Mathematics - Complex Number Question with Solution | TestHub

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