If a complex number z statisfies the equation x+√2|z+1|+i=0, then |z| is equal to :

Given equation is $$\begin{aligned} & z+\sqrt{2}|z+1|+i=0 \\ & \text { put } z=x+i y \text { in the given equation. } \\ & (x+i y)+\sqrt{2}|x+i y+1|+i=0 \\ & \Rightarrow x+i y+\sqrt{2}\left[\sqrt{(x+1)^2+y^2}\right]+i=0 \end{aligned}$$​z+2​∣z+1∣+i=0 put z=x+iy in the given equation. (x+iy)+2​∣x+iy+1∣+i=0⇒x+iy+2​[(x+1)2+y2​]+i=0​ Now, equating real and imaginary part, we get $$\begin{aligned} & x+\sqrt{2} \sqrt{(x+1)^2+y^2}=0 \text { and } \\ & y+1=0 \Rightarrow y=-1 \\ & \Rightarrow x+\sqrt{2} \sqrt{(x+1)^2+(-1)^2}=0 \\ & \Rightarrow \sqrt{2} \sqrt{(x+1)^2+1}=-x \\ & \Rightarrow 2\left[(x+1)^2+1\right]=x^2 \\ & \Rightarrow x^2+4 x+4=0 \\ & \Rightarrow x=-2 \end{aligned}$$​x+2​(x+1)2+y2​=0 and y+1=0⇒y=−1⇒x+2​(x+1)2+(−1)2​=0⇒2​(x+1)2+1​=−x⇒2[(x+1)2+1]=x2⇒x2+4x+4=0⇒x=−2​ Thus, $z=-2+i(-1) \Rightarrow|z|=\sqrt{5}$z=−2+i(−1)⇒∣z∣=5​