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3讲　直线与圆锥曲线的位置关系34

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3讲　直线与圆锥曲线的位置关系34，共6页。

考点一 弦长问题
核心提炼
已知A(x1，y1)，B(x2，y2)，直线AB的斜率为k(k≠0)，
则|AB|＝eq \r(x1－x22＋y1－y22)
＝eq \r(1＋k2)|x1－x2|
＝eq \r(1＋k2)eq \r(x1＋x22－4x1x2)，
或|AB|＝eq \r(1＋\f(1,k2))|y1－y2|
＝eq \r(1＋\f(1,k2))eq \r(y1＋y22－4y1y2).
例1 已知点P在圆O：x2＋y2＝4上运动，过点P作x轴的垂线段PD，D为垂足，M为线段PD的中点(当点P为圆与x轴的交点时，规定点M与点P重合)．
(1)求点M的轨迹方程；
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(2)经过点eq \b\lc\(\rc\)(\a\vs4\al\c1(\r(3)，0))作直线l，与圆O相交于A，B两点，与点M的轨迹相交于C，D两点，若|AB|·|CD|＝eq \f(8\r(10),5)，求直线l的方程．
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易错提醒 (1)设直线方程时，需考虑特殊直线，如直线的斜率不存在、斜率为0等．
(2)涉及直线与圆锥曲线相交时，Δ>0易漏掉．
(3)|AB|＝x1＋x2＋p是抛物线过焦点的弦的弦长公式，其他情况该公式不成立．
跟踪演练1 已知双曲线C：eq \f(x2,a2)－eq \f(y2,b2)＝1(a>0，b>0)的实轴长为6，左、右焦点分别为F1，F2，点A在双曲线C上，AF2⊥x轴，且|AF1|＝7.
(1)求双曲线C及其渐近线的方程；
(2)如图，若过点F1且斜率为k(k>0)的直线l与双曲线C及其两条渐近线从左至右依次交于M，P，Q，N四点，且|MN|＝2|PQ|，求k.
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考点二 面积问题
例2 (2023·合肥模拟)已知双曲线C：eq \f(x2,a2)－eq \f(y2,b2)＝1(a>0，b>0)的左、右焦点分别为F1，F2，A为双曲线C的右支上一点，点A关于原点O的对称点为B，满足∠F1AF2＝60°，且|BF2|＝2|AF2|.
(1)求双曲线C的离心率；
(2)若双曲线C过点eq \b\lc\(\rc\)(\a\vs4\al\c1(\r(3)，2))，过圆O：x2＋y2＝b2上一点T(x0，y0)作圆O的切线l，直线l交双曲线C于P，Q两点，且△OPQ的面积为2eq \r(10)，求直线l的方程．
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规律方法 圆锥曲线中求解三角形面积的方法
(1)常规面积公式：S＝eq \f(1,2)×底×高．
(2)正弦面积公式：S＝eq \f(1,2)absin C.
(3)铅锤水平面面积公式：
①过x轴上的定点：S＝eq \f(1,2)a|y1－y2|(a为x轴上定长)；
②过y轴上的定点：S＝eq \f(1,2)a|x1－x2|(a为y轴上定长)．
跟踪演练2 已知椭圆C：eq \f(x2,a2)＋eq \f(y2,b2)＝1(a>b>0)的一个焦点为Feq \b\lc\(\rc\)(\a\vs4\al\c1(－\r(3)，0))，且离心率为eq \f(\r(3),3).
(1)求椭圆C的方程；
(2)若过椭圆C的左焦点，倾斜角为60°的直线与椭圆交于A，B两点，O为坐标原点，求△AOB的面积．
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考点三 中点弦问题
核心提炼
已知A(x1，y1)，B(x2，y2)为圆锥曲线E上两点，AB的中点C(x0，y0)，直线AB的斜率为k.
若E的方程为eq \f(x2,a2)＋eq \f(y2,b2)＝1(a>b>0)，
则k＝－eq \f(b2,a2)·eq \f(x0,y0)；
若E的方程为eq \f(x2,a2)－eq \f(y2,b2)＝1(a>0，b>0)，
则k＝eq \f(b2,a2)·eq \f(x0,y0)；
若E的方程为y2＝2px(p>0)，则k＝eq \f(p,y0).
例3 (2023·呼和浩特模拟)已知抛物线T：y2＝2px(p>0)和椭圆C：eq \f(x2,4)＋eq \f(y2,2)＝1，过抛物线T的焦点F的直线l交抛物线于A，B两点，线段AB的中垂线交椭圆C于M，N两点．
(1)若F恰是椭圆C的焦点，求p的值；
(2)若p∈N*，且MN恰好被AB平分，求△OAB的面积．
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规律方法 处理中点弦问题常用的求解方法
跟踪演练3 (2023·萍乡模拟)已知双曲线C：eq \f(x2,a2)－eq \f(y2,b2)＝1(a>0，b>0)的右焦点为F(eq \r(6)，0)，且C的一条渐近线经过点D(eq \r(2)，1)．
(1)求C的标准方程；
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(2)是否存在过点P(2,1)的直线l与C交于不同的A，B两点，且线段AB的中点为P.若存在，求出直线l的方程；若不存在，请说明理由．
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