  The geometry of algebraic curves over finite fields is a rich area of
research. In previous work, the authors investigated a particular aspect of the
geometry over finite fields of the classical unit circle, namely how the number
of solutions of the circle equation depends on the characteristic $p$ and the
degree $n\geq 1$ of the finite field $\mathbb{F}_{p^n}$. In this paper, we make
a similar study of the geometry over finite fields of the quadratic curves
defined by the quadratic equations in two variables for the classical conic
sections. In particular the quadratic equation with mixed term is interesting,
and our results display a rich variety of possibilities for the number of
solutions to this equation over a finite field.
