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

A(a,b,c)
Points P, Q & R are `(a,b,-c), (-a,b,c)`
& `(a,-b,c)`
Respectively
Controied of `DeltaPQRT` is `((a)/(3), (b)/(3), (c)/(3))G`
`rArr A, O, G` are collinear `rArr` Area of `DeltaAOG = 0`