Question

If the centroid of the tetrahedron $OABC$, where $A,B,C$ are given by $(\alpha ,5,6),(1,\beta ,4),(3,2,\gamma )$ respectively be $1,-1,2$, then value of ${\alpha }^2+{\beta }^2+{\gamma }^2$ equals

• A. ${\alpha }^2+{\beta }^2$
• B. ${\gamma }^2+{\beta }^2$
• C. ${\alpha }^2+{\gamma }^2$
• D. None of these

Answer 1

The correct option is B ${\gamma }^2+{\beta }^2$

The vertices of tetrahedron are $(0,0,0),(\alpha ,5,6),(1,\beta ,4),(3,2,\gamma )$

The centroid of tetrahedron is $(\frac{\alpha +4}{4},\frac{\beta +7}{4},\frac{\gamma +10}{4})$

Given that centroid is $1,-1,2$

By equating , we get $\alpha =0,\beta =-11,\gamma =-2$

Since $\alpha =0$ , ${\alpha }^2+{\beta }^2+{\gamma }^2={\beta }^2+{\gamma }^2$

Therefore the correct option is $B$