Answer to Question #254162 in Complex Analysis for luzuko

Question

It is given that 

z+2i=iz+λ, (w/z)=2+2i, Imw=8

where z and w are complex numbers, and λ is a real constant. Which of the following is the value of λ? 

1. λ=4
2. λ=8
3. λ=3
4. λ=-3
5. λ=-4

Answer 1

Let $z=a+bi, w=c+8i.$ Then

$$a+bi+2i=i(a+bi)+\lambda, \lambda\in \R$$

$$b+2=a$$

$$a=-b+\lambda$$

$$b=a-2$$

$$\lambda=a+a-2$$

$$(w/z)=\dfrac{c+8i}{a+bi}=2+2i$$

$$c+8i=2a+2bi+2ai-2b$$

Then

$$c=2a-2b$$

$$8=2b+2a$$

$$a+b=4$$

Substitute

$$a+a-2=4=>a=3$$

$$\lambda=3+3-2=4$$

5. $\lambda=4$