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

x y z = 6 k^{3}

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x y z = 8 k^{3}

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x + y + z = 6 k

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x^{3} + y^{3} + z^{3} = 64 k^{3}

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Solution

The correct option is C $x y z = 6 k^{3}$
Let $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$ be a plane
$\to O A = a^i$ , $\to O B = b^j$ , $\to O C = c^k$
Centroid of tetrahedron $(x^i + y^j + z^k) = \frac{\to O O + \to O A + \to O B + \to O C}{4} = \frac{a^i + b^j + c^k}{4}$
Therefore, $a = 4 x$ , $b = 4 y$ and $c = 4 z$
Now, volume of tetrahedron $= 64 k^{3} = \frac{[\to O A \to O B \to O C]}{6} = \frac{a b c}{6} = \frac{(4 x) (4 y) (4 z)}{6}$
$\Rightarrow x y z = 6 k^{3}$
Ans: A

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