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2　定点(定直线)问题32

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2　定点(定直线)问题32，共3页。试卷主要包含了证明等内容，欢迎下载使用。

(1)求双曲线C的方程；
(2)过点P的动直线l与C的左、右两支分别交于A，B两点，若点M在线段AB上，满足eq \f(|AP|,|AM|)＝eq \f(|BP|,|BM|)，证明：点M在定直线上．
思路分析
❶利用离心率和eq \(PF1,\s\up6(—→))·eq \(PF2,\s\up6(—→))＝6求方程
❷设直线方程y－1＝kx－3并联立
❸利用比例关系eq \f(|AP|,|AM|)＝eq \f(|BP|,|BM|)列式
❹将根与系数的关系代入化简
❺消去参数得点在定直线上
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[子题1] (2023·信阳模拟)已知椭圆C：eq \f(x2,4)＋eq \f(y2,2)＝1的左、右顶点分别为A1，A2，过点D(1,0)的直线l与椭圆C交于异于A1，A2的M，N两点．若直线A1M与直线A2N交于点P，证明：点P在定直线上，并求出该定直线的方程．
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[子题2] (2023·岳阳模拟)已知双曲线C：x2－eq \f(y2,3)＝1，P为双曲线的右顶点，设直线l不经过P点且与C相交于A，B两点，若直线PA与直线PB的斜率之和为－1.证明：直线l恒过定点．
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规律方法 动线过定点问题的两大类型及解法
(1)动直线l过定点问题，解法：设动直线方程(斜率存在)为y＝kx＋t，由题设条件将t用k表示为t＝mk，得y＝k(x＋m)，故动直线过定点(－m,0)．
(2)动曲线C过定点问题，解法：引入参变量建立曲线C的方程，再根据其对参变量恒成立，令其系数等于零，得出定点．
1.(2023·襄阳模拟)过抛物线x2＝2py(p>0)内部一点P(m，n)作任意两条直线AB，CD，如图所示，连接AC，BD并延长交于点Q，当P为焦点并且AB⊥CD时，四边形ACBD面积的最小值为32.
(1)求抛物线的方程；
(2)若点P(1,1)，证明：点Q在定直线上运动，并求出定直线方程．
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2．(2023·全国乙卷)已知椭圆C：eq \f(y2,a2)＋eq \f(x2,b2)＝1(a>b>0)的离心率是eq \f(\r(5),3)，点A(－2,0)在C上．
(1)求C的方程；
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(2)过点(－2,3)的直线交C于P，Q两点，直线AP，AQ与y轴的交点分别为M，N，证明：线段MN的中点为定点．
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