Question

A variable plane makes with coordinate planes a tetrahedron of constant volume $64k^3$ . The locus of the centroid of the tetrahedron is

• A. $xyz=6k^3$
• B. $xyz=8k^3$
• C. $x+y+z=6k$
• D. $x^3+y^3+z^3=64k^3$

Answer 1

Answer: (A)

$xyz=6k^3$

Let $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$ be a plane
$\overrightarrow{OA}=a\hat{i}$, $\overrightarrow{OB}=b\hat{j}$, $\overrightarrow{OC}=c\hat{k}$
Centroid of tetrahedron $(x\hat{i}+y\hat{j}+z\hat{k})=\frac{\overrightarrow{OO+}\overrightarrow{OA}+\overrightarrow{OB}+\overrightarrow{OC}}{4}=\frac{a\hat{i}+b\hat{j}+c\hat{k}}{4}$
Therefore, $a=4x$, $b=4y$ and $c=4z$
Now, volume of tetrahedron $=64k^3=\frac{\left[\overrightarrow{OA}\overrightarrow{OB}\overrightarrow{OC}\right]}{6}=\frac{abc}{6}=\frac{\left(4x\right)\left(4y\right)\left(4z\right)}{6}$
$\Rightarrow xyz=6k^3$
Ans: A