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

  • question_answer The locus of \[z\]satisfying the inequality \[{{\log }_{1/3}}|z+1|\,>\] \[{{\log }_{1/3}}|z-1|\] is

    A) \[R\,(z)<0\]

    B) \[R\,(z)>0\]

    C) \[I\,(z)<0\]

    D) None of these

    Correct Answer: A

    Solution :

    We know that \[{{\log }_{a}}m>{{\log }_{a}}n\]Þ \[m>n\]or \[m<n\], according as  \[a>1\]or \[0<a<1\]. Hence for \[z=x+iy\] \[{{\log }_{(1/3)}}|z+1|\,>\,{{\log }_{(1/3)}}|z-1|\Rightarrow |z+1|\]\[<\,|z-1|\]\[\left\{ \because 0<\frac{1}{3}<1 \right\}\] Þ\[|x+iy+1|<|x+iy-1|\] Þ  \[{{(x+1)}^{2}}+{{y}^{2}}<{{(x-1)}^{2}}+{{y}^{2}}\] Þ \[4x<0\,\Rightarrow x<0\,\,\Rightarrow \operatorname{Re}(z)<0\]

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