Explanation:
\(AC\) power gain
\({\rm{ = }}\frac{{{\rm{change}}\,\,{\rm{in}}\,\,{\rm{output}}\,\,{\rm{power}}}}{{{\rm{change}}\,\,{\rm{in}}\,\,{\rm{input}}\,\,{\rm{power}}}}\)\( = \frac{{\Delta \,{V_c} \times \Delta \,{i_c}}}{{\Delta \,{V_i} \times \Delta \,{i_b}}}\)
\( = \left( {\frac{{\Delta \,{V_c}}}{{\Delta \,{V_i}}}} \right) \times \left( {\frac{{\Delta \,{i_c}}}{{\Delta \,{i_b}}}} \right) = {A_V} \times {\beta _{AC}}\)
Where, \({A_V}\) is voltage gain and \(({\beta _{AC}})\) is \(AC\) current gain. Also,
\({A_V} = {\beta _{AC}} \times {\rm{resistance}}\;{\rm{gain}}\,\left( { = \frac{{{R_o}}}{{{R_i}}}} \right)\)
Voltage gain \({\rm{ = }}\beta \times {\rm{impedance}}\,\,{\rm{gain}}\)
\( \Rightarrow \;\quad 50 = \beta \times \frac{{200}}{{100}} \Rightarrow \beta = 25\)
Also , power gain \({\rm{ = }}{\beta ^2} \times {\rm{impedance}}\,\,{\rm{gain}}\)
\( = {25^2} \times \frac{{200}}{{100}} = 1250\)