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

Question

If a complex number $z$ satisfies the equation $z+\sqrt2|z+1|+i=0$, then $|z|$ is equal to:

Accepted Answer

Let $z=x+iy$: real/imaginary parts give $y=-1$ and $x+\sqrt2\sqrt{(x+1)^2+1}=0$. Squaring: $2[(x+1)^2+1]=x^2 \implies x^2+4x+4=0 \implies x=-2$.

$$z=-2-i \implies |z|=\sqrt5.$$

Solutions · 1