Find the modulus of the complex number $z$ that lies in the region $|z-1| \leq |z-i|$ and $|z-(2+2i)| \leq 1$ for which $\arg(z)$ is least.
I am having trouble obtaining the required answer ($\sqrt(7)$)
I first drew a rough sketch of the region that $z$ is in.
I believe the red point is the critical point in the region for which $\arg(z)$ is least. However, this has coordinates $(2, 1)$, I believe. And the modulus here is $\sqrt{5}$... and not $\sqrt{7}$
What have I done wrong?
On
The point you're searching is the point of contact $T$ of the tangent to the lower half circle that can be drawn through the origin $O$.
Now $OT^2$ is the power of a point $A$ w.r.t. a circle $\mathscr C$ is given by the formula $$P(A,\mathscr C)=d^2-r^2,$$ where $d$ is the distance from $A$ to the centre of $\mathscr C$, so here, by Pythagoras, $$OT^2=(4+4)-1.$$
The answer is indeed $\sqrt7$ (and your drawing is not correct). Note that the distance from the red dot to the center of the circle is $1$, by the definition of the circle, and that the distance from the center of the circle to the origin is $\sqrt8$. But the three ponts that I mentioned form a right triangle. By Pythagoras' theorem, the distance that you're after is $\sqrt{8-1}=\sqrt7$.