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11th Class Mathematics Complex Numbers and Quadratic Equations Question Bank Critical Thinking

  • question_answer
    For any two complex numbers \[{{z}_{1}}\]and\[{{z}_{2}}\] and any real numbers a and b; \[|(a{{z}_{1}}-b{{z}_{2}}){{|}^{2}}+|(b{{z}_{1}}+a{{z}_{2}}){{|}^{2}}=\] [IIT 1988]

    A) \[({{a}^{2}}+{{b}^{2}})(|{{z}_{1}}|+|{{z}_{2}}|)\]

    B) \[({{a}^{2}}+{{b}^{2}})(|{{z}_{1}}{{|}^{2}}+|{{z}_{2}}{{|}^{2}})\]

    C) \[({{a}^{2}}+{{b}^{2}})(|{{z}_{1}}{{|}^{2}}-|{{z}_{2}}{{|}^{2}})\]

    D) None of these

    Correct Answer: B

    Solution :

    \[|(a{{z}_{1}}-b{{z}_{2}}){{|}^{2}}+|(b{{z}_{1}}+a{{z}_{2}}){{|}^{2}}\] \[={{a}^{2}}|{{z}_{1}}{{|}^{2}}+{{b}^{2}}|{{z}_{2}}{{|}^{2}}-2\operatorname{Re}(ab)|{{z}_{1}}{{\overline{z}}_{2}}|+{{b}^{2}}|{{z}_{1}}{{|}^{2}}+\] \[{{a}^{2}}|{{z}_{2}}{{|}^{2}}+2\operatorname{Re}(ab)|{{\overline{z}}_{1}}{{z}_{2}}|\] \[=({{a}^{2}}+{{b}^{2}})(|{{z}_{1}}{{|}^{2}}+|{{z}_{2}}{{|}^{2}})\]


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