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

The correct option is B $-2-i$

Since $z+\sqrt{2}|z+1|+i=0$

$\therefore x+i(y+1)+\sqrt{2}|x+iy+1|=0$

By comparing real and imaginary parts

$\Rightarrow y+1=0$ $(\because |x+iy+1|$ is real $)$

$\Rightarrow y=-1$

$\therefore x+\sqrt{2}|x-i+1|=0$

$\Rightarrow x^2=2({(x+1)}^2+1)=2(x^2+2x+2)$

$\Rightarrow x^2+4x+4=0\Rightarrow {(x+2)}^2=0$

$\Rightarrow x=-2$

$\therefore z=-2-i$