Mutual position of two smooth quadrics over finite fields

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