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JEE Main & Advanced Mathematics Complex Numbers and Quadratic Equations Question Bank Mock Test - Complex Numbers and Quadratic Equations

  • question_answer
    If \[{{z}_{1}}\],\[{{z}_{2}}\],\[{{z}_{3}}\]are the vertices of an equilateral triangle ABC such that \[\left| {{z}_{1}}-i \right|\]=\[\left| {{z}_{2}}-i \right|\]=\[\left| {{z}_{3}}-i \right|\],then \[\left| {{z}_{1}}+{{z}_{2}}+{{z}_{3}} \right|\]equals to

    A) \[3\sqrt{3}\]     

    B) \[\sqrt{3}\]

    C) 3                    

    D) \[\frac{1}{3\sqrt{3}}\]

    Correct Answer: C

    Solution :

    [c] Given that.
    \[\left| {{z}_{1}}-i \right|=\left| {{z}_{2}}-i \right|=\left| {{z}_{3}}-i \right|\]
    Hence, \[{{z}_{1}}\],\[{{z}_{2}}\],\[{{z}_{3}}\], lie on the circle whose center is i.
    Also cirucmcenter coincides.
    \[\therefore \frac{{{z}_{1}}+{{z}_{2}}+{{z}_{3}}}{3}=i\]
    \[\Rightarrow \left| {{z}_{1}}+{{z}_{2}}+{{z}_{3}} \right|=3\]
     


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