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$In a three dimensional co-ordinate system P,Q and R are images of a point A(a,b,c) in the xy, yz and zx planes respectively.$
$If G is the centroid of triangle PQR then area of triangle AOG is (O is the origin)$

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$0$

$a^{2} + b^{2} + c^{2}$

$2 (a^{2} + b^{2} + c^{2})$

$\frac{2}{3} (a^{2} + b^{2} + c^{2})$

answer is A.

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Detailed Solution

$P = (a, b, 0), Q = (0, b, c) a n d R = (a, 0, c) \Rightarrow G = (\frac{2 a}{3}, \frac{2 b}{3}, \frac{2 c}{3}) = \frac{2}{3} (a, b, c) \Rightarrow O, G, A a r e c o l l i n e a r . \Rightarrow a r e a o f t r i a n g l e A O G = 0$

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