A) \[{{x}^{2}}+{{y}^{2}}\]
B) \[\sqrt{{{x}^{2}}+{{y}^{2}}}\]
C) \[x+iy\]
D) \[x-iy\]
Correct Answer: D
Solution :
\[\sqrt{a+ib}=x+yi\,\Rightarrow \,{{\left( \sqrt{a+i\,b} \right)}^{2}}={{(x+yi)}^{2}}\] \[\Rightarrow \,a={{x}^{2}}-{{y}^{2}},\,b=2xy\] and hence \[\sqrt{a-ib}=\sqrt{{{x}^{2}}-{{y}^{2}}-2xyi}\]\[=\sqrt{{{(x-yi)}^{2}}}\]\[=x-iy\] Note: In the question, it should have been given that \[a,\,b,\,x,\,\,y\,\in \,R.\]
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