Question

A variable plane forms a tetrahedron of constant volume 64$K^3$ with the co-ordinate planes and the origin, then locus of the centroid of the tetrahedron is

Answer 1

Vertices on co-ordinate of tetrahedron OABC,

$\begin{array}{c}So,co-ordinateofO\left(0,0,0\right),A\left(x,0,0\right),B\left(0,y,0\right),C\left(0,0,z\right)\\ \overrightarrow{OA}=x\hat{i},\overrightarrow{OB}=y\hat{j},\overrightarrow{OC}=z\hat{k}\\ Valueoftetrahedron\Rightarrow \frac{1}{6}\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]\\ v=\frac{1}{6}\left\{\overrightarrow{a}.\left(\overrightarrow{b}\times \overrightarrow{c}\right)\right\}\\ v=\frac{1}{6}\left\{x\hat{i}.\left(y\hat{j}\times 3\hat{k}\right)\right\}\Rightarrow \frac{1}{6}\left\{x\hat{i}.\left(yzj\hat{i}\right)\right\}\left\{i.\hat{i}=1\right\}\\ v=\frac{xyz}{6}\\ xyz=6\times 64k^3\to \left(i\right)\left[v=64k^3\leftarrow Given\right]\\ Centroidis\left(\frac{x}{4},\frac{y}{4},\frac{z}{4}\right)\\ Letthecentre\left(x,y,z\right)\end{array}$

Comparing both, we get,

$\frac{x}{4}=x,\Rightarrow x=4x,y=4y,z=4z$

Put the value in equation (i), we get

$\begin{array}{c}4x_1,4y_1,4z_1=6\times 64k^3\Rightarrow 6_1{x}_1{y}_1{z}_1=6\times 64k^3\\ x_1{y}_1{z}_1=6k^3\\ xyz=6k^3\end{array}$