Question

The number of complex numbers z such that |z−1|=|z+1|=|z−i| is ?

Answer 1

introduction to pure mathematics. I am pretty sure I am close to the answer but I can't quite decide why this proves that there is no complex numbers:

|z|=|z+i√5|=1|z|=|z+i√5|=1

cos²Θ+(isinΘ+i5–√)²−−−−−−−−−−−−−−−−−−−√=1–√cos²Θ+(isinΘ+i5)²=1

cos²Θ−sin²Θ−25–√sinΘ−5=1cos²Θ−sin²Θ−25sinΘ−5=1

cos²Θ−(1−cos²Θ)−25–√sinΘ−5=1cos²Θ−(1−cos²Θ)−25sinΘ−5=1

2cos²Θ−2√5sinΘ−5=0