Question

Find a complex number z satisfying the equation $z+\sqrt{2}|z+1|+i=0$

• A. $2-i$
• B. $-2-i$
• C. $\sqrt{2}-i$
• D. None of these

Answer 1

The correct option is B $-2-i$

Let $z=x+iy$. Then,

$z+\sqrt{2}|z+1|+i=0$

$x+iy+\sqrt{2}|(x+1)+iy|+i=0$

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

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

$x+\sqrt{2(x+1{)}^2+2}=0$ and $y=-1$

$\sqrt{2(x+1{)}^2+2}=-x$ and $y=-1$

$2(x+1{)}^2+2=x^2$ and $y=-1$

$x^2+4x+4=0$ and $y=-1$

$(x+2{)}^2=0$ and $y=-1$

$x=-2$ and $y=-1$

Hence, $z=x+iy=-2-i$