What is the distance between a point outside of circle and any point inside the circle?

Question

Point $P$ is located at distance $d$ from a circle with radius $r$ (that is $d+r$ from the center of circle). What would be the expected value of the distance between the point $P$ and any random (uniform) point in the circle and why?

Answer 1

According to this post

https://en.wikipedia.org/wiki/Circular_segment

we need to calculate $d+x$ (notice that our $d$ is not equal to the one in the post) in order to get $\theta$ and finally $s$.

Let's start by calculating $x$, in the picture there are two right triangles that share a side, let's call it $y$, then

$$\left\{\begin{matrix}R^2&=&(d+x)^2+y^2\\r^2&=&(r-x)^2+y^2\end{matrix}\right.\Rightarrow R^2-(d+x)^2=r^2-(r-x)^2\Rightarrow x=\frac{R^2-d^2}{2(d+r)}$$

thus $$\theta=2\arccos\bigl(\frac{d+x}{R}\bigl)=2\arccos\bigl(\frac{R^2-d^2}{2R(d+r)}+\frac{d}{R}\bigl)$$ and finally $$l(R)=2R\arccos\bigl(\frac{R^2-d^2}{2R(d+r)}+\frac{d}{R}\bigl)$$