Question

A variable plane forms a tetrahedron of constant volume $64K^3$ with the coordinate planes and the origin. The locus of the centroid of the tetrahedron is

• A. $x^3+y^3+z^3=6K^2$
• B. $xyz=6k^3$
• C. $x^2+y^2+z^2=4K^2$
• D. $x^{-2}+y^{-2}+z^{-2}=4k^{-2}$

Answer 1

Answer: (B)

$xyz=6k^3$

Let the variable plane intersect the coordinate axes $A(a,0,0),B(0,b,0),C(0,0,c)$

Then the equation of the plane will be $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1\to \left(1\right)$

Let $P(\alpha ,\beta ,\gamma )$ be the centroid of tetrahedron $OABC$

then$\alpha =\frac{a}{4},\beta =\frac{b}{4},\gamma =\frac{c}{4}$

$\therefore$ volume of tetrahedron=$\frac{1}{3}$(Area of $\lambda$ AOB) $\times OC$

$⟹64k^3=\frac{\left(4\alpha \right)\left(4\beta \right)\left(4\gamma \right)}{6}$

$\therefore k^3=\frac{\left(\alpha \right)\left(\beta \right)\left(\gamma \right)}{6}$

$\therefore$ required locus of $P(\alpha ,\beta ,\gamma )$ is $xyz=6k^3$