Given z=2cosθ+i(1-2sinθ) ,How to prove that the real part of #1/(z+2-i)# is a constant for -180<θ<180?

1 Answer
P dilip_k
Oct 22, 2017

Given complex number

#=1/(z+2-i)#

#=1/(2costheta+i -2i sintheta+2-i)#

#=1/(2costheta -2isintheta+2)#

#=1/(2(costheta -isintheta+1))#

#=(1+costheta+isintheta)/(2(costheta -isintheta+1)(1+costheta+isintheta))#

#=(1+costheta+isintheta)/(2(1+cos^2theta +2costheta+sin^2theta))#

#=(1+costheta+isintheta)/(2(2+2costheta))#

#=(1+costheta)/(4(1+costheta))+isintheta/(4(1+costheta))#

From above expression it is obvious that
if #(1+costheta)!=0# or #-180^@< theta < 180^@ # then the real part of the given complex number will be a constant.

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