If |z+1|=z+2(1+i), find z. - Vidyadip

Question

If $|\text{z}+1|=\text{z}+2(1+\text{i}),$ find z.

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Answer

Let $\text{z}=\text{x}+\text{iy}$ $|\text{z}+1|=\text{z}+2(1+\text{i})$ $\Rightarrow|\text{x}+\text{iy}+1|=\text{x}+\text{iy}+2+2\text{i}$ $\Rightarrow\sqrt{(\text{x}+1)^2+\text{y}^2}=(\text{x}+2)\text{i}(\text{y}+2)$ Comparing, real and imaginary parts, we get $\text{x}+2=\sqrt{\text{x}^2+2\text{x}+1+\text{y}^2}$ and $\text{y}+2=0$ $\text{y}+2=0$ $\Rightarrow\text{y}=-2$ & $(\text{x}+2)^2=\text{x}^2+2\text{x}+1+\text{y}^2$ $\Rightarrow\text{x}^2+4\text{x}+4=\text{x}^2+2\text{x}+1+\text{y}^2$ $\Rightarrow2\text{x}+3=\text{y}^2$ $\Rightarrow2\text{x}+3=(-2)^2$ $\Rightarrow2\text{x}+3=4$ $\Rightarrow2\text{x}=1$ $\Rightarrow\text{x}=\frac{1}{2}$ $\therefore\text{z}=\text{x}+\text{iy}=\frac{1}{2}-2\text{i}$

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