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JEE Main & Advanced Mathematics Complex Numbers and Quadratic Equations Question Bank Conjugate, Modulus and Argument of complex number

  • question_answer
    If z and \[\omega \]are two non-zero complex numbers such that \[|z\omega |\,=1\] and \[arg(z)-arg(\omega )=\frac{\pi }{2},\] then \[\bar{z}\omega \] is equal to [AIEEE 2003]

    A) 1

    B) - 1

    C) i

    D) - i

    Correct Answer: D

    Solution :

    \[|z|\,|\omega |\,=1\]               ......(i) and \[arg\,\left( \frac{z}{\omega } \right)=\frac{\pi }{2}\,\,\,\Rightarrow \,\,\frac{z}{\omega }=i\] Þ \[\left| \frac{z}{\omega } \right|=1\]  .....(ii) From equation (i) and (ii) \[|z|\,=\,|\omega |\,=1\] and \[\frac{z}{\omega }+\frac{{\bar{z}}}{{\bar{\omega }}}=0;\,\,\,z\bar{\omega }+\bar{z}\omega =0\] \[\bar{z}\omega =-z\bar{\omega }=\frac{-z}{\omega }\bar{\omega }\,\omega \]; \[\bar{z}\omega =-\,i\,|\omega {{|}^{2}}=-i.\].


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