If 3a = 4b = 6c and a + b + c = 27 29 , then a^2 + b^2 + c^2 is ?

Answer: Option C

$$\begin{array}{rl}& a+b+c=27\sqrt{29}\\ & \Rightarrow 2c+\frac{3}{2}c+c=27\sqrt{29}\\ & \Rightarrow \frac{9}{2}c=27\sqrt{29}\\ & \Rightarrow c=6\sqrt{29}\\ & \therefore \sqrt{a^2+b^2+c^2}\\ & =\sqrt{{\left(a+b+c\right)}^2-2\left(ab+bc+ca\right)}\\ & =\sqrt{{\left(27\sqrt{29}\right)}^2-2\left(2c\times \frac{3}{2}c+\frac{3}{2}c\times c+c\times 2c\right)}\\ & =\sqrt{\left(729\times 29\right)-2\left(3c^2+\frac{3}{2}{c}^2+2c^2\right)}\\ & =\sqrt{\left(729\times 29\right)-2\times \frac{13}{2}{c}^2}\\ & =\sqrt{\left(729\times 29\right)-13{\left(6\sqrt{29}\right)}^2}\\ & =\sqrt{29\left(729-468\right)}\\ & =\sqrt{29\times 261}\\ & =\sqrt{29\times 29\times 9}\\ & =29\times 3\\ & =87\end{array}$$