If the origin is the centroid of the triangle with vertices P(2a, 2, 6), Q(-4, 3b, -10) and R(8, 14, 2c), find the values of a,b and c. Also, determine the value of $a^{2} + b^{2} - c^{2}$ .

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Solution

Coordinates of the centroid of $△$ PQR = $(\frac{2 a - 4 + 8}{3}, \frac{2 + 3 b + 14}{3}, \frac{6 - 10 + 2 c}{3})$

$[∵ coordinate of centroid = (\frac{x_{1} + x_{2} + x_{3}}{3}, \frac{y_{1} + y_{2} + y_{3}}{3}, \frac{z_{1} + z_{2} + z_{3}}{3})]$

= $(\frac{2 a + 4}{3}, \frac{3 b + 16}{3}, \frac{2 c - 4}{3})$

$∴ (0, 0, 0) = (\frac{2 a + 4}{3}, \frac{3 b + 16}{3}, \frac{2 c - 4}{3})$

$\Rightarrow \frac{2 a + 4}{3} = 0, \frac{3 b + 16}{3} = 0 and \frac{2 c - 4}{3} = 0 [∵ centroid of origin = (0, 0, 0)]$

$\Rightarrow 2 a + 4 = 0, 3 b + 16 = 0 and 2 c - 4 = 0$

$\Rightarrow a = - 2, b = - \frac{16}{3} and c = 2$

$∴ a^{2} + b^{2} - c^{2} = (- 2)^{2} + {(\frac{- 16}{3})}^{2} - (2)^{2}$ = $4 + \frac{256}{9} - 4 = \frac{256}{9}$

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If the origin is the centroid of the triangle with vertices P(2a, 2, 6), Q(-4, 3b, -10) and R(8, 14, 2c), find the values of a,b and c. Also, determine the value of a2+b2−c2.

If the origin is the centroid of the triangle with vertices P(2a, 2, 6), Q(-4, 3b, -10) and R(8, 14, 2c), find the values of a,b and c. Also, determine the value of $a^{2} + b^{2} - c^{2}$ .