Given two irreducible conics C and D over a finite field Fq with q odd, we show that there are q²/4+O(q3/2) points P in P²(Fq) 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 Pⁿ⁻¹ with n odd, where the answer is qⁿ⁻¹/4+O(qn-3/2).
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Asgarli et al. (2024) studied this question.
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