1. Let z=x+iy,x,y∈R
Then
Im(z−i)=Im(x+iy−i)=y−1
∣z+i∣=∣x+iy+i∣=x2+(y+1)2​ We have the equation
y−1=x2+(y+1)2​
x2+(y+1)2​≥0=>y≥1 If y≥1, then
x2+(y+1)2​≥y+1>y−1 Therefore, the equation
y−1=x2+(y+1)2​ has no solution.
2. Let z=x+iy,x,y∈R
Then
Im(z−i)=Im(x+iy−i)=y−1
∣z+1∣=∣x+1+iy∣=(x+1)2+y2​ We have the equation
y−1=(x+1)2+y2​
(x+1)2+y2​≥0=>y≥1 If y≥1, then
(x+1)2+y2​≥y>y−1 Therefore, the equation
y−1=(x+1)2+y2​ has no solution.
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