Abstract Given two irreducible conics C and D over a finite field $$\mathbb {F}_q$$ F q with q odd, we show that there are $$q^2/4+O(q^{3/2})$$ q 2 / 4 + O ( q 3 / 2 ) points P in $$\mathbb {P}^2(\mathbb {F}_q)$$ P 2 ( 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 $$\mathbb {P}^{n-1}$$ P n - 1 with n odd, where the answer is $$q^{n-1}/4+O(q^{n-\frac{3}{2}})$$ q n - 1 / 4 + O ( q n - 3 2 ) .