# The equation of the plane that meets the coordinated axis at the points $A,B,C$ respectively so that the centroid of the $\Delta ABC$ is $(\alpha,\beta,\gamma)$ is ?

$(a)\: \alpha x+\beta y+\gamma z=1 \\(b)\: \large\frac{x}{\alpha}+\frac{y}{\beta}+\frac{z}{\gamma}=3 \\(c)\: \large\frac{x}{\alpha}+\frac{y}{\beta}+\frac{z}{\gamma}=1 \\ (d) \:\alpha x+\beta y+\gamma z=3$

Let the plane cut the coordinate axis at the points $A(a,0,0),\:B(0,b,0)\:and\:\:C(0,0,c)$
$\therefore$ The eqn. of the plane is $\large\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$
$\Rightarrow\:$Centroid of the $\Delta \:ABC$ is $\big(\large\frac{a}{3},\frac{b}{3},\frac{c}{3}\big)$
Given: centroid = $(\alpha,\beta,\gamma)$
$\Rightarrow\:a/3=\alpha,b/3=\beta\:and\:c/3=\gamma$
$\Rightarrow\:a=3\alpha,\:b=3\beta\:and\:c=3\gamma$
$\therefore$ Eqn. of the plane is $\large\frac{x}{\alpha}+\frac{y}{\beta}+\frac{z}{\gamma}=3$
edited Sep 29, 2014