A plane cuts the coordinate axes $X, Y, Z$ at $A$, $B, C$ respectively such that the centroid of the…
A plane cuts the coordinate axes $X, Y, Z$ at $A$, $B, C$ respectively such that the centroid of the $\triangle A B C$ is $(6,6,3)$. Then the equation of that plane is
$x+y+z-6=0$
$x+2 y+z-18=0$
$2 x+y+z-18=0$
$x+y+2 z-18=0$
Solution
Let the equation of the plane is
$
\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1
$
By the definition of the centroi $\left(\frac{a}{3}, \frac{b}{3}, \frac{c}{3}\right)=(6,6,3)$
Therefore,
$
\begin{aligned}
a & =18 \\
b & =18 \\
c & =9
\end{aligned}
$
The equation of the plane becomes
$
\begin{array}{rlrl}
& & \frac{x}{18}+\frac{y}{18}+\frac{z}{9} & =1 \Rightarrow \frac{x+y+2 z}{18}=1 \\
\Rightarrow \quad & x+y+2 z & =18 \\
\Rightarrow \quad x+y+2 z-18 & =0
\end{array}
$