Question

If $z$ is a complex number satisfying the relation $|z+1|=z+2(1+i)$ then $z$ is

• A. $\frac{1}{2}(1+4i)$
• B. $\frac{1}{2}(3+4i)$
• C. $\frac{1}{2}(1-4i)$
• D. $\frac{1}{2}(3-4i)$

Answer 1

The correct option is C $\frac{1}{2}(1-4i)$

in given equation left side is always positive and real than right side must be real
let take $z=x+iy$
in right side imaginary part is
$y+2=0,y=-2$
$|x-2i+1|=x+2$
On squaring we get,
$(x+1{)}^2+4=(x+2{)}^2$
$\Rightarrow 2x+5=4x+4$
$\Rightarrow x=\frac{1}{2}$
$z=\frac{1}{2}-2i$
Hence, option C is correct.