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　微重点11　圆锥曲线中二级结论的应用29

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　微重点11　圆锥曲线中二级结论的应用29，共6页。试卷主要包含了焦点弦定理等内容，欢迎下载使用。

考点一 焦点弦问题
核心提炼
1．已知F1，F2分别为椭圆的左、右焦点，直线 l过左焦点F1与椭圆(焦点在 x 轴上)交于A，B两点，设 ∠AF1F2＝α，e为椭圆的离心率，p为椭圆的焦点到对应准线的距离，则p＝eq \f(a2,c)－c＝eq \f(b2,c).
(1)椭圆焦半径公式：|AF1|＝eq \f(ep,1－e·cs α)，|BF1|＝eq \f(ep,1＋e·cs α)，eq \f(1,|AF1|)＋eq \f(1,|BF1|)＝eq \f(2,ep).
(2)椭圆焦点弦弦长公式：|AB|＝|AF1|＋|BF1|＝eq \f(2ep,1－e2·cs2α).
(3)焦点三角形的面积公式：P为椭圆上异于长轴端点的一点，F1，F2为其左、右焦点且∠F1PF2＝θ，则＝b2·tan eq \f(θ,2).
2．已知F1，F2分别为双曲线的左、右焦点，直线 l过左焦点F1与双曲线(焦点在 x 轴上)交于A，B两点，设 ∠AF1F2＝α，e为双曲线离心率，p为双曲线的焦点到对应准线的距离，则p＝c－eq \f(a2,c)＝eq \f(b2,c).
图1 图2
(1)若直线与双曲线交于一支(如图1)，则|AF1|＝eq \f(ep,1＋e·cs α)，|BF1|＝eq \f(ep,1－e·cs α)，eq \f(1,|AF1|)＋eq \f(1,|BF1|)＝eq \f(2,ep).
若直线与双曲线交于两支(如图2)，则|AF1|＝eq \f(ep,e·cs α＋1)，|BF1|＝eq \f(ep,e·cs α－1)，eq \b\lc\|\rc\|(\a\vs4\al\c1(\f(1,|AF1|)－\f(1,|BF1|)))＝eq \f(2,ep).
(2)双曲线焦点弦弦长公式：若直线与双曲线交于一支，则|AB|＝|AF1|＋|BF1|＝eq \f(2ep,1－e2·cs2α).
若直线与双曲线交于两支，则|AB|＝||AF1|－|BF1||＝eq \f(2ep,e2·cs2α－1).
(3)焦点三角形的面积公式：P为双曲线上异于实轴端点的一点，F1，F2为其左、右焦点且∠F1PF2＝θ，则＝eq \f(b2,tan \f(θ,2)).
3.已知直线 l过焦点F与抛物线(焦点在 x 轴上)交于A，B两点，设 ∠AFx＝α，e为抛物线离心率，p为抛物线的焦点到对应准线的距离．
(1)抛物线焦半径公式：|AF|＝eq \f(ep,1－e·cs α)＝eq \f(p,1－cs α)，|BF|＝eq \f(ep,1＋e·cs α)＝eq \f(p,1＋cs α)，eq \f(1,|AF|)＋eq \f(1,|BF|)＝eq \f(2,ep)＝eq \f(2,p).
(2)抛物线焦点弦弦长公式：|AB|＝|AF|＋|BF|＝eq \f(2ep,1－e2·cs2α)＝eq \f(2p,sin2α).
4．焦点弦定理
已知焦点在 x轴上的椭圆或双曲线或抛物线，经过其焦点F的直线交曲线于 A，B两点，直线AB的倾斜角为α，eq \(AF,\s\up6(→))＝λeq \(FB,\s\up6(→))，则曲线的离心率满足等式|ecs α|＝eq \b\lc\|\rc\|(\a\vs4\al\c1(\f(λ－1,λ＋1))).
例1 (2023·滨州模拟)过椭圆T: eq \f(x2,2)＋y2＝1上的焦点F作两条互相垂直的直线l1，l2，l1交椭圆于A，B两点，l2交椭圆于C，D两点，则|AB|＋|CD|的取值范围是( )
A.eq \b\lc\[\rc\](\a\vs4\al\c1(\f(8\r(3),3)，3\r(3))) B.eq \b\lc\[\rc\](\a\vs4\al\c1(\f(8\r(2),3)，3\r(3)))
C.eq \b\lc\[\rc\](\a\vs4\al\c1(\f(8\r(2),3)，3\r(2))) D.eq \b\lc\[\rc\](\a\vs4\al\c1(\f(8\r(3),3)，3\r(2)))
易错提醒 (1)要注意公式中α的含义．
(2)公式中的加减符号易混淆．
(3)直线与双曲线交于一支和两支的公式不一样．
跟踪演练1 已知双曲线x2－y2＝2，点F1，F2为其左、右焦点，点P为双曲线上一点，若∠F1PF2＝60°，则△F1PF2的面积为( )
A．2 B．2eq \r(2) C.eq \r(3) D．2eq \r(3)
考点二 等角的性质
核心提炼
1．已知椭圆eq \f(x2,a2)＋eq \f(y2,b2)＝1(a>b>0)，过长轴上任意一点N(t,0)的弦的端点A，B与对应的点Geq \b\lc\(\rc\)(\a\vs4\al\c1(\f(a2,t)，0))的连线所成的角被焦点所在的直线平分，即∠OGA＝∠OGB(如图1)．
图1 图2 图3
2．已知双曲线eq \f(x2,a2)－eq \f(y2,b2)＝1(a>0，b>0)，过实轴所在直线上任意一点N(t,0)的弦的端点 A，B与对应点Geq \b\lc\(\rc\)(\a\vs4\al\c1(\f(a2,t)，0))的连线所成的角被焦点所在的直线平分，即∠NGA＝∠NGB(如图2)．
3．已知抛物线 y2＝2px(p>0)，过抛物线对称轴上任意一点N(a,0)的一条弦的端点 A，B与对应点G(－a,0)的连线所成角被对称轴平分，即∠OGA＝∠OGB(如图3)．
例2 (2023·洛阳模拟)在平面直角坐标系Oxy中，曲线C：x2＝6y与直线l：y＝kx＋3交于M，N两点．
(1)当1________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
(2)y轴上是否存在点P，使得当k变动时，总有∠OPM＝∠OPN？若存在，求以线段OP为直径的圆的方程；若不存在，请说明理由．
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
规律方法 根据等角性质，存在某定点满足条件，快速算出此点的坐标，这给算出准确答案提供了依据．
跟踪演练2 椭圆C：eq \f(x2,a2)＋eq \f(y2,b2)＝1(a>b>0)的离心率为eq \f(\r(2),2)，过点P(0,1)的动直线l与椭圆相交于A，B两点，当直线l平行于x轴时，直线l被椭圆C截得的线段长为2eq \r(6).
(1)求椭圆C的方程；
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
(2)在y轴上是否存在异于点P的定点Q，使得直线l变化时，总有∠PQA＝∠PQB？若存在，求出点Q的坐标；若不存在，请说明理由．
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
考点三 切线、切点弦方程
核心提炼
1．已知点P(x0，y0)为椭圆(或双曲线)上任一点，则过点P与圆锥曲线相切的切线方程为椭圆中eq \f(x0x,a2)＋eq \f(y0y,b2)＝1，双曲线中eq \f(x0x,a2)－eq \f(y0y,b2)＝1.
2．若点P(x0，y0)是椭圆(或双曲线)外一点，过点P(x0，y0)作椭圆(或双曲线)的两条切线，切点分别为A，B，则切点弦AB的直线方程是椭圆中eq \f(x0x,a2)＋eq \f(y0y,b2)＝1，双曲线中eq \f(x0x,a2)－eq \f(y0y,b2)＝1.
例3 过点Q(－1，－1)作已知直线l：y＝eq \f(1,4)x＋1的平行线，交双曲线eq \f(x2,4)－y2＝1于点M，N.
(1)证明：Q是线段MN的中点；
(2)分别过点M，N作双曲线的切线l1，l2，证明：三条直线l，l1，l2相交于同一点；
(3)设P为直线l上一动点，过P作双曲线的切线PA，PB，切点分别为A，B，证明：点Q在直线AB上．
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
规律方法 运用联想，由过已知圆上和圆外的点的切线方程联想到过圆锥曲线上和圆锥曲线外的切线方程，触类旁通，实现知识的内迁，使知识更趋于系统化，取得事半功倍的效果．
跟踪演练3 (2023·锦州模拟)已知椭圆E：eq \f(x2,a2)＋eq \f(y2,b2)＝1(a>b>0)经过点eq \b\lc\(\rc\)(\a\vs4\al\c1(0，\r(2)))，且离心率为eq \f(\r(6),3).F为椭圆E的左焦点，点P为直线l：x＝3上的一点，过点P作椭圆E的两条切线，切点分别为A，B，连接AB，AF，BF.
(1)求证：直线AB过定点M，并求出定点M的坐标；
(2)记△AFM，△BFM的面积分别为S1和S2，当|S1－S2|取最大值时，求直线AB的方程．
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________