If a² + b² + c² + 96 = 8(a + b - 2c), then $$\sqrt {ab - bc - ca} $$   is equal to

Solution(By Examveda Team)

\[\begin{align} & {{a}^{2}}+{{b}^{2}}+{{c}^{2}}+96=8\left( a+b-2c \right) \\ & \Rightarrow {{a}^{2}}+{{b}^{2}}+{{c}^{2}}+96=8a+8b-16c \\ & \Rightarrow {{a}^{2}}-2\times 4a+{{b}^{2}}-2\times 4b+{{c}^{2}}+2\times 8c+96=0 \\ & \Rightarrow \left( {{a}^{2}}-2\times 4a+16 \right)+\left( {{b}^{2}}-2\times 4b+16 \right)+\left( {{c}^{2}}+2\times 8c+64 \right)=16+16+64-96 \\ & \Rightarrow {{\underset{\begin{smallmatrix} \downarrow \\ 0 \end{smallmatrix}}{\mathop{\left( a-4 \right)}}\,}^{2}}+{{\underset{\begin{smallmatrix} \downarrow \\ 0 \end{smallmatrix}}{\mathop{\left( b-4 \right)}}\,}^{2}}+{{\underset{\begin{smallmatrix} \downarrow \\ 0 \end{smallmatrix}}{\mathop{\left( c-8 \right)}}\,}^{2}}=0 \\ & a-4=0,\,\,a=4 \\ & b-4=0,\,\,b=4 \\ & c-8=0,\,\,c=8 \\ & \sqrt{ab-bc-ca} \\ & =\sqrt{{{a}^{2}}-ac+ca} \\ & =a \\ & =4 \\ \end{align}\]