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4　探究性问题30

母题突破4探究性问题母题(2023·廊坊质检)已知椭圆C：eq\f(x2,a2)＋eq\f(y2,b2)＝1(a>b>0)经过点A(－2,0)，且两个焦点及短轴两顶点围成四边形的面积为4.(1)求椭圆C的方程和离心率；(2)设P，Q为椭圆C上两个不同的点，直线AP与y轴交于点E，直线AQ与y轴交于点F，且P，O，Q三点共线．其中O为坐标原点．问：在x轴上是否存在点M，使得∠AME＝∠EFM？若存在，求点M的坐标；若不存在，请说明理由．思路分析❶代入点，结合面积求方程和离心率❷设点P，Q，表示出直线AP，AQ的方程❸求出E，F的坐标❹由∠AME＝∠EFM得eq\o(ME,\s\up6(→))·eq\o(MF,\s\up6(→))＝0❺利用向量运算求点M的坐标________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________[子题1](2023·西安模拟)已知椭圆C：eq\f(x2,4)＋eq\f(y2,3)＝1，过点Teq\b\lc\(\rc\)(\a\vs4\al\co1(\r(3)，0))的直线交该椭圆于P，Q两点，若直线PQ与x轴不垂直，在x轴上是否存在定点S(s,0)，使得∠PST＝∠QST恒成立？若存在，求出s的值；若不存在，请说明理由．________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________[子题2]已知双曲线C：eq\f(y2,a2)－eq\f(x2,b2)＝1(a>0，b>0)，直线l在x轴上方与x轴平行，交双曲线C于A，B两点，直线l交y轴于点D.当l经过C的焦点时，点A的坐标为(6,4)．(1)求C的方程；(2)设OD的中点为M，是否存在定直线l，使得经过M的直线与C交于P，Q两点，与线段AB交于点N(N，D不重合)，eq\o(PM,\s\up6(→))＝λeq\o(PN,\s\up6(→))，eq\o(MQ,\s\up6(→))＝λeq\o(QN,\s\up6(→))均成立？若存在，求出l的方程；若不存在，请说明理由．________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________规律方法探索性问题的求解策略(1)若给出问题的一些特殊关系，要探索一般规律，并能证明所得规律的正确性，通常要对已知关系进行观察、比较、分析，然后概括一般规律．(2)若只给出条件，求“不存在”“是否存在”等语句表述问题时，一般先对结论给出肯定的假设，然后由假设出发，结合已知条件进行推理，从而得出结论．1．已知抛物线C：x2＝2py(p>0)，点P(2,8)在抛物线上，直线y＝kx＋2交C于A，B两点，M是线段AB的中点，过M作x轴的垂线交C于点N.(1)求点P到抛物线焦点的距离；(2)是否存在实数k使得eq\o(NA,\s\up6(→))·eq\o(NB,\s\up6(→))＝0，若存在，求k的值；若不存在，请说明理由．________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________2.(2023·池州模拟)如图，点A为椭圆E：eq\f(x2,4)＋y2＝1的上顶点，圆C：x2＋y2＝1，过坐标原点O的直线l交椭圆E于M，N两点．(1)求直线AM，AN的斜率之积；(2)设直线AM：y＝kx＋1(k≠0)，AN与圆C分别交于点P，Q，记直线MN，PQ的斜率分别为k1，k2，探究是否存在实数λ，使得k1＝λk2？若存在，求出λ的值；若不存在，请说明理由．____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________