complex numbers

Vali

Junior Member
Joined
Feb 27, 2018
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Let u=2+2i\displaystyle u=2+2i and A=\displaystyle A = { zC:z1zi\displaystyle z \in \mathbb{C} : \vert z-1 \vert \leq \vert z-i \vert and zu1\displaystyle \vert z-u \vert \leq 1 }

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

The correct answer should be 7\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?
rdVMiDr.png
 
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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