I am attempting to determine the locus of z for arg &#x2061;<!-- ⁡ --> ( z 2

Let $n$ points be placed uniformly at random on the boundary of a circle of circumference 1.

These $n$ points divide the circle into $n$ arcs.

Let $Z_i$ for $1\le i\le n$ be the length of these arcs in some arbitrary order, and let $X$ be the number of $Z_i$ that are at least $\frac{1}{n}$.

What is $E[X]$ and $Var[X]$?

Any hints will be appreciated. Thanks..

(By the way this problem is exercise 8.12 from the book 'Probability and Computing' by Mitzenmacher and Upfal)

Let
$f(z)={\displaystyle \frac{z(z-a)}{(z-b)(z-c)(z-d)}}$
be a complex rational function and let $S=\{z\in \mathbb{C}:|z|=1\}$ be the unit circle.

My question:
For which (non-zero complex numbers) $a,b,c$ and $d$ there exist an arc $C$ of $S$ such that $f(C)\subseteq S$ ?

If the arc length and chord length between two points (two points on a circle that constitute a minor arc ) in a circle are known , find radius of the circle?

What angle is subtended at the center of a circle of radius 2 km by an arc of length 9 m?

I am trying to figure out what the best method is to go about finding this locus.
$\arg {\displaystyle \frac{z-a}{z-b}}=\theta$
I am aware that it must be part of an arc of a circle that passes through the points $a$ and $b$. The argument means that the vector $(z-a)$ leads the vector $(z-b)$ by $\theta$ and so by the argument must lie on an arc of a circle due to the converse of 'angles in the same segment theorem'.

My question is, how do I know what the circle will look like, ie. the centre of the circle and the radius. If I don't need to know this, how will I know what the circle looks like.

Finally, if I changed the locus to
$\arg {\displaystyle \frac{z-a}{z-b}}=-\theta$
Would this just be the other part of the same circle as previously or a different circle?

Perhaps you could help by showing my how it would work with the question
$\arg \frac{z-2j}{z+3}=\pi /3$

How to find the center of a circle with given an arbitrary arc. we only have the arc nothing else. Is there any known equation or way to complete the circle.

Today, I ran into a calculus class and the professor was calculating the perimeter of a circle using arc lenghts. Even though they seemed to be on the right track, there is a problem:

They are using sinx and cosx functions, which are actually defined using the perimeter of a circle, aren't they? So, in my opinion, that was not a real proof since they were using what they want to show.

Or am I missing something? Was that a real proof?

Edit: I don't know but most probably they defined sinx and cosx in the usual way, because it was a calculus class.