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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
    \[{{z}_{1}}\] and \[{{z}_{2}}\] are the roots of \[3{{z}^{2}}+3z+b=0\]. If  \[O(0),\]\[A({{z}_{1}}),\] \[B({{z}_{2}})\] form an equilateral triangle, then the value of b is

    A) \[-1\]        

    B) 1

    C) 0            

    D) does not exist

    Correct Answer: B

    Solution :

    \[{{z}_{1}}+{{z}_{2}}=-1,\,\,{{z}_{1}}{{z}_{2}}=\frac{b}{3}\] \[{{0}^{2}}+{{z}_{1}}^{2}+{{z}_{2}}^{2}=0\times {{z}_{1}}+0\times {{z}_{2}}+{{z}_{1}}{{z}_{2}}\] \[\Rightarrow \,\,\,{{({{z}_{1}}+{{z}_{2}})}^{2}}-2{{z}_{1}}{{z}_{2}}={{z}_{1}}{{z}_{2}}\Rightarrow 1=3{{z}_{1}}{{z}_{2}}=3\frac{b}{3}\] \[\Rightarrow \,\,\,b=1\]


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