Answer to Question #222027 in Algebra for jceka

Question #222027

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


1
Expert's answer
2021-08-02T17:28:48-0400

Let z=x+yi, thenx+i(3+y)=(x+5)+i(y2)By definition of magnitude we have thatx2+(3+y)2=(x+5)2+(y2)2=10x10y=20    xy=2(1)Also for the second equation, we have that x+(y4)i=x+(2+y)i=x2+(y4)2=x2+(2+y)2Simplifying the equation, we have thaty=1From eqn(1), we have that x=-2+y    x=1Therefore z=1+i\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 $z=-1+i$}


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