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If |z^2-1|=|z|^2+1 , then show that z ...

If `|z^2-1|=|z|^2+1` , then show that `z` lies on the imaginary axis.

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`|Z^2-1|=|Z|^2+1`
Square both sides
`|Z^2-1|^2=(|Z|^2+1)^2`
`(Z^2-1)(Z^2-1)=(Z Z+1)^2`
`Z^2*Z^2-Z^2-Z^2+1=(z Z)^2+1+2Z Z`
`Z^2+z^2+2zZ=0`
`(Z+z)^2=0`
`Z=x+iy`
...
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