Question

If a variable plane forms a tetrahedron of constant volume $64k^3$ with the coordinate planes, find the locus of the centroid of the tetrahedron.

• A. $xyz=32k^3$
• B. $xyz=6k^3$
• C. $xyz=12k^3$
• D. None of these

Answer 1

The correct option is B $xyz=6k^3$

Let the variable plane intersects 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

$\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$ ...(1)

Let $P(\alpha ,\beta ,\gamma )$ be the centroid of tetrahedron $OABC,$ then $\alpha =\frac{a}{4},\beta =\frac{b}{4},\gamma =\frac{c}{4}$ or $a=4\alpha ,b=4\beta ,c=4\gamma .$

$\Rightarrow$ Volume of tetrahedron $=\frac{1}{3}$ $($ Area of $△OAB).OC$

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

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

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

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