The plane $2 x+3 y+4 z=1$ meets $X$-axis in $A$, Y -axis in B and Z -axis in C . Then the centroid of…

The plane $2 x+3 y+4 z=1$ meets $X$-axis in $A$, Y -axis in B and Z -axis in C . Then the centroid of $\triangle A B C$ is
  1. $(2,3,4)$
  2. $\left(\frac{1}{2}, \frac{1}{3}, \frac{1}{4}\right)$
  3. $\left(\frac{1}{6}, \frac{1}{9}, \frac{1}{12}\right)$
  4. $\left(\frac{3}{2}, \frac{3}{3}, \frac{3}{4}\right)$

Solution

Note that $\begin{aligned} & \mathrm{A} \equiv\left(\frac{1}{2}, 0,0\right), \mathrm{B} \equiv\left(0, \frac{1}{3}, 0\right), \mathrm{C} \equiv\left(0,0, \frac{1}{4}\right) \\ & \therefore \quad \text { Cendroid }=\left(\frac{\frac{1}{2}+0+0}{3}, \frac{0+\frac{1}{3}+0}{3}, \frac{0+0+\frac{1}{4}}{3}\right) \\ &=\left(\frac{1}{6}, \frac{1}{9}, \frac{1}{12}\right) \end{aligned}$

Asked in: MHT CET 2024 (11 May Shift 1)

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