Question

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

Answer 1

We have given that a complex number $z$ satisfy these two equation

$$|z − 4i| = |z + 2i| \hspace{1cm}(1)\\ |z + 3i| = |z + 5 − 2i|\hspace{1cm}(2)$$

Let's assume the solution is $z=z_0=x_0+y_0$ ,thus equation (1) will be,

$$|z_0-4i|=|z_0+2i|\\ \implies |z_0-4i|^2=|z_0+2i|^2\\ \implies (z_0-4i)(\overline{z_0}+4i)=(z_0+2i)(\overline{z_0}-2i) \hspace{1cm}(\because |z|^2=z\overline{z})\\ \implies 6i(z_0-\overline{z_0})=-12 \hspace{1cm}(\because z_0=x_0+y_0)\\ \implies -12y_0=-12 \implies y_0=1$$

Hence,

$$z_0=x_0+i \hspace{1cm}(3)$$

And similarly, from equation (2) we get,

$$|z_0 + 3i| = |z_0+ 5 − 2i|\\ \implies |(x_0+i) + 3i|^2 = |(x_0+i) + 5 − 2i|^2 \hspace{1cm}(\because z_0=x_0+i)\\ \implies |x_0 + 4i|^2 = |(x_0 + 5) − i|^2\\ \implies x_{0}^2 + 16 = (x_0 + 5)^2 + 1\\ \implies 10x_0=-10 \implies x_0=-1$$

Thus,

$$z_0=-1+i$$

Therefore, required complex number

$$z=z_0=-1+i$$

which satisfy the equation $(1) \: \& (2)$. Hence we are done.