Answer to Question #222027 in Algebra for jceka

Question

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

Answer 1

$\text{Let $z=x+yi$, then}\\|x+i(3+y)|=|(x+5)+i(y-2)|\\\text{By definition of magnitude we have that}\\x^2+(3+y)^2=(x+5)^2+(y-2)^2\\=10x-10y=-20\\\implies x-y=-2-(1) \\\text{Also for the second equation, we have that }\\|x+(y-4)i|=|x+(2+y)i| \\=x^2+(y-4)^2=x^2+(2+y)^2\\\text{Simplifying the equation, we have that}\\y=1\\\text{From eqn(1), we have that x=-2+y}\\\implies x=-1\\\text{Therefore $z=-1+i$}$