Solve the equation |z| = z + 1 + 2i. - Vidyadip

Question

Solve the equation$ |z| = z + 1 + 2i.$

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Answer

Given that$, |z| = z + 1 + 2i$
$|z| = (z + 1) + 2i$
Squaring both sides
$|z|^2 = |z + 1|^2 + 4i^2 + 4(z + 1)i$
$\Rightarrow |z|^2 = |z|^2 + 1 + 2z - 4 + 4(z + 1)i$
$\Rightarrow 0 = -3 + 2z + 4(z + 1)i$
$\Rightarrow 3 - 2z - 4(z + 1)i = 0$
$\Rightarrow 3 - 2(x + yi) - 4[x + yi + 1]i = 0$
$\Rightarrow 3 - 2x - 2yi - 4xi - 4yi^2 - 4i = 0$
$\Rightarrow 3 - 2x + 4y - 2yi - 4i - 4xi = 0$
$\Rightarrow (3 - 2x + 4y) - i(2y + 4x + 4) = 0$
$\Rightarrow 3 - 2x + 4y = 0$
$\Rightarrow 2x - 4y = 3$
And $4x + 2y + 4 = 0$
$\Rightarrow 2x + y = -2$
Solving $eq. (i)$ and $(ii),$ we get
$\text{y}=-1$ and $\text{x}=-\frac{1}{2}$
Hence$,$ the value of $\text{z}=\text{x}+\text{yi}=\Big(-\frac{1}{2}-\text{i}\Big)$

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