A) \[x-\]axis
B) \[y-\]axis
C) The straight line \[x=a\]
D) None of these
Correct Answer: B
Solution :
We have \[\left| \frac{z-a}{z+\bar{a}} \right|=1\] Þ \[|z-a|\,=\,|z+\overline{a}|\]Þ \[|z-a{{|}^{2}}=|z+\overline{a}{{|}^{2}}\] Þ \[(z-a)(\overline{z-a})=(z+\overline{a})(\overline{z+\overline{a}})\] Þ \[(z-a)(\overline{z}-\overline{a})=(z+\overline{a})(\overline{z}+a)\] Þ \[z\overline{z}-z\overline{a}-a\overline{z}+a\overline{a}=z\overline{z}+za+\overline{a}\overline{z}+\overline{a}a\] Þ \[za+z\overline{a}+\overline{a}\overline{z}+a\overline{z}=0\,\,\,\,\Rightarrow (a+\overline{a})(z+\overline{z})=0\] Þ \[z+\overline{z}=0\,\,(\because a+\overline{a}=2\operatorname{Re}(a)\ne 0)\] Þ \[2\operatorname{Re}(z)=0\]Þ \[2x=0\]Þ\[x=0\] Which is the equation of y-axis.
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