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

Step-by-step solution

Given equation is aligned & z+ 2 |z+1|+i=0 & put z=x+i y in the given equation. & (x+i y)+ 2 |x+i y+1|+i=0 & x+i y+ 2 [ (x+1)^2+y^2 ]+i=0 aligned Now, equating real and imaginary part, we get aligned & x+ 2 (x+1)^2+y^2 =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 ]=x^2 & x^2+4 x+4=0 & x=-2 aligned Thus, z=-2+i(-1) |z|= 5