Proof of trasformation effect (in complex plane)

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I have to show that the transformation, $enspace ztofrac1zenspace$ , in the complex plane, transforms lines or circles in lines or circles (not respectively).



I'm completely new on complex plane, It's geometry and transformations. I've looked around and understood what the reciprocal of a complex number is and where it gets mapped to but I still don't see how you get a circle out of a line or vice versa.



EDIT: I have to use the following equations:



line: $$enspace azenspace+enspaceoverlineazenspace+enspace c = 0enspacetext with ainmathbbC,enspace binmathbbR$$



circumference:$$azoverlinezenspace+enspace bzenspace+enspaceoverlinebzenspace+enspace c = 0enspacetext with enspace a, binmathbbC, enspace cinmathbbR$$




complex-numbers




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  • With the new information at hand, I have deleted my answer. If I have enough time, I'll post a new one.
    – Niki Di Giano
    2 days ago










  • Ok, thanks a lot
    – IDK
    2 days ago














up vote
0
down vote

favorite
1












I have to show that the transformation, $enspace ztofrac1zenspace$ , in the complex plane, transforms lines or circles in lines or circles (not respectively).



I'm completely new on complex plane, It's geometry and transformations. I've looked around and understood what the reciprocal of a complex number is and where it gets mapped to but I still don't see how you get a circle out of a line or vice versa.



EDIT: I have to use the following equations:



line: $$enspace azenspace+enspaceoverlineazenspace+enspace c = 0enspacetext with ainmathbbC,enspace binmathbbR$$



circumference:$$azoverlinezenspace+enspace bzenspace+enspaceoverlinebzenspace+enspace c = 0enspacetext with enspace a, binmathbbC, enspace cinmathbbR$$




complex-numbers




share|cite|improve this question





















  • With the new information at hand, I have deleted my answer. If I have enough time, I'll post a new one.
    – Niki Di Giano
    2 days ago










  • Ok, thanks a lot
    – IDK
    2 days ago












up vote
0
down vote

favorite
1









up vote
0
down vote

favorite
1






1





I have to show that the transformation, $enspace ztofrac1zenspace$ , in the complex plane, transforms lines or circles in lines or circles (not respectively).



I'm completely new on complex plane, It's geometry and transformations. I've looked around and understood what the reciprocal of a complex number is and where it gets mapped to but I still don't see how you get a circle out of a line or vice versa.



EDIT: I have to use the following equations:



line: $$enspace azenspace+enspaceoverlineazenspace+enspace c = 0enspacetext with ainmathbbC,enspace binmathbbR$$



circumference:$$azoverlinezenspace+enspace bzenspace+enspaceoverlinebzenspace+enspace c = 0enspacetext with enspace a, binmathbbC, enspace cinmathbbR$$




complex-numbers




share|cite|improve this question













I have to show that the transformation, $enspace ztofrac1zenspace$ , in the complex plane, transforms lines or circles in lines or circles (not respectively).



I'm completely new on complex plane, It's geometry and transformations. I've looked around and understood what the reciprocal of a complex number is and where it gets mapped to but I still don't see how you get a circle out of a line or vice versa.



EDIT: I have to use the following equations:



line: $$enspace azenspace+enspaceoverlineazenspace+enspace c = 0enspacetext with ainmathbbC,enspace binmathbbR$$



circumference:$$azoverlinezenspace+enspace bzenspace+enspaceoverlinebzenspace+enspace c = 0enspacetext with enspace a, binmathbbC, enspace cinmathbbR$$





complex-numbers





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited 2 days ago
























asked 2 days ago









IDK

215




215











  • With the new information at hand, I have deleted my answer. If I have enough time, I'll post a new one.
    – Niki Di Giano
    2 days ago










  • Ok, thanks a lot
    – IDK
    2 days ago
















  • With the new information at hand, I have deleted my answer. If I have enough time, I'll post a new one.
    – Niki Di Giano
    2 days ago










  • Ok, thanks a lot
    – IDK
    2 days ago















With the new information at hand, I have deleted my answer. If I have enough time, I'll post a new one.
– Niki Di Giano
2 days ago




With the new information at hand, I have deleted my answer. If I have enough time, I'll post a new one.
– Niki Di Giano
2 days ago












Ok, thanks a lot
– IDK
2 days ago




Ok, thanks a lot
– IDK
2 days ago















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