Question

what does the equation |z+1-i|=|z+i-1| represent

Answer 1

Let $z=x+iy$

$|x+iy+1-i|=|x+iy+i-1|$

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

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

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

$x-y=-x+y$

$y=x$

so the given equation represents the straight line through origin : $\boxed{y=x}$