Question #262543

Determine complex number(s) z such that z- i , iz- i and z-iz have the same modulus


1
Expert's answer
2021-11-10T10:13:03-0500

zi=izi=ziz|z-i|=|iz-i|=|z-iz|

z=x+iyz=x+iy

zi=x+i(y1)z-i=x+i(y-1)

izi=y+i(x1)iz-i=-y+i(x-1)

ziz=(x+y)+i(yx)z-iz=(x+y)+i(y-x)


zi=x2+(y1)2|z-i|=\sqrt{x^2+(y-1)^2}

izi=y2+(x1)2|iz-i|=\sqrt{y^2+(x-1)^2}

ziz=(x+y)2+(yx)2|z-iz|=\sqrt{(x+y)^2+(y-x)^2}


x2+(y1)2=y2+(x1)2x^2+(y-1)^2=y^2+(x-1)^2

2y=2x-2y=-2x

x=yx=y


y2+(x1)2=(x+y)2+(yx)2y^2+(x-1)^2=(x+y)^2+(y-x)^2

2x22x+1=2x22x^2-2x+1=2x^2

x=y=1/2x=y=1/2


z=0.5+0.5iz=0.5+0.5i


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