Question

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

• A. x + 2y + 4z = 12
• B. 4x + 2y + z = 12
• C. x + 2y + 4z = 3
• D. 4x + 2y + z = 3

Answer 1

Answer: (B)

The equation of the plane meets the coordinate axes at $A,B,C$ is

$\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$ ...(i)
Let $OA=a,OB=b,OC=c$
Also, the centroid of $\Delta ABC$ 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 $4x+2y+z=12$