Question

Question #d266f

Answer 1

`|z-(2+i)|<|z+1|`

by replacing `z` by `x+iy`,

`Rightarrow |x+iy-(2+i)|<|x+iy+1|`

by separating real parts and imaginary parts,

`Rightarrow |(x-2)+i(y-1)|<|(x+1)+iy|`

by find the norms,

`Rightarrow sqrt{(x-2)^2+(y-1)^2} < sqrt{(x+1)^2+y^2}`

by squaring,

`Rightarrow (x-2)^2+(y-1)^2<(x+1)^2+y^2`

by multiplying out,

`Rightarrow x^2-4x+4+y^2-2y+1 < x^2+2x+1 + y^2`

by cleaning up a bit,

`Rightarrow -2y < 6x-4`

by dividing by -2,

`Rightarrow y > -3x+2`

Hence, this is a region above the line `y=-3x+2`.

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I hope that this was helpful.