Average distance between circumference and a point?

Question

What would be the average distance from the point $P=(a,b)$ (outside the circle), to any point on the circumference with center at $(0,0)$ and radius $r$ be?

Answer 1

Use the SAS (side-angle-side) formula for the green distance, $d = \sqrt{1^2 + r^2 - 2 r \cos \theta}$, and integrate over $\theta$ (using symmetry)

$$2 \int\limits_{\theta = 0}^\pi \sqrt{1^2 + r^2 - 2 r \cos \theta}\ d \theta = 4 (r+1) E\left(\frac{4 r}{(r+1)^2}\right)$$

where the $E$ is the elliptic integral function.

Can you polish this ensuring normalization? (Think about the length of the arc integrated over.)