Answer to Question #116863 in Complex Analysis for Amoah Henry

Question

Determine the complex number z which satisfies the equations |z + 3i| = |z + 5 − 2i| and |z − 4i| = |z + 2i| simultaneously

Answer 1

We know that:

$|z|=\sqrt{Re^2(z)+Im^2(z)}$

So, we have:

$\begin{cases} Re^2(z)+(Im(z)+3)^2=(Re(z)+5)^2+(Im(z)-2)^2 \\ Re^2(z)+(Im(z)-4)^2=Re^2(z)+(Im(z)+2)^2 \\ \end{cases}$

From the second equation it follows that

$Im(z)=1$

Substituting into the first equation

$Re(z)=-1=>z=-1+i$