Let z ∈ C be such that |z|

Engineering

Mathematics

Conjugate Modulus and Argument of a Complex Number

Geometry of Complex Number

Question

Let z ∈ C be such that |z| < 1. If $ω = \frac{5 + 3 z}{5 (1 - z)}$ , then

5 Re(ω) > 1

5 Re(ω) > 4

4Im(ω) > 5

5Im(ω) < 1

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$W = \frac{5 + 3 z}{5 (1 - z)}$

⇒ 5w – 5wz = 5 + 3z

$\Rightarrow z = \frac{5 w - 5}{3 + 5 w}$

Given |z| < 1

$\Rightarrow | \frac{5 w - 5}{3 + 5 w} | < 1$

⇒ |5w – 5| < |3 + 5w|

$\Rightarrow | w - 1 | < | \frac{3}{5} + w |$