# complex numbers

#### Vali

##### Junior Member
Let $$\displaystyle u=2+2i$$ and $$\displaystyle A =$$ { $$\displaystyle z \in \mathbb{C} : \vert z-1 \vert \leq \vert z-i \vert$$ and $$\displaystyle \vert z-u \vert \leq 1$$ }

Find the module of $$\displaystyle z \in A$$ so that the argument of $$\displaystyle z$$ has a minimum value.

The correct answer should be $$\displaystyle \sqrt{7}$$

Haven't encountered this type of exercises before. I guess the geometric representation of complex numbers will help.The problem is that I have no idea how to represent this geometrically.All I know that from the second condition I get a circle, but why?
Can you explain me this sketch?

#### Dr.Peterson

##### Elite Member
Do you recognize that |z - u| is the distance from u to z? The second inequality therefore says that z is less than 1 unit away from u. Sounds like a circle to me!

Then, from the picture, we can draw in radius MA and use the Pythagorean Theorem to find |OA|.