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

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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

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

Posted by

Deependra Verma

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