Question

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

Answer 1

The Correct Option is B

Let the equation of the plane is
$\frac{x}{\alpha} + \frac{y}{\beta} + \frac{z}{\gamma} = 1 $
Then, $A \left(\alpha,0,0\right),B\left(0,\beta,0\right)$ and $C\left(0,0,\gamma\right)$
Since, the points on the coordinate axes,
The centroid of the triangle is (1, 2, 4).
$\therefore \frac{\alpha}{3} = 1 \Rightarrow \alpha=3$
$\frac{\beta}{3} = 2 \Rightarrow \beta=6 $
and $\frac{\gamma}{3} =4 \Rightarrow \gamma =12$
$\therefore$ The equation of the plane is
$\frac{x}{3}+\frac{y}{6} +\frac{z}{12}=1$
$\Rightarrow 4x+2y+z =12$