complex numbers

Vali

Junior Member
Joined
Feb 27, 2018
Messages
87
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
Joined
Nov 12, 2017
Messages
3,650
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|.
 
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