question_answer

Let \[{{z}_{1}},{{z}_{2}}\] be two complex numbers represented by points on the circles \[~\left| z \right|=1\] and \[\left| z \right|\text{=}2\] respectively, then which of the following is incorrect

A) \[\max \left| 2{{z}_{1}}+{{z}_{2}} \right|=4\]

B) \[\min \left| {{z}_{1}}-{{z}_{2}} \right|=1\]

C) \[\left| {{z}_{2}}+\frac{1}{{{z}_{1}}} \right|\le 3\]

D) None of these

Correct Answer: D

Solution :

\[\left| 2{{z}_{1}}+{{z}_{2}}\left| \le  \right|2{{z}_{1}} \right|+\left| {{z}_{2}} \right|=2\left| {{z}_{1}} \right|+\left| {{z}_{2}} \right|\]

\[2\times 1+2=4\]

From the figure, \[\left| {{z}_{1}}-{{z}_{2}} \right|\]is least when \[0,{{z}_{1}},{{z}_{2}}\] are collinear.

\[\therefore \left| {{z}_{1}}-{{z}_{2}} \right|=1\]

Again, \[\left| {{z}_{2}}+\frac{1}{{{z}_{1}}} \right|\le \left| {{z}_{2}} \right|+\left| \frac{1}{{{z}_{1}}} \right|=2+\left| \frac{1}{{{z}_{1}}} \right|=2+\frac{1}{1}=3\]