A) \[|{{z}_{1}}{{|}^{2}}+|{{z}_{2}}{{|}^{2}}\]
B) \[2(|{{z}_{1}}{{|}^{2}}+|{{z}_{2}}{{|}^{2}})\]
C) \[2(z_{1}^{2}+z_{2}^{2})\]
D) \[4{{z}_{1}}{{z}_{2}}\]
Correct Answer: B
Solution :
We know, \[|{{z}_{1}}+{{z}_{2}}{{|}^{2}}=|{{z}_{1}}{{|}^{2}}+|{{z}_{2}}{{|}^{2}}+2\operatorname{Re}\,({{z}_{1}}{{\bar{z}}_{2}})\] \[|{{z}_{1}}-{{z}_{2}}{{|}^{2}}=|{{z}_{1}}{{|}^{2}}+|{{z}_{2}}{{|}^{2}}-2\operatorname{Re}\,({{z}_{1}}{{\bar{z}}_{2}})\] \[\Rightarrow \,\,|{{z}_{1}}+{{z}_{2}}{{|}^{2}}+|{{z}_{1}}-{{z}_{2}}{{|}^{2}}=2(|{{z}_{1}}{{|}^{2}}+|{{z}_{2}}{{|}^{2}})\]
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