On a sketch of an Argand diagram, shade the region whose points represent complex numbers satisfying the inequalities $|z-2-i|\leq1$ and $|z-i|\leq|z-2|$. Calculate the maximum value of arg $z$ for points lying in the shaded region.
I've already sketched the diagram. Can anyone explain how to find the maximum value with an aid of diagram? Thanks.
Best Answer
$|z-2-i|\le 1$ is the disk centre $2+i$ radius 1. $|z-i|\le |z-2|$ is the half-plane to the left of the perpendicular bisector of the line joining the points 2 and $i$. We want the point in the intersection of these two regions which has the largest arg. That is obviously $X$ where $OX$ is tangent to the circle.
We have $\arg z=2\arg(2+i)=2\tan^{-1}0.5$.