1 . 已知双曲线
的焦距为8,右焦点为
,直线
与双曲线在一、三象限的交点分别为
,且
.
(1)求双曲线
的方程及
的面积;
(2)直线
与双曲线
交于
两点,若直线
与
轴分别交于点
,且
.证明:
为定值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83bf4fd84818abac17a9d21237ac5ce5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04f19a2e857946e5dec7c134838ee074.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6bce3d91ca23b86d8c6625f2632e437.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a52c17a41524894bb6932b11181e6e7.png)
(1)求双曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5adccd1dd14171c8c29d4a3836728c0f.png)
(2)直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9bb854ce93ff37f79994b2f392c76974.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5df8904b700810cfd3519798668aa35f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a8454989732716850cb57ca15f8ef596.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/27de0204d3e883f79084fe6f600c09ed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
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解题方法
2 . 已知倾斜角为
的直线
与椭圆
交于
两点,
为
中点,
为坐标原点,则直线
的斜率为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15615de1a6df206dbd081251f676578e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb4402aeb853b22f20992156957ef0fd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/abd13974aebe38eb2a1d744a01ea5aa5.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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解题方法
3 . 已知抛物线
的焦点为
,直线
交抛物线于
两点,
为抛物线
的准线与
轴的交点,直线
分别交抛物线于
两点(点
异于点
,
),
为坐标原点,则实数
的取值范围为__________ ;![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7b21b94a2bb768ccbc65c33f9c4ed923.png)
__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1bb4dd4670828f75bc573b52cdd02e1d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c654f3bb56ece42d29a9f14cca37bf40.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7789a500686c7a73770404ead6af0590.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3eb2d74932ede98011219d319f5b9995.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6bce3d91ca23b86d8c6625f2632e437.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6bce3d91ca23b86d8c6625f2632e437.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7b21b94a2bb768ccbc65c33f9c4ed923.png)
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4 . 已知椭圆
短轴长为2,椭圆
上一点
到
距离的最大值为3.
![](https://img.xkw.com/dksih/QBM/2024/5/26/3504894981390336/3520242638839808/STEM/32c129353263403684caca44141869ef.png?resizew=151)
(1)求
的取值范围;
(2)当椭圆
的离心率达到最大时,过原点
斜率为
的直线
与
交于
两点,
分别与椭圆
的另一个交点为
.
①是否存在实数
,使得
的斜率
等于
?若存在,求出
的值;若不存在,说明理由;
②记
与
交于点
,求线段
长度的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7e5578ca83f5bd5c285994061b9c015.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9059420851d219dba2189975f8eb5178.png)
![](https://img.xkw.com/dksih/QBM/2024/5/26/3504894981390336/3520242638839808/STEM/32c129353263403684caca44141869ef.png?resizew=151)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(2)当椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d7708c156249bb475564027f73313fa8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4f2ec2e454afe8452a6f1714a986aa38.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/235557c7e459c7100787e436826c198e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f9f963b16dcebd808b41842febc329a6.png)
①是否存在实数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7e08d5c04f0431fb57b33a01717b599.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e4b2e1a496f0df7918efed76a51f34c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
②记
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a5f1641947153c80b987320885a2b57.png)
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解题方法
5 . 已知抛物线
的焦点为
,其准线
与
轴的交点为
,过点
的直线与C交于
,
两点,点
为点
在
上的射影,线段
与
轴的交点为
,线段
的延长线交
于点
,则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/159a2d4945049c5a8f525f18763eba91.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/907d5147cea4c9ce855074864fe54506.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f50b3ae183997b707d16eb4e7f6712fa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
A.![]() |
B.![]() |
C.直线![]() ![]() |
D.![]() ![]() ![]() |
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解题方法
6 . 已知椭圆
的离心率为
,且过点
.圆
的切线l与椭圆E相交于A,B两点.
(1)求椭圆E的方程;
(2)直线OA,OB的斜率存在为
,
,直线l的斜率存在为k,若
,求直线l的方程;
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7e5578ca83f5bd5c285994061b9c015.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d5989c84e320b504511f23eeb6e7357.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80991c1f0c963104740e50cfff6f29a8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/52b33328faae2d2d4921900e97424de5.png)
(1)求椭圆E的方程;
(2)直线OA,OB的斜率存在为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6defc43285a40f7ccb74c1cc04265eba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/423b7ae39db552e60ee8b1d27312306f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a08a512b63d60ae9ef4315ed8615542.png)
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解题方法
7 . 在平面直角坐标系中,点
在运动过程中,总满足关系式
.
(1)求点
的轨迹
的方程;
(2)过点
作两条斜率分别为
的直线
和
,分别与
交于
和
,线段
和
的中点分别为
,若
,证明直线
过定点.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c62b58e1ce45cfd3fe723345eaf411f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/17aa130296d594a23b0a7a864fc33320.png)
(1)求点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
(2)过点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fd3b260036958c271fee22820b05fdb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4f5fac15de56be6dfb7ba2429b54cea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e9b0f5f44abbc6544a2f672b025b013.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f6f17bc385bafb37e8f964e5eb99cd0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d762c4e0c2e788c94066aeea1530f4e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/227c1d105f7abf228e7a4f3097ae93f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2026c8a047f60c7b84f4078466dcce6f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/077aaf808a6243d4af30a3eb9320fb99.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e42887d9bf31c1dd99f13c39e63c9ab9.png)
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昨日更新
|
57次组卷
|
4卷引用:四川省南充高中2023-2024学年高三下学期第十三次月考理科数学试卷(附答案)
解题方法
8 . 已知椭圆
的左、右焦点分别为
,点
在椭圆
上,点
与点
关于原点对称,四边形
的面积为4.
(1)求椭圆
的方程;
(2)若直线
与椭圆
交于
两点.与
轴交于点
.试判断是否存在
,使得
为定值?若存在,求出
的值;若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/851a5d6ec23256f9b4a9e98aa92945fe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d2a97987f71835f519b462f5b8f5957.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a692d8ace5a3c7217023c4b71dddcdb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dfffd420523729074995e9e55f464d4c.png)
(1)求椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)若直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/723e3e82e054474bff639701bf504fb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6bce3d91ca23b86d8c6625f2632e437.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/886becb1b4afcdebfa835261dc346ac9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a423a6529a1bb6c0c96027d40a7817d2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
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解题方法
9 . 如图,直线
与直线
,分别与抛物线
交于点A,B和点C,D(A,D在x轴同侧).当
经过T的焦点F且垂直于x轴时,
.
(2)线段AC与BD交于点H,线段AB与CD的中点分别为M,N
①求证:M,H,N三点共线;
②若
,求四边形ABCD的面积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16660ffd67194f17709d0b35f85ba095.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/25445686787e27c15ce3cbe20bbf2ea7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/430cd4dfec1c0932fe44320a3ef85171.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e9b0f5f44abbc6544a2f672b025b013.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a983df57f48039f3c03303a8ed2fb543.png)
(2)线段AC与BD交于点H,线段AB与CD的中点分别为M,N
①求证:M,H,N三点共线;
②若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/782f7729242188e0a9fbb12d3984512a.png)
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解题方法
10 . 已知抛物线
上一点Q到焦点F的距离为2,点Q到y轴的距离为
.
(1)求抛物线C的方程;
(2)过F的直线交抛物线C于A,B两点,过点B作x轴的垂线交直线AO(O是坐标原点)于D,过A作直线DF的垂线与抛物线C的另一交点为E,直线
与
交于点G.求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/70792da1e2be7212a3f76b8d1c999bff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8794dad05cff3e19d6ba8f1658aa8422.png)
(1)求抛物线C的方程;
(2)过F的直线交抛物线C于A,B两点,过点B作x轴的垂线交直线AO(O是坐标原点)于D,过A作直线DF的垂线与抛物线C的另一交点为E,直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/43ab7ed9b0df4db720f6f6c3e32f7c1d.png)
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