2024学生版大二轮数学新高考提高版（京津琼鲁辽粤冀鄂湘渝闽苏浙黑吉晋皖云豫新甘贵赣桂）专题六　第4讲　母题突破1　范围、最值问题33

这是一份2024学生版大二轮数学新高考提高版（京津琼鲁辽粤冀鄂湘渝闽苏浙黑吉晋皖云豫新甘贵赣桂）专题六　第4讲　母题突破1　范围、最值问题33，共4页。
母题突破1 范围、最值问题
母题 (2023·全国甲卷)已知直线x－2y＋1＝0与抛物线C：y2＝2px(p>0)交于A，B两点，|AB|＝4eq \r(15).
(1)求p；
(2)设F为C的焦点，M，N为C上两点，eq \(FM,\s\up6(→))·eq \(FN,\s\up6(→))＝0，求△MFN面积的最小值．
思路分析
❶联立方程利用弦长求p
❷设直线MN：x＝my＋n和点M，N的坐标
❸利用eq \(FM,\s\up6(→))·eq \(FN,\s\up6(→))＝0，得m，n的关系
❹写出S△MFN的面积
❺利用函数性质求S△MFN面积的最小值
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[子题1] (2023·武汉模拟)已知椭圆C：eq \f(x2,4)＋y2＝1，椭圆C的右顶点为A，若点P，Q在椭圆C上，且满足直线AP与AQ的斜率之积为eq \f(1,20)，求△APQ面积的最大值．
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[子题2] (2023·深圳模拟)已知双曲线C：x2－y2＝1，设点A为C的左顶点，若过点(3,0)的直线l与C的右支交于P，Q两点，且直线AP，AQ与圆O：x2＋y2＝1分别交于M，N两点，记四边形PQNM的面积为S1，△AMN的面积为S2，求eq \f(S1,S2)的取值范围．
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规律方法 求解范围、最值问题的常见方法
(1)利用判别式来构造不等关系．
(2)利用已知参数的范围，在两个参数之间建立函数关系．
(3)利用隐含或已知的不等关系建立不等式．
(4)利用基本不等式．
1．(2023·佛山模拟)在平面直角坐标系中，点O为坐标原点，Meq \b\lc\(\rc\)(\a\vs4\al\c1(－1，0))，N(1,0)，Q为线段MN上异于M，N的一动点，点P满足eq \f(|PM|,|QM|)＝eq \f(|PN|,|QN|)＝2.
(1)求点P的轨迹E的方程；
(2)点A，C是曲线E上两点，且在x轴上方，满足AM∥NC，求四边形AMNC面积的最大值．
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2．(2023·温州模拟)已知抛物线C1：y2＝4x－4与双曲线C2：eq \f(x2,a2)－eq \f(y2,4－a2)＝1(1