question_answer

If z satisfies the equation \[\left( \frac{z-2}{z+2} \right)\left( \frac{\overline{z}-2}{\overline{z}+2} \right)=1\], then minimum value of \[\left| z \right|\] is equal to

A)  0        

B)  2

C)  4                                

D)  6

Correct Answer: A

Solution :

\[(z-2)(\bar{z}-2)=(z+2)(\bar{z}+2)\] \[{{\left| z-2 \right|}^{2}}{{\left| z+2 \right|}^{2}}\] \[\Rightarrow \left| z-2 \right|=\left| z+2 \right|\] \[\Rightarrow z\] lies on imaginary axis. \[\therefore {{\left| {{z}_{1}} \right|}_{\min }}=0\]