If $|z-i|<1$ what can we deduce about $|z-1|$ and $|z+i|$

Question

If $|z-i|<1$ what can we deduce about how large or small $|z-1|$ and $|z+i|$ could be?

I tried drawing a diagram to get a feel but I don't know how to do anything more.

I feel like $ 1< |z+i|<3$ based on diagram but how can I go about the $|z-1|$

Answer 1

A diagram will certainly help you.

It also helps to think of $|z_1-z_2|$ as being the distance between two points $z_1$ and $z_2$ plotted in the argand diagram.

So you know from $|z-i|<1$ that $z$ lives somewhere 1 unit or less away from the point represented by $i$. That gives you a circle as the locus of points $z$.

$|z-1|$ is the distance from $z$ to the point represented by $1$. What is the closest point on the circle to $1$ and what is the furthest point on the circle from $1$?

Answer 2

First, draw the disk corresponding to the inequality. Then determine the maximum and minimum distance between a point in that disk and the point $1$ (for $z-1$) or $-i$ (for $z+i$).

Answer 3

If $|z-i| < 1$, then, using triangle inequality:

$|z-1| \leq |z-i| + |i - 1| < 1 + \sqrt{2} $.

$|z+i| \leq |z-i| + |i + i| < 1 + \sqrt{2}$.

Which is also observable in a diagram.