If ab = 21 and ( a + b )^2 ( a - b )^2 = 254 , then the value of a2 + b2 + 3ab is?

Answer: Option B

\begin{aligned} & \frac{{{{\left( {a + b} \right)}^2}}}{{{{\left( {a - b} \right)}^2}}} = \frac{{25}}{4} \\ & \Rightarrow \frac{{a + b}}{{a - b}} = \frac{5}{2} \\ & \Rightarrow {\text{By Componendo & Dividendo}} \\ & \Rightarrow \frac{{a + b + a - b}}{{a + b - a + b}} = \frac{{5 + 2}}{{5 - 2}} \\ & \Rightarrow \frac{{2a}}{{2b}} = \frac{7}{3} \\ & \Rightarrow \frac{a}{b} = \frac{7}{3} \\ & {\text{Now, the value of}} \\ & \Rightarrow {a^2} + {b^2} + 3ab \\ & \Rightarrow {7^2} + {3^2} + 3.7.3 \\ & \Rightarrow 49 + 9 + 63 \\ & \Rightarrow 121 \\\end{aligned}