Question

If a complex number $z$ statisfies the equation $z+\sqrt{2}\left|z+1\right|+i=0,$ then $\left|z\right|$ is equal to:

• A. $2$
• B. $\sqrt{3}$
• C. $\sqrt{5}$
• D. $1$

Answer 1

Answer: (C)

Given equation is
$z+\sqrt{2}\left|z+1\right|+i=0$
put $z=x+iy$ in the given equation.
$\left(x+iy\right)+\sqrt{2}\left|x+iy+1\right|+i=0$
$\Rightarrow x+iy+\sqrt{2}\left[\sqrt{{\left(x+1\right)}^2+y^2}\right]+i=0$
Now, equating real and imaginary part, we get
$x+\sqrt{2}\sqrt{{\left(x+1\right)}^2+y^2}=0$ and
$y+1=0\Rightarrow y=-1$
$\Rightarrow x+\sqrt{2}\sqrt{{\left(x+1\right)}^2+{\left(-1\right)}^2}=0\left(\because y=-1\right)$
$\Rightarrow \sqrt{2}\sqrt{{\left(x+1\right)}^2+1}=-x$
$\Rightarrow 2\left[{\left(x+1\right)}^2+1\right]=x^2$
$\Rightarrow x^2+4x+4=0$
$\Rightarrow x=-2$
Thus, $z=-2+i\left(-1\right)\Rightarrow \left|z\right|=\sqrt{5}$