$| \frac{1 - i z}{z - i} |$ =1, Find the locus of z.

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Solution

$\begin{matrix} ∣ ∣ ∣ \frac{i - i z}{z - i} ∣ ∣ ∣ = 1 ∣ ∣ ∣ \frac{1 - i (x + i y)}{x + i y - i} ∣ ∣ ∣ = 1 | 1 + y - i x | = | x + i (y - 1) | {(1 + y)}^{2} + x^{2} = x^{2} + {(y - 1)}^{2} 1 + 2 y = 1 - 2 y 4 y = 0 y = 0 z = 0 \Rightarrow I_{n} (z) = 0 H e n c e, z i s p u r e l y r e a l . \end{matrix}$

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