b
(b) \(\because {T_B} > {T_A}\) ==> Heat will flow \(B\) to \(A\) via two paths \((i)\) \(B\) to \(A\) \((ii)\) and along \(BCA\) as shown.

Rate of flow of heat in path \(BCA\) will be same

i.e. \({\left( {\frac{Q}{t}} \right)_{BC}} = {\left( {\frac{Q}{t}} \right)_{CA}}\)

\( \Rightarrow \frac{{k(\sqrt 2 T - {T_C})A}}{a} = \frac{{k({T_C} - T)A}}{{\sqrt 2 a}}\)

\( \Rightarrow \frac{{{T_C}}}{T} = \frac{3}{{1 + \sqrt 2 }}\)