,\beta ,\gamma )\text{ be the centroid of tetrahedron }OABC\text{ , then }$

       $\left(\alpha ,\beta ,\gamma \right) =\left(\frac{a}{4},\frac{b}{4},\frac{c}{4}\right)$

    $\alpha =\frac{a}{4},\beta =\frac{b}{4}\text{ and }\gamma =\frac{c}{4}$

$\begin{array}{l}a=4\alpha ,b=4\beta ,c=4\gamma \\ \text{ Now, volume of tetrahedron }\\ =\left[\text{ Area of }\mathrm{△}AOB\right]OC\\ \Rightarrow 64k^3=\frac{1}{3}\left(\frac{1}{2}ab\right)c=\frac{abc}{6}\mid \\ \Rightarrow 64k^3=\frac{\left(4\alpha \right)\left(4\beta \right)\times \left(4\gamma \right)}{3\times 2}\end{array}$

$\begin{array}{l}{k}^3=\frac{\alpha \beta \gamma }{6}\\ \text{ Hence, locus of }P(\alpha ,\beta ,\gamma )\text{ is }xyz=6k^3\end{array}$