Question

Geometric representation of S={Im(z-i)=|z+I|}

Answer 1

1. Let $z=x+iy, x, y \in \R$

Then

$$Im(z-i)=Im(x+iy-i)=y-1$$

$$|z+i|=|x+iy+i|=\sqrt{x^2+(y+1)^2}$$

We have the equation

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

$$\sqrt{x^2+(y+1)^2}\geq0=>y\geq1$$

If $y\geq1,$ then

$$\sqrt{x^2+(y+1)^2}\geq y+1>y-1$$

Therefore, the equation

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

has no solution.

2. Let $z=x+iy, x, y \in \R$

Then

$$Im(z-i)=Im(x+iy-i)=y-1$$

$$|z+1|=|x+1+iy|=\sqrt{(x+1)^2+y^2}$$

We have the equation

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

$$\sqrt{(x+1)^2+y^2}\geq0=>y\geq1$$

If $y\geq1,$ then

$$\sqrt{(x+1)^2+y^2}\geq y>y-1$$

Therefore, the equation

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

has no solution.