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 $4x+2y+z=12$

Let the equation of the plane is
$\frac{x}{\alpha }+\frac{y}{\beta }+\frac{z}{\gamma }=1$
Then, $A(\alpha ,0,0),B(0,\beta ,0)$ and $C(0,0,\gamma )$
are 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
$\frac{x}{3}+\frac{y}{6}+\frac{z}{12}=1$
$\Rightarrow 4x+2y+z=12$