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

  • question_answer If \[{{z}^{2}}+z|z|+|z{{|}^{2}}=0\], then the locus of \[z\] is

    A) A circle

    B) A straight line

    C) A pair of straight lines

    D) None of these

    Correct Answer: C

    Solution :

    \[{{z}^{2}}+z|z|+|z{{|}^{2}}=0\]Þ\[{{\left( \frac{z}{|z|} \right)}^{2}}+\frac{z}{|z|}+1=0\] Þ\[\frac{z}{|z|}=\omega ,{{\omega }^{2}}\] Þ\[z=\omega |z|\]or \[z={{\omega }^{2}}|z|\] Þ\[x+iy=|z|\left( \frac{-1}{2}+\frac{i\sqrt{3}}{2} \right)\]or \[x+iy=|z|\left( \frac{-1}{2}-\frac{i\sqrt{3}}{2} \right)\] Þ\[x=-\frac{1}{2}|z|,y=|z|\frac{\sqrt{3}}{2}\]  or \[x=-\frac{|z|}{2},y=-\frac{|z|\sqrt{3}}{2}\] Þ \[y+\sqrt{3}x=0\]or \[y-\sqrt{3}x=0\] Þ \[{{y}^{2}}-3{{x}^{2}}=0\].

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