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  • Number of complex number satisfying |z|=1 and |z/z(bar) +zbar/z|=1, is

Number of complex number satisfying |z|=1 and |z/z(bar) +zbar/z|=1, is

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Solution: left | z ight |=1Rightarrow z=(cos heta +isin heta ) for some heta in (0,2pi )

             Now,           left | z ight |=1Rightarrow left | z ight |^2=1Rightarrow zarz=1

            Thus,      z/arz+arz/z=z^2+1/z^2

               Rightarrow      (cos heta +isin heta )^2+(cos heta -isin heta )^2=2(cos2 heta )

              herefore      left | z/arz+arz+z ight |=1Rightarrow left | cos2 heta ight |=1/2

             Rightarrow     cos2 heta =pm 1/2

          2 heta =pi /3,2pi /3,4pi /3,5pi /3,7pi /3,8pi /3,10pi /3,11pi /3

         Hence , there are 8 value of z.  

Posted by

Deependra Verma

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