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