Question

If origin is the centroid of a $ \Delta PQR $ with vertices $ P(2a,2,6),Q(-4,3b,- 10) $ and $ (8,14,2c) $ , then the value of $ a,b $ and $ c $ are respectively.

• A. $ - 2 , 2 , 2 $
• B. $ -2, 2 , - \frac {16}{3} $
• C. $ -2, - \frac {16}{3}, 2 $
• D. $ -\frac {16}{3}, -2,2 $

Answer 1

Answer: (C)

Coordinates of centroid of $\Delta PQR$
$\equiv\left(\frac{2a-4+8}{3}, \frac{2 +3b +14}{3}, \frac{6-10 + 2c}{3}\right)$
But it is given that origin is the centroid of $\Delta PQR$
$\therefore \left(0,0,0\right) = \left(\frac{2a+4}{3}, \frac{3b+16}{3}, \frac{2c-4}{3}\right) $
On comparing both sides, we get
$\frac{2a+4}{3} = 0, \frac{3b+16}{3} = 0$ and $\frac{2c-4}{3} = 0$
$\Rightarrow 2a = -4, 3b= -16$ and $2c = 4 $
$\Rightarrow a = -2, b = - \frac{16}{3}$ and $c = 2$
$ \therefore \left( a, b, c\right)\equiv \left(-2, -\frac{16}{3}, 2\right)$