In a three-dimensional coordinate system, `P ,Q ,a n dR` are images of a point `A(a ,b ,c)` in the `x-y ,y-za n dz-x` planes, respectively. If `G` is the centroid of triangle `P Q R ,` then area of triangle `A O G` is (`O` is the origin) a. `0` b. `a^2+b^2+c^2` c. `2/3(a^2+b^2+c^2)` d. none of these

Options

A. 0
B. `a^(2)+b^(2)+c^(2)`
C. `2/3(a^(2)+b^(2)+c^(2))`
D. none of these

Answer

A

Explanation

Point A is (a,b,c)
`rArr` Points P,Q and R are (a,b,-c), (-a,b,c) and (a,-b.c) respectively.
`rArr` Centroid of triangle PQR is `G(a/3, b/3, c/3)`
Clearly, A,O and G are collinear
`rArr` Area of triangle AOG is zero.