Answer to Question #262543 in Complex Analysis for luka

Question

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

Answer 1

$|z-i|=|iz-i|=|z-iz|$

$z=x+iy$

$z-i=x+i(y-1)$

$iz-i=-y+i(x-1)$

$z-iz=(x+y)+i(y-x)$

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

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

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

$x^2+(y-1)^2=y^2+(x-1)^2$

$-2y=-2x$

$x=y$

$y^2+(x-1)^2=(x+y)^2+(y-x)^2$

$2x^2-2x+1=2x^2$

$x=y=1/2$

$z=0.5+0.5i$