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  4. A plane meets the coordinate axes at A, B, C so that the centroid of the triangle A B C is (1,2,4). Then, the equation of the plane is

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Q. A plane meets the coordinate axes at $A, B, C$ so that the centroid of the triangle $A B C$ is $(1,2,4)$. Then, the equation of the plane is

EAMCETEAMCET 2010

Solution:

The equation of the plane meets the coordinate axes at $A, B, C$ is
image
$\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$ ...(i)
Let $O A=a, O B=b, O C=c$
Also, the centroid of $\Delta A B C$ is $\left(\frac{a}{3}, \frac{b}{3}, \frac{c}{3}\right)$.
which is equal to $(1,2,4)$
ie, $\frac{a}{3}=1 \Rightarrow a=3$
$\frac{b}{3}=2 \Rightarrow b=6$
$\frac{c}{3}=4 \Rightarrow c=12$
From Eq. (i), $\frac{x}{3}+\frac{y}{6}+\frac{z}{12}=1$
or $4 x+2 y+ z=12$