Let z be a complex number such that |z + 2| = |x – 2| and . Then is equal to: [2026]

Question

Let z be a complex number such that |z + 2| = |x – 2| and $\arg (\frac{z + 3}{z - i}) = \frac{π}{4}$ . Then $| z |^{2}$ is equal to: [2026]

Smart Achievers · Accepted Answer

(1)

We have |z + 2| = |z – 2 $\Rightarrow$ y-axis, i.e., x = 0 and $\arg (\frac{z + 3}{z - i}) = \frac{π}{4}$

$\Rightarrow \arg (\frac{x + i y + 3}{x + i y - i}) = \frac{π}{4}$

$\Rightarrow \arg (\frac{3 + i y}{i (y - 1)}) = \frac{π}{4}$

$\Rightarrow \arg (\frac{- y + 3 i}{1 - y}) = \frac{π}{4}$

$\Rightarrow \tan^{- 1} (\frac{3}{- y}) = \frac{π}{4}$

$\Rightarrow - \frac{3}{y} = 1 \Rightarrow y = - 3$

$∴ z = - 3 i$

Thus, $| z |^{2} = 9$

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