Question

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

• A. $xyz=k^3$
• B. $xyz=2k^3$
• C. $xyz=12k^3$
• D. $xyz=6k^3$

Answer 1

The correct option is D

$xyz=6k^3$

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 $(\alpha ,\beta ,\gamma )$ be the centroid of tetrahedron OABC. Then,
$\alpha =\frac{a}{4},\beta =\frac{b}{4},\gamma =\frac{c}{4}ora=4\alpha ,b=4\beta ,c=4\gamma .\Rightarrow Volumeoftetrahedron=\frac{1}{3}\times \left(Areaof\Delta AOB\right)\times OC\Rightarrow 64k^3=\frac{1}{3}\left(\frac{1}{2}ab\right)c=\frac{abc}{6}\Rightarrow 64k^3=\frac{\left(4\alpha \right)\left(4\beta \right)\left(4\gamma \right)}{6}\Rightarrow k^3=\frac{\alpha \beta \gamma }{6}$
Therefore, the required locus of the centroid P is $xyz=6k^3$