Find the complex number satisfying

According to the question,

We have,

z + √2 |(z + 1)| + i = 0 … (1)

Substituting z = x + iy, we get

⇒ x + iy + √2 |x + iy + 1| + i = 0

$\Rightarrow \mathrm{x}+\mathrm{i}(1+\mathrm{y})+\sqrt{2}\left[\sqrt{(\mathrm{x}+1)^{2}+\mathrm{y}^{2}}\right]=0$

$\Rightarrow \mathrm{x}+\mathrm{i}(1+\mathrm{y})+\sqrt{2} \sqrt{\left(\mathrm{x}^{2}+2 \mathrm{x}+1+\mathrm{y}^{2}\right)}=0$

Comparing real and imaginary parts to zero, we get

$\Rightarrow x+\sqrt{2} \sqrt{x^{2}+2 x+1+y^{2}}=0$.......(2)

And,

$\mathrm{y}+1=0$

$\Rightarrow \mathrm{y}=-1$

Substituting $y=-1$ into equation (2), we get

$\Rightarrow x+\sqrt{2} \sqrt{x^{2}+2 x+1+1}=0$

$\Rightarrow \sqrt{2} \sqrt{x^{2}+2 x+2}=-x$

⇒ 2x2 + 4x + 4 = x2

⇒ x2 + 4x + 4 = 0

⇒ (x + 2)2 = 0

⇒ x = -2

Hence, z = x + iy

= – 2 – i