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Question Number 126848 by mohammad17 last updated on 24/Dec/20

Answered by mathmax by abdo last updated on 24/Dec/20

$$\mid \mathrm{z}-1+\mathrm{i}\mid =1\mathrm{put}\mathrm{z}=\mathrm{x}+\mathrm{iy}\Rightarrow$$$$\mid \mathrm{z}-1+\mathrm{i}\mid =\mid \mathrm{x}+\mathrm{iy}-1+\mathrm{i}\mid =\mid \mathrm{x}-1+\mathrm{i}(\mathrm{y}+1)\mid =\sqrt{{(\mathrm{x}-1)}^2+{(\mathrm{y}+1)}^2}$$$$\mid \mathrm{z}-1+\mathrm{i}\mid =1\Rightarrow {(\mathrm{x}-1)}^2+{(\mathrm{y}+1)}^2=1\mathrm{so}\mathrm{the}\mathrm{locus}\mathrm{is}\mathrm{the}\mathrm{circle}$$$$\mathrm{with}\mathrm{centre}\mathrm{w}(1,-1)\mathrm{and}\mathrm{radius}\mathrm{R}=1$$

Commented by mohammad17 last updated on 24/Dec/20

$$pleassirhelpmeinallqestion$$

Answered by mathmax by abdo last updated on 25/Dec/20

$$\mid \mathrm{z}-3\mathrm{i}\mid <2\mathrm{let}\mathrm{z}=\mathrm{x}+\mathrm{iy}$$$$\mid \mathrm{z}-3\mathrm{i}\mid <2\Rightarrow \mid \mathrm{x}+\mathrm{iy}-3\mathrm{i}\mid <2\Rightarrow \mid \mathrm{x}+\mathrm{i}(\mathrm{y}-3)\mid <2\Rightarrow \sqrt{{\mathrm{x}}^2+{(\mathrm{y}-3)}^2}<2\Rightarrow$$$${\mathrm{x}}^2+{(\mathrm{y}-3)}^2<4\Rightarrow \mathrm{locus}\mathrm{is}\mathrm{intrior}\mathrm{of}\mathrm{circle}\mathrm{with}\mathrm{centre}\mathrm{w}\left({\mathrm{o}}^{\prime }3\right)$$$$\mathrm{and}\mathrm{radius}2$$

Answered by mathmax by abdo last updated on 25/Dec/20

$$\mathrm{Re}(\mathrm{z}+1)=2\mathrm{let}\mathrm{z}=\mathrm{x}+\mathrm{iy}\Rightarrow \mathrm{Res}(\mathrm{z}+1)=\mathrm{Res}(\mathrm{x}+\mathrm{iy}+1)=2\Rightarrow$$$$\mathrm{x}+1=2\Rightarrow \mathrm{x}=1\mathrm{so}\mathrm{the}\mathrm{locus}\mathrm{is}\mathrm{line}\mathrm{x}=1$$

Answered by mathmax by abdo last updated on 25/Dec/20

$$\mid \mathrm{z}+\mathrm{i}\mid <2\mathrm{and}\mid \mathrm{z}-\mathrm{i}\mid >3\mathrm{let}\mathrm{z}=\mathrm{x}+\mathrm{iy}$$$$\mid \mathrm{z}+\mathrm{i}\mid <2\Rightarrow \mid \mathrm{x}+\mathrm{iy}+\mathrm{i}\mid <2\Rightarrow \sqrt{{\mathrm{x}}^2+{(\mathrm{y}+1)}^2}<2\Rightarrow {\mathrm{x}}^2+{(\mathrm{y}+1)}^2<4$$$$\Rightarrow \mathrm{locus}\mathrm{is}\mathrm{Interior}\mathrm{of}\mathrm{circle}\mathrm{C}(\mathrm{w},\mathrm{r})/\mathrm{w}(0,-1)\mathrm{and}\mathrm{r}=2$$$$\mid \mathrm{z}-\mathrm{i}\mid >3\Rightarrow \mid \mathrm{x}+\mathrm{iy}-\mathrm{i}\mid >3\Rightarrow \mid \mathrm{x}+\mathrm{i}(\mathrm{y}-1)\mid >3\Rightarrow \sqrt{{\mathrm{x}}^2+{(\mathrm{y}-1)}^2}>3\Rightarrow$$$${\mathrm{x}}^2+{(\mathrm{y}-1)}^2>9\Rightarrow \mathrm{locus}\mathrm{is}\mathrm{Int}(\mathrm{C}({\mathrm{w}}^{{}^{\prime }}(\mathrm{o},1),3)\Rightarrow$$$$\mathrm{locus}\mathrm{is}\mathrm{In}\left(\mathrm{C}(\mathrm{w},2)\right)\cap \mathrm{Int}\left(\mathrm{C}({\mathrm{w}}^{{}^{\prime }},3)\right)$$

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