Question

Complex number $z$ satisfying $|z+1|=z+2(1+i),$ is

• A. $1-2i$
• B. $\frac{1}{2}-2i$
• C. $2-\frac{i}{2}$
• D. $\frac{1}{2}+2i$

Answer 1

The correct option is B $\frac{1}{2}-2i$

Given,
$|z+1|=z+2(1+i)$
Let $z=x+iy$ where $x,y\in R,$ then
$|(x+iy)+1|=(x+iy)+2(1+i)\Rightarrow \sqrt{(x+1{)}^2+y^2}=(x+2)+i(y+2)$
Equating real part and imaginary part from both the sides, we get
$y+2=0\Rightarrow y=-2\dots \left(i\right)$
and $\sqrt{(x+1{)}^2+y^2}=x+2$
$\Rightarrow \sqrt{(x+1{)}^2+(-2{)}^2}=x+2$
$($from equation $\left(i\right))$
$\Rightarrow x^2+2x+5=(x+2{)}^2$
$\Rightarrow x^2+2x+5=x^2+4x+4$
$\Rightarrow x=\frac{1}{2}$
Hence, required complex number is $z=\frac{1}{2}-2i.$