A) the real axis
B) the imaginary axis
C) a circle
D) an ellipse
Correct Answer: B
Solution :
Given that \[|{{z}^{2}}-1|=|z{{|}^{2}}+1\] \[\Rightarrow \] \[|{{z}^{2}}+(-1)|=|{{z}^{2}}|+|-1|\] It shows that the origin,\[-1\]and\[{{z}^{2}}\]lies on a line and\[{{z}^{2}}\]and\[-1\]lies on one side of the origin, therefore\[{{z}^{2}}\]is a negative number. Hence z will be purely imaginary. So we can say that z lies on y-axis. Alternate Solution We know that, if \[|{{z}_{1}}+{{z}_{2}}|=|{{z}_{1}}|+|{{z}_{2}}|\] \[\Rightarrow \] \[\arg ({{z}_{1}})=\arg ({{z}_{2}})\] \[\because \] \[|{{z}_{2}}+(-1)|=|{{z}^{2}}|+|-1|\] \[\Rightarrow \] \[\arg ({{z}^{2}})=\arg (-1)\] \[\Rightarrow \] \[2\arg (z)=\pi \] \[(\because \arg (-1)=\pi )\] \[\Rightarrow \] \[\arg (z)=\frac{\pi }{2}\] \[\Rightarrow \]\[z\]lies on y-axis (imaginary axis).
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