Question

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

• A. $x^3+y^3+z^3=6K^3$
• 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 variable cut coordinate axes at $A(a,0,0),B(0,b,0),C(0,0,c)$

Then equation of the plane will be $\frac{x}{a}=\frac{y}{b}=\frac{z}{c}=1$

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

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

Volume of tetrahedron $=\frac{1}{3}$ $($area of $△AOB).OC$

$\Rightarrow 64k^3=\frac{1}{3}\left(\frac{1}{2}ab\right)c=\frac{abc}{6}$

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

$\Rightarrow \frac{64\times 6k^3}{64}=\alpha \beta \gamma$

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