Question

The complex number which satisfies the equation $z+\sqrt{2}|z+1|+i=0$ is

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

Answer 1

Answer: (B)

Since $z+\sqrt{2}|z+1|+i=0$
$\therefore x+i(y+1)+\sqrt{2}|x+iy+1|=0$
$\therefore y+1=0$
$(\because |x+iy+1|$ is real $)$
$\therefore y=-1$
$\therefore x+\sqrt{2}|x-i+1|=0$
$\Rightarrow x^2=2\left[(x+1{)}^2+1\right]$
$=2\left(x^2+2x+2\right)$
$\therefore x^2+4x+4=0$
$\Rightarrow (x+2{)}^2=0$
or $x=-2$
$\therefore z=-2-i.$