The complex number $2 + 2i$ is denoted by $u$.
Sketch an Argand diagram showing the points representing the complex numbers $1$, $i$ and $u$. Shade
the region whose points represent the complex numbers $B$ which satisfy both the inequalities
$|z-1|\le|z-i|$ and $|z-u|\le1$.Using your diagram, calculate the value of $|z|$ for the point in this region for which $\arg z$ is least.
I've already solved the first part; I'm stuck at the last part which is the value of $|z|$. Can anyone explain and proceed for me? Thanks in advance.
Best Answer
Here is a diagram of the problem:
We see that the line containing the origin and the point of least argument $p$ in $B$ forms a tangent to the circle $|z-u|=1$. The segment of this line between the two points has length $2\sqrt2$ and is the hypotenuse of a right triangle with one leg of length 1 (corresponding to the circle's radius). Therefore the length of the other leg, which is also the modulus of $p$ and the answer to the question, is $$\sqrt{(2\sqrt2)^2-1^2}=\sqrt7$$