A) \[|{{z}_{1}}+{{z}_{2}}|\]
B) \[|{{z}_{1}}-{{z}_{2}}|\]
C) \[|{{z}_{1}}+{{z}_{2}}|\]
D) \[|{{z}_{1}}|-|{{z}_{2}}|\]
Correct Answer: C
Solution :
R.H.S = \[\frac{1}{2}|{{(\sqrt{{{z}_{1}}}+\sqrt{{{z}_{2}}})}^{2}}|+\frac{1}{2}|{{(\sqrt{{{z}_{1}}}-\sqrt{{{z}_{2}}})}^{2}}|\] \[=\frac{1}{2}|\sqrt{{{z}_{1}}}+\sqrt{{{z}_{2}}}{{|}^{2}}+\frac{1}{2}|\sqrt{{{z}_{1}}}-\sqrt{{{z}_{2}}}{{|}^{2}}\] \[\{\because |{{z}^{2}}|=|z{{|}^{2}}\}\] \[=\frac{1}{2}2\,[|\sqrt{{{z}_{1}}}{{|}^{2}}+|\sqrt{{{z}_{2}}}{{|}^{2}}]=|{{z}_{1}}|+|{{z}_{2}}|\]
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