Question

hallar z PERTENECE A LOS numeros complejos TAL QUE i/z y i-z tengan el mismo modulo

Answer 1

$\left|\dfrac{i}{z}\right|=|i-z|\\ \\ \\ \dfrac{|i|}{|z|}=|i-z|\\ \\ \\ \dfrac{1}{|z|}=|i-z|\\ \\ \\ 1=|z|\cdot|i-z|\\ \\ 1=|z(i-z)|\\ \\ 1=\left|zi-z^2\right|\\ \\ \left|ai-b-(a+bi)^2\right|=1\\ \\ |ai-b-(a^2+2abi-b^2)|=1\\ \\ |(b^2-a^2-b)+(a-2ab)i|=1\\ \\ (b^2-a^2-b)^2+(a-2ab)^2=1\\ \\ a^4 + (2b^2 - 2b + 1)a^2 + (b^4 - 2b^3 + b^2-1)=0\\ \\ a=\pm\sqrt{\dfrac{-(2b^2 - 2b + 1)\pm\sqrt{(2b^2 - 2b + 1)^2-4(b^4 - 2b^3 + b^2-1)}}{2}}$


$a=\pm\sqrt{\dfrac{-(2b^2 - 2b + 1)+\sqrt{4b^2-4b+5}}{2}}\\ \\ \\ -(2b^2 - 2b + 1)+\sqrt{4b^2-4b+5}\geq 0\\ \\ \\ \dfrac{1}{2}(1-\sqrt{5})\leq b\leq\dfrac{1}{2}(1+\sqrt{5})\\ \\ \\ \text{Donde: }a=\text{Re}(z)\,,\, b=\text{Im}(z)$