Question

solve the equation |z|=z+1+2i​

Answer 1

Answer:

The answer is $z=x+iy=\frac{3}{2}-2i$

Step-by-step explanation:

Given equation is $|z|=z+1+2i$

Let $z=x+iy\Rightarrow |z|=\sqrt{x^2+y^2}$, so equation becomes

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

Equate real and imaginary parts

        $\sqrt{x^2+y^2}=x+1 \text{ and }0=y+2\Rightarrow y=-2$

    Square first equation both sides

       $x^2+y^2=x^2+2x+1\\\Rightarrow 4=2x+1\\\Rightarrow x=\frac{3}{2}$

Hence the solution is $z=x+iy=\frac{3}{2}-2i$