If the centroid of tetrahedron OABC where A B C are given by (a 2 3) (1 b 2) and (2 1 c) respectively is (1 2 -1) then distance of P(a b c) from origin is

If the centroid of tetrahedron OABC where A,B,C are given by (a,2,3),(1,b,2) and (2,1,c) respectively is (1,2,−2), then distance of P(a,b,c) from origin is

If (1, 1, 1) is the centroid of the triangle with vertices (2a, 3, 7), (-4, 3b, -8) and (8, 4, 6c), find a, b, c.

If the centroid of the triangle formed by points P ( a , b ) , Q ( b , c ) and R ( c , a ) is at the origin, what is the value of (a^2)/(b c)+(b^2)/(c a)+(c^2)/(a b) ?

If origin is the centroid of a triangle ABC having vertices A(a,1,3), B(-2,b,-5) and C(4,7, c) , then the values of a, b, c are

If the volume of tetrahedron A B C D is 1 cubic units, where A(0,1,2),B(-1,2,1) and C(1,2,1), then the locus of point D is a. x+y-z=3 b. y+z=6 c. y+z=0 d. y+z=-3

If the volume of tetrahedron A B C D is 1 cubic units, where A(0,1,2),B(-1,2,1)a n dC(1,2,1), then the locus of point D is a. x+y-z=3 b. y+z=6 c. y+z=0 d. y+z=-3

OABC is a tetrahedron where O is the origin and A,B,C have position vectors veca,vecb,vecc respectively prove that circumcentre of tetrahedron OABC is (a^2(vecbxxvecc)+b^2(veccxxveca)+c^2(vecaxxvecb))/(2[veca vecb vecc])

Two planes P_1 and P_2 pass through origin. Two lines L_1 and L_2 also passingthrough origin are such that L_1 lies on P_1 but not on P_2, L_2 lies on P_2 but not on P_1 A,B, C are there points other than origin, then prove that the permutation [A', B', C'] of [A, B, C] exists. Such that: (a) A lies on L1, B lies on P1 not on L1, C does not lie on P1 . (b) A lies on L2, B lies on P2 not on L2, C' does not lies on P2.

If in a /_\ABC, a,b,c are in A.P. and P_(1),P_(2),P_(3) are th altitude from the vertices A,B and C respectively then

Two systems of rectangular axes have the same origin. If a plane cuts them at distance a ,b ,c and a^prime ,b^(prime),c ' from the origin, then a. 1/(a^2)+1/(b^2)+1/(c^2)+1/(a^('2))+1/(b^('2))+1/(c^('2))=0 b. 1/(a^2)-1/(b^2)-1/(c^2)+1/(a^('2))-1/(b^('2))-1/(c^('2))=0 c. 1/(a^2)+1/(b^2)+1/(c^2)-1/(a^('2))-1/(b^('2))-1/(c^('2))=0 d. 1/(a^2)+1/(b^2)+1/(c^2)+1/(a^('2))+1/(b^('2))+1/(c^('2))=0