Determine the complex number z which satisfies the equations |z + 3i| = |z + 5 − 2i| and |z − 4i| = |z + 2i| simultaneously
"\\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)\n\\\\\\text{Also for the second equation, we have that }\\\\|x+(y-4)i|=|x+(2+y)i|\n\\\\=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 $z=-1+i$}"
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