$ A = $ { $ z \in \mathbb{C} : \vert z-1 \vert \leq \vert z-i \vert $ and $ \vert z-u \vert \leq 1 $ }

Question

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

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

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

Haven't encountered this type of exercises before. I guess the geometric representation of complex numbers will help.

Answer 1

$A$ is the region below the line $y=x$ and inside or on the circle of radius $1$ centre $(2,2)$

Draw a tangent to the circle from the origin. You simply need the length of this tangent which is $\sqrt{7}$ by Pythagoras