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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  \[{{z}_{1}}\] and \[{{z}_{2}}\] are two complex numbers satisfying the equation  \[\left| \frac{{{z}_{1}}+{{z}_{2}}}{{{z}_{1}}-{{z}_{2}}} \right|\]=1, then \[\frac{{{z}_{1}}}{{{z}_{2}}}\] is a number which is

    A) Positive real

    B) Negative real

    C) Zero or purely imaginary

    D)  None of these

    Correct Answer: C

    Solution :

    Given \[\left| \frac{{{z}_{1}}+{{z}_{2}}}{{{z}_{1}}-{{z}_{2}}} \right|=1\]Þ \[\frac{{{z}_{1}}+{{z}_{2}}}{{{z}_{1}}-{{z}_{2}}}=\cos \theta +i\sin \theta \](say) Þ \[\frac{{{z}_{1}}}{{{z}_{2}}}=\frac{1+\cos \theta +i\sin \theta }{-1+\cos \theta +i\sin \theta }=-i\cot \frac{\theta }{2}\] which is zero, if  \[\theta =n\pi (n\in I),\] and is otherwise purely imaginary.


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