Question

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

Answer 1

$\begin{cases} | z+3i| = | z+5-2i| \\ | z - 4i | = | z + 2i | \end{cases}$

$z=x+iy$

$|x+iy+3i| =|x+iy+5-2i|$

$\sqrt{x^2+(y+3)^2}=\sqrt{(x+5)^2+(y-2)^2}$

$x^2+y^2+6y+9=x^2+10x+25+y^2-4y+4$

$10y=10x+20\implies y=x+2$

$| z - 4i | = | z + 2i |$

 $|x+iy-4i| =|x+iy+2i|$

$\sqrt{x^2+(y-4)^2}=\sqrt{(x^2+(y+2)^2}$

$x^2+y^2-8y+16=x^2+y^2+4y+4$

$-8y+16=4y+4\implies12y=12\implies y=1$

$\begin{cases} y=x+2 \\ y=1 \end{cases}$

$x=-1$    $y=1$

$z=i-1$