Complex Loci with Arguments












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$begingroup$


I need to find the locus of points (on an Argand diagram) such that:



(i) $arg(z-(-1-4i)) + arg(z-(5+8i)) =0$



(ii) $arg(z-(-1-4i)) + arg(z-(5+8i)) = pi/2$



I could not see a way to solve these problems other than plotting arbitrary points and trying to observe a general pattern.



I am aware that $arg(z-(-1-4i)) - arg(z-(5+8i)) = pi/2$ is a semicircle, and for other angles, say $pi/3$ or $pi/4 $, part of the arc of a circle, but this is only because I knew that this equation represents the locus of points that made a certain angle between the two complex numbers. I was unable to find a similar representation, however, for (i) and (ii).



I am also interested in whether problems (i) and (ii) can be generalised to any angle between $0$ to $pi$. Any help here would be greatly appreciated.










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$endgroup$

















    0












    $begingroup$


    I need to find the locus of points (on an Argand diagram) such that:



    (i) $arg(z-(-1-4i)) + arg(z-(5+8i)) =0$



    (ii) $arg(z-(-1-4i)) + arg(z-(5+8i)) = pi/2$



    I could not see a way to solve these problems other than plotting arbitrary points and trying to observe a general pattern.



    I am aware that $arg(z-(-1-4i)) - arg(z-(5+8i)) = pi/2$ is a semicircle, and for other angles, say $pi/3$ or $pi/4 $, part of the arc of a circle, but this is only because I knew that this equation represents the locus of points that made a certain angle between the two complex numbers. I was unable to find a similar representation, however, for (i) and (ii).



    I am also interested in whether problems (i) and (ii) can be generalised to any angle between $0$ to $pi$. Any help here would be greatly appreciated.










    share|cite|improve this question











    $endgroup$















      0












      0








      0


      0



      $begingroup$


      I need to find the locus of points (on an Argand diagram) such that:



      (i) $arg(z-(-1-4i)) + arg(z-(5+8i)) =0$



      (ii) $arg(z-(-1-4i)) + arg(z-(5+8i)) = pi/2$



      I could not see a way to solve these problems other than plotting arbitrary points and trying to observe a general pattern.



      I am aware that $arg(z-(-1-4i)) - arg(z-(5+8i)) = pi/2$ is a semicircle, and for other angles, say $pi/3$ or $pi/4 $, part of the arc of a circle, but this is only because I knew that this equation represents the locus of points that made a certain angle between the two complex numbers. I was unable to find a similar representation, however, for (i) and (ii).



      I am also interested in whether problems (i) and (ii) can be generalised to any angle between $0$ to $pi$. Any help here would be greatly appreciated.










      share|cite|improve this question











      $endgroup$




      I need to find the locus of points (on an Argand diagram) such that:



      (i) $arg(z-(-1-4i)) + arg(z-(5+8i)) =0$



      (ii) $arg(z-(-1-4i)) + arg(z-(5+8i)) = pi/2$



      I could not see a way to solve these problems other than plotting arbitrary points and trying to observe a general pattern.



      I am aware that $arg(z-(-1-4i)) - arg(z-(5+8i)) = pi/2$ is a semicircle, and for other angles, say $pi/3$ or $pi/4 $, part of the arc of a circle, but this is only because I knew that this equation represents the locus of points that made a certain angle between the two complex numbers. I was unable to find a similar representation, however, for (i) and (ii).



      I am also interested in whether problems (i) and (ii) can be generalised to any angle between $0$ to $pi$. Any help here would be greatly appreciated.







      complex-numbers locus






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      edited Oct 7 '18 at 6:53







      NKRsolutions

















      asked Oct 6 '18 at 19:47









      NKRsolutionsNKRsolutions

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      588






















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          $begingroup$

          Denote the points $A(-1-4i),; B(5+8i),; M(z).$



          (i) is equivalent to $$arg(z-(-1-4i)) =- arg(z-(5+8i))$$ This signifies that the direction from $A$ towards $M$ is opposite to the one from $B$ towards $M.$ The locus of points $M(z)$ is the segment $AB$ except $A,B.$






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            0












            $begingroup$

            Denote the points $A(-1-4i),; B(5+8i),; M(z).$



            (i) is equivalent to $$arg(z-(-1-4i)) =- arg(z-(5+8i))$$ This signifies that the direction from $A$ towards $M$ is opposite to the one from $B$ towards $M.$ The locus of points $M(z)$ is the segment $AB$ except $A,B.$






            share|cite|improve this answer









            $endgroup$


















              0












              $begingroup$

              Denote the points $A(-1-4i),; B(5+8i),; M(z).$



              (i) is equivalent to $$arg(z-(-1-4i)) =- arg(z-(5+8i))$$ This signifies that the direction from $A$ towards $M$ is opposite to the one from $B$ towards $M.$ The locus of points $M(z)$ is the segment $AB$ except $A,B.$






              share|cite|improve this answer









              $endgroup$
















                0












                0








                0





                $begingroup$

                Denote the points $A(-1-4i),; B(5+8i),; M(z).$



                (i) is equivalent to $$arg(z-(-1-4i)) =- arg(z-(5+8i))$$ This signifies that the direction from $A$ towards $M$ is opposite to the one from $B$ towards $M.$ The locus of points $M(z)$ is the segment $AB$ except $A,B.$






                share|cite|improve this answer









                $endgroup$



                Denote the points $A(-1-4i),; B(5+8i),; M(z).$



                (i) is equivalent to $$arg(z-(-1-4i)) =- arg(z-(5+8i))$$ This signifies that the direction from $A$ towards $M$ is opposite to the one from $B$ towards $M.$ The locus of points $M(z)$ is the segment $AB$ except $A,B.$







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Nov 26 '18 at 21:51









                user376343user376343

                3,3833826




                3,3833826






























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