Centroid of a Tetrahedron

Q.

If centroid of the tetrahedron OABC, where coordinates of $A,B,C$ are $(a,2,3),(1,b,3)$ and $(2,1,c)$ respectively be $(1,2,3),$ then find the distance of a point $(a,b,c)$ from the origin, where O is the origin.

1. 

2. 7

3. 

4. 

Q. A variable plane which remains at a constant distance p from the origin cuts the coordinate axes in A, B, C. The locus of the centroid of the tetrahedron OABC is $y^2{z}^2+z^2{x}^2+x^2{y}^2=k{x}^2{y}^2{z}^2$, where k is equal to

1. $9p^2$
2. $\frac{9}{p^2}$
3. $\frac{7}{p^2}$
4. $\frac{16}{p^2}$

Q. If centroid of the tetrahedron , whose vertices are given by (0, 0, 0), (a, 2, 3), (1, b, 2) and (2, 1, c) be (1, 2, –1), then distance of P(a, b, c) from origin is equal to

1. $\sqrt{107}$
2. $\sqrt{14}$
3. $\frac{\sqrt{107}}{\sqrt{14}}$
4. $10$

Q.

If centroid of the tetrahedron OABC, where coordinates of $A,B,C$ are $(a,2,3),(1,b,3)$ and $(2,1,c)$ respectively be $(1,2,3),$ then find the distance of a point $(a,b,c)$ from the origin, where O is the origin.

1. $\sqrt{50}$

2. $\sqrt{76}$

3. 7

4. $\sqrt{75}$