If the origin is the centroid of the triangle $P Q R$ with vertices $P (2 a, 2, 6), Q (- 4, 3 b, - 10)$ and $R (8, 14, 2 c),$ then find the values of $a, b$ and $c$

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Solution

The coordinates of the centroid of $△ P Q R$
$= (\frac{2 a - 4 - 8}{3}, \frac{2 + 3 b + 14}{3}, \frac{6 - 10 + 2 c}{3}) = (\frac{2 a + 4}{3}, \frac{3 b + 16}{3}, \frac{2 c - 4}{3})$
It is given that origin is the centroid of $△ P Q R$
$∴$ $(0, 0, 0) = (\frac{2 a + 4}{3}, \frac{3 b + 16}{3}, \frac{2 c - 4}{3})$
$\Rightarrow \frac{2 a + 4}{3} = 0, \frac{3 b + 16}{3} = 0 and \frac{2 c - 4}{3} = 0$
$\Rightarrow a = - 2, b = - \frac{16}{3}$ and $c = 2$

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