Question

If the complex number $z$ satisfies $z+\sqrt{2}|z+1|+i=0$, then $z$ is :

• A. $-2-i$
• B. $1+i$
• C. $-3-2i$
• D. $1+2i$

Answer 1

The correct option is A $-2-i$

Let $z=x+iy$
Given $z+\sqrt{2}|z+1|+i=0$
$\Rightarrow (x+iy)+\sqrt{2}|(x+iy)+1|+i=0$
$\Rightarrow x+\sqrt{2}\left(\sqrt{(x+1{)}^2+y^2}\right)+i(y+1)=0$
Comparing real and imaginary part, we get
$y=-1$ and
$x+\sqrt{2}\left(\sqrt{(x+1{)}^2+(-1{)}^2}\right)=0$
$\Rightarrow 2\left(\right(x+1{)}^2+1)=(-x{)}^2$
$\Rightarrow x^2+4x+4=0$
$\Rightarrow (x+2{)}^2=0$
$\therefore x=-2$
Hence $z=-2-i$