Question

If $|z^2-1|=|z{|}^2+1,$ then z lies on

• A. the real axis
• B. the imaginary axis
• C. a circle
• D. an ellipse

Answer 1

Answer: (B)

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).