Question

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

• A. $3-2i$
• B. $2-\frac{3}{2}i$
• C. $2+\frac{3}{2}i$
• D. $\frac{3}{2}-2i$

Answer 1

The correct option is D $\frac{3}{2}-2i$

$\left|z\right|=z+1+2i$

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

$⟹\sqrt{x^2+y^2}=x+1$ and $0=2+y$ or $y=-2$

$⟹\sqrt{x^2+4}=x+1$

$\Rightarrow x^2+4=x^2+2x+1$

$\Rightarrow 2x=3$

$\therefore x=\frac{3}{2}$

$⟹x+iy=\frac{3}{2}-2i$
Ans: D