JEE Main & Advanced Mathematics Complex Numbers and Quadratic Equations Question Bank Integral power of iota, Algebraic operations and Equality of complex numbers

  • question_answer  Let \[{{z}_{1}},{{z}_{2}}\] be two complex numbers such that \[{{z}_{1}}+{{z}_{2}}\] and \[{{z}_{1}}{{z}_{2}}\] both are real, then [RPET 1996]

    A) \[{{z}_{1}}=-{{z}_{2}}\]

    B) \[{{z}_{1}}={{\bar{z}}_{2}}\]

    C) \[{{z}_{1}}=-{{\bar{z}}_{2}}\]

    D) \[{{z}_{1}}={{z}_{2}}\]

    Correct Answer: B

    Solution :

    Let \[{{z}_{1}}=a+ib,{{z}_{2}}=c+id\], then  \[{{z}_{1}}+{{z}_{2}}\] is real    Þ \[(a+c)+i(b+d)\]is real Þ \[b+d=0\]     Þ \[d=-b\]            .....(i) \[{{z}_{1}}{{z}_{2}}\] is real        Þ \[(ad-bd)+i(ac+bc)\]is real Þ  \[ad+bc=0\] Þ \[a(-b)+bc=0\]Þ \[a=c\] \\[{{z}_{1}}=a+ib=c-id={{\bar{z}}_{2}}\] \[(\because a=c\]and \[b=-d)\]

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