Let z \in C with Im(z) = 10 and it satisfies \frac{2 z - n}{2 z + n}=2i

Engineering

Mathematics

Introduction to Complex Number

Question

Let z $\in$ C with Im(z) = 10 and it satisfies $\frac{2 z - n}{2 z + n}$ =2i – 1 for some natural number n. Then :

n = 40 and Re(z) = –10

n = 40 and Re(z) = 10

n = 20 and Re(z) = 10

n = 20 and Re(z) = –10

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Let z = x + 10i

Given $\frac{2 z - n}{2 z + n} = 2 i - 1$

$\Rightarrow \frac{2 (x + 10 i) - n}{2 (x + 10 i) + n} = 2 i - 1$

⇒ (2x – n) + 20i = (2i – 1) [(2x + n) + 20i]

Comparing real and imaginary part

⇒ 2x – n = 2(–20) – (2x + n) and 20 = 2(2x + n) – 20

⇒ 2x – n = –40 – 2x – n and 20 = 4x + 2n – 20

⇒ 4x = –40 and 4x + 2n = 40

⇒ x = –10 and –40 + 2n = 40

⇒ n = + 40

⇒ n = 40 and Re(z) = –10