Centroid of a Tetrahedron

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 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. $\sqrt{75}$

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 OABC, where A, B, C are given by (a, 2, 3), (1, b, 2) and (2, 1, c) respectively be (1, 2, -1), then distance of P(a, b, c) from origin is equal to

1. $\sqrt{107}$

2. $\sqrt{14}$

3. $\sqrt{\frac{107}{14}}$

4. $Noneofthese$

Q.

If a variable plane forms a tetrahedron of constant volume $64k^3$ with the co-ordinate planes, then the locus of the centroid of the tetrahedron is

1. $xyz=k^3$

2. $xyz=2k^3$

3. $xyz=12k^3$

4. $xyz=6k^3$