Mathematics - 3D-Coordinate Geometry Question with Solution | TestHub

Let co-ordinate of the points A, B, C be $(\mathrm{a},0,0),(0,\mathrm{b},0)$ and $(0,0,\mathrm{c})$ respectively. Then equation of the sphere $\mathrm{OABC}$ is $x^2+y^2+z^2-ax-by-cz=0$ Radius of this sphere is equal to $2\mathrm{k}$ $\therefore {\mathrm{a}}^2+{\mathrm{b}}^2+{\mathrm{c}}^2=4(2\mathrm{k}{)}^2=16{\mathrm{k}}^2$ Also let (z,y, z) be the co-ordinates of the centroid of the tetrahedron $\mathrm{OABC}$, then $\mathrm{x}=\frac{1}{4}\mathrm{a},\mathrm{y}=\frac{1}{4}\mathrm{b},\mathrm{z}=\frac{1}{4}\mathrm{c}$ so that $\mathrm{a}=4\mathrm{x},\mathrm{b}=4\mathrm{y},\mathrm{c}=4\mathrm{z}$ Eliminating a, b, c from (1) and (2), the required locus is $\begin{array}{rl}& 16x^2+16y^2+16z^2=16k^2\\ & \text{ or }{x}^2+y^2+z^2=k^2\end{array}$