On distinct perpendicular bisectors and pinned distances in finite fields

Given a set of points $P \subset \mathbb F_q^2$ such that $|P|\geq q^{3/2}$ it is established that $|P|$ determines $\Omega(q^2)$ distinct perpendicular bisectors. It is also proven that, if $|P| \geq q^{4/3}$, then for a positive proportion of points $a \in P$, we have $$|\{\| a- b\|: b \in P\}|=\Omega(q),$$ where $\|a- b\|$ is the distance between points $a$ and $b$. The latter result represents an improvement on a result of Chapman et al. (arxiv:0903.4218).