Question

A plane meets the coordinate axes at A, B and C such that the centroid of ∆ABC is the point (a, b, c). If the equation of the plane is \[\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = k,\] then k = 

 

Options

•  1

•  2

•  3

•  None of these

   

Answer 1

 3

\[\text{ Let } \alpha, \beta \text{ and } \gamma  \text{ be the intercepts of the given plane on the coordinate axes } .\]
\[\text{ Then, the plane meets the coordinate axes at } \]
\[A \left( \alpha, 0, 0 \right), B \left( 0, \beta, 0 \right) \text{ and } C = \left( 0, 0, \gamma \right)\]
\[\text{ Given that the centroid of the triangle }  =\left( a, b, c \right)\]
\[\Rightarrow\left( \frac{\alpha + 0 + 0}{3}, \frac{0 + \beta + 0}{3}, \frac{0 + 0 + \gamma}{3} \right)=\left( a, b, c \right)\]
\[\Rightarrow\left( \frac{\alpha}{3}, \frac{\beta}{3}, \frac{\gamma}{3} \right)=\left( a, b, c \right)\]
\[\Rightarrow\frac{\alpha}{3}=a,\frac{\beta}{3}=b,\frac{\gamma}{3}=c\]
\[ \Rightarrow \alpha = 3a, \beta = 3b, \gamma = 3c . . . \left( 1 \right)\]
\[\text{ Equation of the plane whose intercepts on the coordinate axes are }  \alpha,\beta\text{ and} \gamma \text{ is} \]
\[\frac{x}{\alpha} + \frac{y}{\beta} + \frac{z}{\gamma} = 1\]
\[ \Rightarrow \frac{x}{3a} + \frac{y}{3b} + \frac{z}{3c} = 1.................... [\text{ From } (1)]\]
\[ \Rightarrow \frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 3\]