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

  • question_answer
    Suppose  \[{{z}_{1}},\,{{z}_{2}},\,{{z}_{3}}\] are the vertices of an equilateral triangle inscribed in the circle \[|z|\,=2\]. If  \[{{z}_{1}}=1+i\sqrt{3},\] then values of  \[{{z}_{3}}\] and \[{{z}_{2}}\] are respectively [IIT 1994]

    A) \[-2,\,1-i\sqrt{3}\]

    B) \[2,\,1+i\sqrt{3}\]

    C) \[1+i\sqrt{3},-2\]

    D) None of these

    Correct Answer: A

    Solution :

    One of the number must be a conjugate of        \[{{z}_{1}}=1+i\sqrt{3}\,\,i.e.\,{{z}_{2}}=1-i\sqrt{3}\] or \[{{z}_{3}}={{z}_{1}}{{e}^{i2\pi /3}}\]and \[{{z}_{2}}={{z}_{1}}{{e}^{-i2\pi /3}}\] \[{{z}_{3}}=(1+i\sqrt{3})\left[ \cos \left( \frac{2\pi }{3} \right)+i\sin \frac{2\pi }{3} \right]=-2\] Aliter: Obviously \[|z|=2\] is a circle with centre \[O(0,\,0)\] and radius 2. Therefore, \[OA=OB=OC\] and this is satisfied by (a) because two vertices of any triangle cannot be same.

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