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. $\lambda=4$