A plane meets the coordinate axes in points A, B, C and the centroid of the triangle ABC is.$(\alpha ,\beta ,\gamma )$ Then equation of the plane is.

Let the equation of the required plane be 

$\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$……….(i)

It meets co-ordinate axes in points

$\mathrm{A}(\mathrm{a},0,0),\mathrm{B}(0,\mathrm{b},0),\mathrm{C}(0,0,\mathrm{c})$

The centroid of $\mathrm{△}\mathrm{ABC}\text{ is }\left(\frac{\mathrm{a}}{3},\frac{\mathrm{b}}{3},\frac{\mathrm{c}}{3}\right)$

$\begin{array}{r}\Rightarrow \frac{\mathrm{a}}{3}=\alpha ,\frac{\mathrm{b}}{3}=\beta ,\frac{\mathrm{c}}{3}=\gamma \\ \Rightarrow \mathrm{a}=3\alpha ,\mathrm{b}=3\beta ,\mathrm{c}=3\gamma \end{array}$

Hence the required plane is

$\frac{x}{3\alpha }+\frac{y}{3\beta }+\frac{z}{3\gamma }=1\text{ i.e., }\frac{x}{\alpha }+\frac{y}{\beta }+\frac{z}{\gamma }=3\text{. }$