2024學生版大二輪數學新高考提高版（京津瓊魯遼粵冀鄂湘渝閩蘇浙黑吉晉皖云豫新甘貴贛桂）專題六　第4講　母題突破1　范圍、最值問題33

第4講圓錐曲線的綜合問題[考情分析]1.圓錐曲線的綜合問題是高考考查的重點內容，常見的熱點題型有范圍、最值問題，定點、定直線、定值問題及探索性問題.2.以解答題的形式壓軸出現(xiàn)，難度較大．母題突破1范圍、最值問題母題(2023·全國甲卷)已知直線x－2y＋1＝0與拋物線C：y2＝2px(p>0)交于A，B兩點，|AB|＝4eq\r(15).(1)求p；(2)設F為C的焦點，M，N為C上兩點，eq\o(FM,\s\up6(→))·eq\o(FN,\s\up6(→))＝0，求△MFN面積的最小值．思路分析?聯(lián)立方程利用弦長求p?設直線MN：x＝my＋n和點M，N的坐標?利用eq\o(FM,\s\up6(→))·eq\o(FN,\s\up6(→))＝0，得m，n的關系?寫出S△MFN的面積?利用函數性質求S△MFN面積的最小值________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________[子題1](2023·武漢模擬)已知橢圓C：eq\f(x2,4)＋y2＝1，橢圓C的右頂點為A，若點P，Q在橢圓C上，且滿足直線AP與AQ的斜率之積為eq\f(1,20)，求△APQ面積的最大值．________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________[子題2](2023·深圳模擬)已知雙曲線C：x2－y2＝1，設點A為C的左頂點，若過點(3,0)的直線l與C的右支交于P，Q兩點，且直線AP，AQ與圓O：x2＋y2＝1分別交于M，N兩點，記四邊形PQNM的面積為S1，△AMN的面積為S2，求eq\f(S1,S2)的取值范圍．________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________規(guī)律方法求解范圍、最值問題的常見方法(1)利用判別式來構造不等關系．(2)利用已知參數的范圍，在兩個參數之間建立函數關系．(3)利用隱含或已知的不等關系建立不等式．(4)利用基本不等式．1．(2023·佛山模擬)在平面直角坐標系中，點O為坐標原點，Meq\b\lc\(\rc\)(\a\vs4\al\co1(－1，0))，N(1,0)，Q為線段MN上異于M，N的一動點，點P滿足eq\f(|PM|,|QM|)＝eq\f(|PN|,|QN|)＝2.(1)求點P的軌跡E的方程；(2)點A，C是曲線E上兩點，且在x軸上方，滿足AM∥NC，求四邊形AMNC面積的最大值．________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________2．(2023·溫州模擬)已知拋物線C1：y2＝4x－4與雙曲線C2：eq\f(x2,a2)－eq\f(y2,4－a2)＝1(1<a<2)相交于A，B兩點，F(xiàn)是C2的右焦點，直線AF分別交C1，C2于C，D(不同于A，B點)兩點，直線BC，BD分別交x軸于P，Q兩點．