What is the answer to |z-i|=0 ?

The answer to |z-i|=0 is z=sqrt(-y^2+2y-1)+yi,z=-sqrt(-y^2+2y-1)+yi;\quad y\in \mathbb{R}

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|z-i|=0

∣ z - i ∣ = 0

z = - y^{2} + 2 y - 1 + y i, z = - - y^{2} + 2 y - 1 + y i; y \in R

Solution steps

∣ z - i ∣ = 0

Substitute

z = x + y i

∣ x + y i - i ∣ = 0

Expand

∣ x + y i - i ∣ : x^{2} + y^{2} - 2 y + 1

x^{2} + y^{2} - 2 y + 1 = 0

Solve for

x : x = - y^{2} + 2 y - 1, x = - - y^{2} + 2 y - 1

Substitute back

z = x + y i

z = - y^{2} + 2 y - 1 + y i, z = - - y^{2} + 2 y - 1 + y i; y \in R

z^{2} = - (\frac{2}{3}) i

x + i y + \frac{1}{x + i y} = 0

x (1 - x) = y^{2} - i y (2 x + 1)

(a^{2} - b^{2} - a + b) + (2 ab - b + a) i = 0

\frac{1 + i}{z} - (1 + 2 i) = i

What is the answer to |z-i|=0 ?
The answer to |z-i|=0 is z=sqrt(-y^2+2y-1)+yi,z=-sqrt(-y^2+2y-1)+yi;\quad y\in \mathbb{R}