Question

A plane meets the coordinate axes at A, B and C, respectively, such that the centroid of triangle ABC is (1, −2, 3). Find the equation of the plane.

Answer 1

$$\begin{gathered} \text{Let}\mathit{\text{a}}\text{,}\mathit{\text{b}}\text{and}\mathit{\text{c}}\text{be the intercepts of the given plane on the coordinate axes.} \\ \text{Then the plane meets the coordinate axes at} \\ A\left(a,0,0\right),B\left(0,b,0\right)\text{and}C\left(0,0,c\right) \\ \text{Given that the centroid of the triangle is}\left(1,-2,3\right) \\ \text{⇒}\left(\frac{a+0+0}{3},\frac{0+b+0}{3},\frac{0+0+c}{3}\right)\text{=}\left(1,-2,3\right) \\ \text{⇒}\left(\frac{a}{3},\frac{b}{3},\frac{c}{3}\right)\text{=}\left(1,-2,3\right) \\ \text{⇒}\frac{a}{3}\text{= 1,}\frac{b}{3}\text{= -2,}\frac{c}{3}\text{= 3} \\ \Rightarrow a=3,b=-6,c=9...\left(1\right) \\ \text{Equation of the plane whose intercepts on the coordinate axes are}\mathit{\text{a}}\text{,}\mathit{\text{b}}\text{and}\mathit{\text{c}}\text{is} \\ \frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1 \\ \Rightarrow \frac{x}{3}+\frac{y}{-6}+\frac{z}{9}=1[\text{From (1)]} \\ \Rightarrow 6x-3y+2x=18 \end{gathered}$$