Exercise:
Describe geometrically, the set of all complex numbers *z* which
satisfy
the following condition

Solution:

Since |*z*-1| >0, we know the set we want to describe does not contain
the point *z*=1. By the triangle inequality, we have

for all *z*. So we want to exclude all points from the plane where the
equality

holds. That is, we want to exclude any *z* whose distance from 1
is equal to 1 plus its distance to the origin. This just means we
have to exclude the negative real axis and the origin. (Draw a picture.)

We can also see this algebraically. Writing *z*= *x*+*iy* we have

and

Setting these equal gives

which reduces to

This can only hold if and *y*=0.

So the set we want is the complex plane with the point *z*=1 and
the segment deleted.

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