Question

A plane meets the coordinate axes in A,B and C such that the centroid of $△$ABC is the point $(\alpha ,\beta ,\gamma )$ . Show that the equation of the plane is : $\frac{x}{\alpha }+\frac{y}{\beta }+\frac{z}{\gamma }=3$

Answer 1

$A(a,0,0),B(0,b,0),C(0,0,c)$ are the points on coordinate axis and centriod $(\alpha ,\beta ,\gamma )$.

According to centroid formula,

$\alpha =\frac{a}{3}$

$a=3\alpha$

$b=3\beta$

$c=3\gamma$

Equation of plane is,

$\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$

$\frac{x}{\alpha }+\frac{y}{\beta }+\frac{z}{\gamma }=3$