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JEE Main & Advanced Mathematics Complex Numbers and Quadratic Equations Question Bank Geometry of complex numbers

  • question_answer
    If  \[{{\log }_{\sqrt{3}}}\left( \frac{|z{{|}^{2}}-|z|+1}{2+|z|} \right)\]\[<2\], then the locus of \[z\] is

    A) \[|z|=5\]

    B) \[|z|<5\]

    C) \[|z|>5\]

    D) None of these

    Correct Answer: B

    Solution :

    We have \[{{\log }_{\sqrt{3}}}\left( \frac{|z{{|}^{2}}-|z|+1}{2+|z|} \right)<2\] Þ \[\frac{|z{{|}^{2}}-|z|+1}{2+|z|}<{{(\sqrt{3})}^{2}}\] Þ \[|z{{|}^{2}}-4|z|-5<0\] Þ \[-1<|z|<5\] Þ  \[|z|<5\]as \[|z|>0\] \  Locus of \[z\] is |z|<5


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