Let $M$ be a Poisson random measure on $E=\mathbb{R} \times[0, \pi)$ with constant intensity $\lambda$. For $(x, \theta) \in E$, denote by $l(x, \theta)$ the line in $\mathbb{R}^{2}$ obtained by rotating the line $\{(x, y): y \in \mathbb{R}\}$ through an angle $\theta$ about the origin.

Consider the line process $L=M \circ l^{-1}$.

(i) What is the distribution of the number of lines intersecting the disc $\left\{z \in \mathbb{R}^{2}:|z| \leqslant a\right\}$ ?

(ii) What is the distribution of the distance from the origin to the nearest line?

(iii) What is the distribution of the distance from the origin to the $k$ th nearest line?