If a variable plane forms a tetrahedron of constant volume $64 k^{3}$ with the co-ordinate planes, then the locus of the centroid of the tetrahedron is

$x y z = k^{3}$

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$x y z = 2 k^{3}$

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$x y z = 12 k^{3}$

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$x y z = 6 k^{3}$

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Solution

The correct option is D
$x y z = 6 k^{3}$

Let the variable plane intersect the co-ordinate axes at A(a, 0, 0), B(0,b, 0) and C(0, 0, c). Then the equation of the plane will be
$\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$ ......................(i)
Let P $(α, β, γ)$ be the centroid of tetrahedron OABC. Then,
$α = \frac{a}{4}, β = \frac{b}{4}, γ = \frac{c}{4} o r a = 4 α, b = 4 β, c = 4 γ . \Rightarrow V o l u m e o f t e t r a h e d r o n = \frac{1}{3} \times (A r e a o f Δ A O B) \times O C \Rightarrow 64 k^{3} = \frac{1}{3} (\frac{1}{2} a b) c = \frac{a b c}{6} \Rightarrow 64 k^{3} = \frac{(4 α) (4 β) (4 γ)}{6} \Rightarrow k^{3} = \frac{α β γ}{6}$
Therefore, the required locus of the centroid P is $x y z = 6 k^{3}$

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If a variable plane forms a tetrahedron of constant volume 64k3 with the co-ordinate planes, then the locus of the centroid of the tetrahedron is

If a variable plane forms a tetrahedron of constant volume $64 k^{3}$ with the co-ordinate planes, then the locus of the centroid of the tetrahedron is