  Given two irreducible conics $C$ and $D$ over a finite field $\mathbb{F}_q$
with $q$ odd, we show that there are $q^2/4+O(q^{3/2})$ points $P$ in
$\mathbb{P}^2(\mathbb{F}_q)$ such that $P$ is external to $C$ and internal to
$D$. This answers a question of Korchm\'{a}ros. We also prove the analogous
result for higher-dimensional smooth quadric hypersurfaces over
$\mathbb{P}^{n-1}$ with $n$ odd, where the answer is
$q^{n-1}/4+O(q^{n-\frac{3}{2}})$.
