Question
If $$\left| {z - \frac{4}{z}} \right| = 2,$$ then the maximum value of $${\left| z \right|}$$ is equal to:
A.
$$\sqrt 5 + 1$$
B.
2
C.
$$2 + \sqrt 2 $$
D.
$$\sqrt 3 + 1$$
Answer :
$$\sqrt 5 + 1$$
Solution :
$$\eqalign{
& {\text{Given}}\,{\text{that}}\,\,\,\,\left| {z - \frac{4}{z}} \right| = 2, \cr
& {\text{Now}}\left| z \right|\,\,\, = \,\,\,\left| {z - \frac{4}{z} + \frac{4}{{ - z}}} \right|\,\,\,\, \leqslant \,\,\,\,\left| {z - \frac{4}{z}} \right| + \frac{4}{{\left| z \right|}} \cr
& \Rightarrow \,\,\left| z \right|\,\,\, \leqslant 2 + \frac{4}{{\left| z \right|}} \cr
& \Rightarrow \,\,{\left| z \right|^2} - 2\left| z \right| - 4 \leqslant 0 \cr
& \Rightarrow \,\,\left( {\left| z \right| - \frac{{2 + \sqrt {20} }}{2}} \right)\,\left( {\left| z \right| - \frac{{2 - \sqrt {20} }}{2}} \right) \leqslant 0 \cr
& \Rightarrow \left( {\left| z \right| - \left( {1 + \sqrt 5 } \right)} \right)\,\left( {\left| z \right| - \left( {1 - \sqrt 5 } \right)} \right) \leqslant 0 \cr
& \Rightarrow \,\left( { - \sqrt 5 + 1} \right)\, \leqslant \,\,\left| z \right|\, \leqslant \,\left( {\sqrt 5 + 1} \right) \cr
& \Rightarrow \,{\left| z \right|_{\max }} = \sqrt 5 + 1 \cr} $$