Question

A variable plane forms a tetrahedron of constant volume $64K^3$ with the coordinate planes and the origin, then locus of the centroid 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 the variable plane forms the tetrahedron having the vertices $(P,0,0),(0,Q,0),(0,0,R)$

Vertices of tetrahedron are $(P,0,0),(0,Q,0),(0,0,R)$ and $(0,0,0)$

Centroid of the tetrahedron is $=(\frac{P}{4},\frac{Q}{4},\frac{R}{4})\equiv (h,k,j)$

Volume of tetrahedron $=\frac{PQR}{6}=64K^3$

$\Rightarrow \frac{P}{4}.\frac{Q}{4}.\frac{R}{4}=6K^3$

$\Rightarrow h.k.j=6K^3$

Therefore, locus of the centroid is $xyz=6K^3$.