Q74 / 120 · VITEEE 2014 · Mathematics

If a plane meets the coordinate axes at $A, B$ and $C$ such that the centroid of the triangle is $(1,2,4)$, then the equation of the plane is

  • A
    $x+2 y+4 z=12$
  • Correct answer
    B
    $4 x+2 y+z=12$
  • C
    $x+2 y+4 z=3$
  • D
    $4 x+2 y+z=3$

Solution

Let the equation of the plane is $\frac{x}{\alpha}+\frac{y}{\beta}+\frac{z}{\gamma}=1$ Then, $\mathrm{A}(\alpha, 0,0), \mathrm{B}(0, \beta, 0)$ and $\mathrm{C}(0,0, \gamma)$ are the points on the coordinate axes, The centroid of the triangle is $(1,2,4)$. $\begin{array}{l} \therefore \frac{\alpha}{3}=1 \Rightarrow \alpha=3 \\ \frac{\beta}{3}=2 \Rightarrow \beta=6 \end{array}$ and $\frac{\gamma}{3}=4 \Rightarrow \gamma=12$ $\therefore \quad$ The equation of the plane is $\begin{aligned} & \frac{x}{3}+\frac{y}{6}+\frac{z}{12}=1 \\ \Rightarrow & 4 x+2 y+z=12 \end{aligned}$

Hence, the correct answer is option: 'B'.

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