1 . 已知椭圆C:
,若椭圆的焦距为4且经过点
,过点
的直线交椭圆于P,Q两点.
(1)求椭圆方程;
(2)求
面积的最大值,并求此时直线
的方程;
(3)若直线
与x轴不垂直,在x轴上是否存在点
使得
恒成立?若存在,求出s的值;若不存在,说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/347b68f42934c74e0d759a67613a1da9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39dad657ef82bcc0e05077a859f16b1c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/25050138e5f3f481c6c2b04e03ad4d5d.png)
(1)求椭圆方程;
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2969ed9ba40bbcb2a5aa64d44f099785.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e430f13f42cf2d44aa0f0e20b959684f.png)
(3)若直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e430f13f42cf2d44aa0f0e20b959684f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2e22ccf383840e92e1aa33a15b874aa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/47fad1530eff87a020a1a947cfb56258.png)
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解题方法
2 . 已知椭圆
的左顶点为
,上顶点为
,离心率为
,
.
(1)求椭圆的方程;
(2)设点
在椭圆上,且异于椭圆的上、下顶点,点
在圆
上,直线
,
的斜率分别为
,
,且
,求证:
(i)
;
(ii)直线
过定点,并求出此定点的坐标.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7dd54b9df3402ad91e2d34c40efe0c7a.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/1174142f3bba761585b6bc2653009b36.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ae5a64bcb77f5f64e4af6930c249a270.png)
(1)求椭圆的方程;
(2)设点
![](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/ef74c4299221a967507c6a179337581a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e69d2b798744645af88a4fa411344a83.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7785afeeaf274892253d04b4f693b367.png)
![](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/c80b04ce48c9ace43276552c77108126.png)
(i)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d85c6b63bef0f632fee2e7e438a4b5cc.png)
(ii)直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
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名校
解题方法
3 . 设
,
分别是椭圆
(
)的左右焦点,过
的直线与椭圆交于
、
两点,若
的周长为16,且
的最小值为2,则椭圆的方程为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5076289823db419f94e9c0c8f4aafd9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3fb78c5f885034612c0e030b920143d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d7aea48c44781a844b5c19191f70f61.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a0c4c098615c6bc7e6dcf72e5b5201a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5076289823db419f94e9c0c8f4aafd9.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/e5eb2485f90dbfd0dfd6e7d179a856f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f4dfec890cdfdda355e19463f3be813.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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解题方法
4 . 已知椭圆
的左、右焦点分别为
,上、下顶点分别为
,且四边形
是面积为8的正方形.
(1)将椭圆
的标准方程;
(2)过点
分别作直线
交椭圆于
两点,设两直线
的斜率分别为
,且
,证明:直线
过定点.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fd486b8796b3454eab219c28ed131683.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d2a97987f71835f519b462f5b8f5957.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7789a500686c7a73770404ead6af0590.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/173ca5f0edcdd00a62de9ecd071a40a4.png)
(1)将椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)过点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03df57efff473b3cfeb8503796b7d6b6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03df57efff473b3cfeb8503796b7d6b6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/90963760acac7bfad3ae03088c6c80b0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c2b1bd378406bcd8156a56469f9300f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
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5 . 设椭圆的中心在坐标原点,对称轴为坐标轴,其中一个焦点在抛物线
的准线上,且椭圆上的任意一点到两个焦点的距离的和等于10,则椭圆的方程为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c3577bf250e9110428ff34f518fbd9f2.png)
A.![]() | B.![]() |
C.![]() | D.![]() |
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6 . 已知椭圆
的离心率为
,右焦点为
.
(1)求椭圆的方程;
(2)设直线
与椭圆交于A,B两点,求
的面积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7dd54b9df3402ad91e2d34c40efe0c7a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d5989c84e320b504511f23eeb6e7357.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/092fd1b1d33979818300cd2e3699bff7.png)
(1)求椭圆的方程;
(2)设直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e747e5274776e352f3d9ff5c2efc4b6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2eaceb8d6c6927e14d9ac7a557a2b11d.png)
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名校
解题方法
7 . 已知点
是椭圆
上一点,
,
分别为椭圆
的左、右焦点,
,当
,
的面积为
.
(1)求椭圆
的方程;
(2)设过点
直线
和椭圆
交于两点
,
,是否存在直线
,使得
与
(
是坐标原点)的面积比值为
. 若存在,求出直线
的方程;若不存在,说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3324199c6751f2e0e6d8542783b0d957.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5076289823db419f94e9c0c8f4aafd9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3fb78c5f885034612c0e030b920143d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/621f16b930274bf27e1a7fe00a1f0f72.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e073c13fdfdd225de88de306c5da23cc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a6787578ab6ec561f02578823b3b762.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8eb6e0fbc21282248f53f3f962e433db.png)
(1)求椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)设过点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3fb78c5f885034612c0e030b920143d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.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/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a2ed07d64bea40348d179123c869490e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ec11ff3a07361d2d2ab9de7a194524c3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2cd51877a2f035c75bb5f353cab2b16.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
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名校
解题方法
8 . 已知椭圆方程
,左右焦点分别
,
.离心率
,长轴长为4.
(1)求椭圆方程.
(2)若斜率为1的直线
交椭圆于A,B两点,与以
,
为直径的圆交于C,
两点.若
,求直线
的方程.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7dd54b9df3402ad91e2d34c40efe0c7a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5076289823db419f94e9c0c8f4aafd9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3fb78c5f885034612c0e030b920143d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d5c7316976a221c051a2c14df80b1347.png)
(1)求椭圆方程.
(2)若斜率为1的直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5076289823db419f94e9c0c8f4aafd9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3fb78c5f885034612c0e030b920143d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7909220d1db380634534190b662ee0a7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d95485530815641d3e7b21b4e73f2b68.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
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2024-01-24更新
|
347次组卷
|
3卷引用:天津市重点校2023-2024学年高二上学期期末联考数学试题
9 . 已知椭圆![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d7aea48c44781a844b5c19191f70f61.png)
的一个顶点为
,左、右焦点为
,
,其中O为坐标原点,过右焦点
的直线交椭圆于P,Q两点,
的周长为
.
(1)求椭圆的标准方程;
(2)已知点C满足
,点B在椭圆上(异于椭圆的顶点),直线
与以C为圆心的圆相切于点M,且M为线段
的中点,求直线
的方程.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d7aea48c44781a844b5c19191f70f61.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf7216d1023a8be68e34375f51a51dda.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/753e1797acdf8e73b7a46e8e6027f2cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5076289823db419f94e9c0c8f4aafd9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3fb78c5f885034612c0e030b920143d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3fb78c5f885034612c0e030b920143d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6380c54a58ed53ebe8977402b42c960.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/354ec8391bdd39377804ee4dab1d8f1c.png)
(1)求椭圆的标准方程;
(2)已知点C满足
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cce2ba3c9c023830673737fb1e95df25.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
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解题方法
10 . 设椭圆
(
)的上顶点为A,左焦点为F,已知椭圆的离心率
,
.
(1)求椭圆方程;
(2)设过点
且斜率为
的直线
与椭圆交于点
(
异于点
),与直线
交于点
,点
关于
轴的对称点为
,直线
与
轴交于点
,若
的面积为
,求直线
的方程.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d7aea48c44781a844b5c19191f70f61.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a0c4c098615c6bc7e6dcf72e5b5201a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/860884c0017c8bceb5b0edff796c144f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5ba3a6326dc83632fa772c1d34cbff9.png)
(1)求椭圆方程;
(2)设过点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.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/eefa44964db83759aff6fc8dd7ef8f28.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ce6c0e9de83f2e64ae33609fc08459d.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/98013a5042685a1db94249e70c62c09a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b282c337227aa697d420b8c3c8d4309.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
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