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JEE Main & Advanced Mathematics Complex Numbers and Quadratic Equations Question Bank Self Evaluation Test - Complex Numbers and Quadratic Equations

  • question_answer
    If \[\omega =\frac{z}{z-\frac{1}{3}i}\] and \[|\omega |=1,\] then z lies on

    A) an ellipse      

    B) a circle

    C) a straight line   

    D) a parabola

    Correct Answer: C

    Solution :

    As given \[\omega =\frac{z}{z-\frac{1}{3}i}\Rightarrow |\omega |=\frac{|z|}{|z-\frac{1}{3}i|}=1\] \[\Rightarrow \,\,\,|z|=\left| z-\frac{1}{3}i \right|\] \[\Rightarrow \] distance of z from origin and point \[\left( 0,\frac{1}{3} \right)\] is same hence z lies on bisector of the line joining points \[(0,0)\] and \[(0,1/3)\]. Hence, z lies on a straight line.


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