The circle with center (1,0) and radius 5 intercepts at points A and B

\(? = w + z = {x_M} + {y_M}\,\,\,\,\,\left[ {M = \left( {{x_M};{y_M}} \right)} \right]\)


\(A,B\,\,\,\,\left\{ \matrix{\\
\,{\left( {x - 1} \right)^2} + {y^2} = {5^2}\,\,\,\,\,\left[ {{\rm{circle}}} \right] \hfill \cr \\
\,y = 2x\,\,\,\,\,\left[ {{\rm{line}}} \right] \hfill \cr} \right.\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,{\left( {x - 1} \right)^2} + {\left( {2x} \right)^2} = {5^2}\,\,\,\, \Rightarrow \,\,\,\,5{x^2} - 2x - 24 = 0\,\,\,\,\left( * \right)\)


\(M = \left( {{{{x_A} + {x_B}} \over 2}\,\,;\,\,{{{y_A} + {y_B}} \over 2}} \right)\,\,\,\mathop = \limits^{M\, \in \,\,{\rm{line}}} \,\,\,\left( {{{{x_A} + {x_B}} \over 2}\,\,;\,\,{{2\,{x_A} + 2\,{x_B}} \over 2}} \right)\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,? = {3 \over 2}\left( {{x_A} + {x_B}} \right)\)


\(\left( * \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,{x_A} + {x_B} = {\rm{sum}}\,\,{\rm{roots}} = - {{\left( { - 2} \right)} \over 5} = {2 \over 5}\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,? = {3 \over 2}\left( {{2 \over 5}} \right) = {3 \over 5}\)


The correct answer is therefore (C).


We follow the notations and rationale taught in the GMATH method.

Regards,
Fabio.