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　培优点8　圆锥曲线中非对称韦达定理的应用27

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　培优点8　圆锥曲线中非对称韦达定理的应用27，共3页。

考点一 分式型
例1 (2023·新高考全国Ⅱ)已知双曲线C的中心为坐标原点，左焦点为(－2eq \r(5)，0)，离心率为eq \r(5).
(1)求C的方程；
(2)记C的左、右顶点分别为A1，A2，过点(－4,0)的直线与C的左支交于M，N两点，M在第二象限，直线MA1与NA2交于点P.证明：点P在定直线上．
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规律方法 非对称结构的常规处理方法有和积转换、配凑、求根公式(暴力法)、曲线方程代换、第三定义等方法，将其转化为对称结构计算．
跟踪演练1 已知椭圆C：eq \f(x2,a2)＋eq \f(y2,b2)＝1(a>b>0)的左、右焦点分别为F1(－c,0)，F2(c,0)，M，N分别为左、右顶点，直线l：x＝ty＋1与椭圆C交于A，B两点，当t＝－eq \f(\r(3),3)时，A是椭圆的上顶点，且△AF1F2的周长为6.
(1)求椭圆C的方程；
(2)设直线AM，BN交于点Q，证明：点Q在定直线上．
(3)设直线AM，BN的斜率分别为k1，k2，证明：eq \f(k1,k2)为定值．
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考点二 比值型
例2 (2023·深圳模拟)在平面直角坐标系Oxy中，已知双曲线C：eq \f(y2,a2)－eq \f(x2,b2)＝1(a>0，b>0)的一条渐近线为y＝eq \f(\r(3),3)x，且点Peq \b\lc\(\rc\)(\a\vs4\al\c1(\r(3)，\r(2)))在C上．
(1)求C的方程；
(2)设C的上焦点为F，过F的直线l交C于A，B两点，且eq \(AF,\s\up6(→))＝7eq \(BF,\s\up6(→))，求l的斜率．
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规律方法 比值型问题适用于x1＝λx2型，可以采用倒数相加，但有时得到的可能不是这种形式，而是x1＝λx2＋k的形式，此时采用待定系数法，例如x1＝－3x2＋4，可以转化x1－1＝－3(x2－1)，得到eq \f(x1－1,x2－1)＝－3，继续采用倒数相加解决．
跟踪演练2 已知点A(0，－2)，椭圆E：eq \f(x2,a2)＋eq \f(y2,b2)＝1(a>b>0)的离心率为eq \f(\r(3),2)，F是椭圆E的右焦点，直线AF的斜率为eq \f(2\r(3),3)，O为坐标原点．
(1)求E的方程；
(2)设过点A的直线l与E相交于P，Q两点，且eq \(AP,\s\up6(→))＝eq \f(1,2)eq \(AQ,\s\up6(→))，求△OPQ的面积及直线l的方程．
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