Solve the equation :
$| z + 1 | = z + 2 (i + 1)$

z = \frac{1}{2} - 2 i

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z = \frac{1}{2} + 2 i

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z = 2 - 2 i

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none of these

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Solution

The correct option is A $z = \frac{1}{2} - 2 i$
Let $z = x + i y$ . Then, $z + 1 = (x + 1) + i y$

Therefore, $| z + 1 | = \sqrt{(x + 1)^{2} + y^{2}}$

Now, $| z + 1 | = z + 2 (1 + i)$

$\sqrt{(x + 1)^{2} + y^{2}} = (x + i y) + 2 (1 + i)$

$\sqrt{(x + 1)^{2} + y^{2}} + 0 i = (x + 2) + (y + 2) i$

$(x + 1)^{2} + y^{2} = (x + 2)^{2}$ and $y = - 2$

$y^{2} = 2 x + 3$ and $y = - 2$

$4 = 2 x + 3$ and $y = - 2$

$x = \frac{1}{2}$ and $y = - 2$

Hence, $z = \frac{1}{2} - 2 i$

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