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 =
(a) 1
(b) 2
(c) 3
(d) None of these

Answer 1

(c) 3

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