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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
    For the complex numbers \[{{z}_{1}}\] and \[{{z}_{2}}\] if \[|1-{{\bar{z}}_{1}}{{z}_{2}}{{|}^{2}}-|{{z}_{1}}-{{z}_{2}}{{|}^{2}}=k(1-|{{z}_{1}}{{|}^{2}})(1-|{{z}_{2}}{{|}^{2}})\] then ?k? equals to

    A) 1                     

    B) \[-1\]  

    C) 2                     

    D) \[-2\]

    Correct Answer: A

    Solution :

    \[{{\left| 1-{{{\bar{z}}}_{1}}{{z}_{2}} \right|}^{2}}-{{\left| {{z}_{1}}-{{z}_{2}} \right|}^{2}}\] \[=\,\,\,(1-{{\bar{z}}_{1}}{{z}_{2}})\,\,(1-{{z}_{1}}{{\bar{z}}_{2}})-({{z}_{1}}-{{z}_{2}})({{\bar{z}}_{1}}-{{\bar{z}}_{2}})\] \[=\,\,1-{{z}_{1}}{{\bar{z}}_{2}}-\,{{\bar{z}}_{1}}{{z}_{2}}+{{z}_{1}}{{\bar{z}}_{1}}{{z}_{2}}{{\bar{z}}_{2}}-({{z}_{1}}{{\bar{z}}_{1}}-{{z}_{1}}{{\bar{z}}_{2}}-{{\bar{z}}_{1}}{{z}_{2}}+{{z}_{2}}{{\bar{z}}_{2}})\] \[=\,\,1+{{z}_{1}}{{\bar{z}}_{1}}\,{{z}_{2}}{{\bar{z}}_{2}}-{{z}_{1}}{{\bar{z}}_{1}}-{{z}_{2}}{{\bar{z}}_{2}}\] \[=\,\,1+{{\left| {{z}_{1}} \right|}^{2}}{{\left| {{z}_{2}} \right|}^{2}}-{{\left| {{z}_{1}} \right|}^{2}}-{{\left| {{z}_{2}} \right|}^{2}}\] \[=\,\,(1-{{\left| {{z}_{1}} \right|}^{2}})(1-{{\left| {{z}_{2}} \right|}^{2}})\]- \[\Rightarrow \,\,\,\,k=1\]


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