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

Question

Aicharya · Accepted Answer

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

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

Solution