Question

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

Answer 1

Let $z=x+iy$
$\because |z|=z+1+2i$
$\Rightarrow |x+iy+1|=x+iy+1+2i$
$\Rightarrow \sqrt{x^2+y^2}=(x+1)+i(y+2)$
Equate the real and imaginary parts,
$\Rightarrow \sqrt{x^2+y^2}=(x+1)\dots \left(1\right)$
and $y+2=0$
$\Rightarrow y=-2\dots \left(2\right)$
Put value of 𝑦 in equation (1)
$\therefore \sqrt{x^2+(-2{)}^2}=(x+1)$
{squaring both of side}
$\Rightarrow x^2+4=x^2+2x+1$
$\Rightarrow x=\frac{3}{2}$
Therefore, $z=\frac{3}{2}-2i$