The complex number z satisfying the equations |z|-4=|z-i|-|z-5i|=0,

Answers (1)

Solution:   We have two equations ,

                   left | z 
ight |-4=0     and   left | z-i 
ight |-left | z+5i 
ight |=0

               Putting z=x+iy , these equations becomes

              Rightarrow           left | x-iy 
ight |=4Rightarrow x^2+y^2=16       .......(1)

             Rightarrow         left | x+iy-i 
ight |=left | x+iy+5i 
ight |

            Rightarrow         x^2+(y-1)^2=x^2+(y+5)^2

            Rightarrow        x^2+y^2-2y+1=x^2+y^2+10y+25Rightarrow y=-2    ......(2)

            Putting y=-2 in (1) , x^2+4=16Rightarrow x=pm 2sqrt3.

      Hence the complex number z satisfying the given  equation are

       z_1=(2sqrt3,-2) and z_2=(-2sqrt3,-2)  that is

        Rightarrow                              z_1=2sqrt3-2i,z_2=-2sqrt3-2i

 

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