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If x + iy = (a + ib) / (a – ib), prove that x² + y² = 1.
If x – iy = √((a – ib)/(c – id)) prove that (x² + y²)² = (a² + b²)/(c² + d²).
If z₁ = 2 – i, z₂ = 1 + i, find |z₁ + z₂ + 1| / |z₁ – z₂ + 1|.
If a + ib = (x + i)² / (2x² + 1), prove that a² + b² = (x² + 1)² / (2x² + 1)².
Find the modulus of (1 + i)/(1 – i) – (1 – i)/(1 + i).
If α and β are different complex numbers with |β| = 1, then find |(β – α) / (1 – ᾱβ)|.
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