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Previous Year Question (PYQs)

Let $z$ be a complex number. Which of the following is a solution of $|z|-z=1+2i$?




Solution

Let $z=x+iy$. Then $|z|-x-iy=1+2i \Rightarrow |z|-x=1,\; -y=2\Rightarrow y=-2$. $|z|=1+x$, and $(1+x)^{2}=x^{2}+4 \Rightarrow x=\tfrac{3}{2}$. Thus $z=\tfrac{3}{2}-2i$.

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