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In a three-dimensional coordinate sys...

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

A

1

B

2

C

3

D

4

Text Solution

Verified by Experts

The correct Answer is:
C
Vertices are `A(a,0),B(-a,0)` and `C(b,c)`
`:.` Centroid is `G((b)/(3),(c)/(3))`
`(AB^(2)+BC^(2)+CA^(2))/(GA^(2)+GB^(2)+GC^(2))`
`=(4a^(2)+(a+b)^(2)+c^(2)+(a-b)^(2)+c^(2))/(((b)/(3)-a)^(2)+((c)/(3))^(2)+((b)/(3)+a)^(2)+((c)/(3))^(2)+((2b)/(3))^(2)+((2c)/(3))^(2))`
`=(4a^(2)+2c^(2)+2a^(2)+2b^(2))/((2b^(2))/(9)+2a^(2)+(6c^(2))/(9)+(4b^(2))/(9))=3`
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