解题方法
1 . 已知椭圆![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca3997e2e743835239010fa32ee3e001.png)
,过原点且倾斜角为
和![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d00ba13bb9d464a1c6609acb530179e3.png)
的两条直线分别交椭圆于
、
和
、
四点.
(1)用
、
、
表示四边形
的面积
;
(2)若
、
为定值,当
时,求
的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca3997e2e743835239010fa32ee3e001.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20ac536dc56e720243eca2f623a6b3c8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d00ba13bb9d464a1c6609acb530179e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f799e2ddc863782fb7816c599bc6e955.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/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
(1)用
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/89bffbbb348ffe8b819e2af068d49b29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
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解题方法
2 . 已知椭圆
的短轴上端点为P,过点P作椭圆互相垂直的两弦
.连接
,试求点P在
上的射影Q的轨迹方程.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc603fea876cbf85b1efcb5bab0d500f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2ec8858389f4c3156a946ba8bf0d8a7b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
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解题方法
3 . 已知抛物线C:
,过点
的直线l交抛物线于P、Q两点,以OP、OQ为邻边作平行四边形OPRQ.
(1)求点R的轨迹方程.
(2)是否存在l,使四边形OPRQ为正方形?证明你的结论.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/745de5ef1fd897d16e37464172d5e8c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a23e87d16c32b5aa4357f481b5808a0.png)
(1)求点R的轨迹方程.
(2)是否存在l,使四边形OPRQ为正方形?证明你的结论.
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解题方法
4 . 双曲线M的中心在原点,并以椭圆
的焦点为焦点,以抛物线
的准线为右准线.
(1)求双曲线M的方程.
(2)设直线l:
与双曲线M相交于A、B两点,若A、B两点关于直线
对称,求k的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/061e7ba66be97eed28385c7a040369cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/178c349ffcb903c4cca9fce87d0fad6f.png)
(1)求双曲线M的方程.
(2)设直线l:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/023ab3a7eb0f59993c7608576e47c0f0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/92b793d7aca7075061058b87fd0c6190.png)
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5 . 已知
的三个顶点都在抛物线
上,且O为抛物线的顶点,抛物线的焦点F满足
,若BC边上的中线所在直线l的方程为
(m,n为常数且
),记
、
、
的面积分别记为
、
、
,则
的值为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7089148c36cb3c39af71de653756396a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d856ed2306a7cc3e64a1edfb84627aa5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d786625f71656562d6e44af9c0d2ee0a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0baedc4d7e690ab3f7d80d30ba0a9efe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/92fb4c7a4bf45e467b2d90f6325056c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bff3b4de4a08843340abc16e10bbd3d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/716189b906950b7e8ee499e9b04f5b2c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e097c8d4c948de063796bd19f85b3a9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e0bd63f55069a3bc870915010b39225.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6899bf9cadae2ccdb14cbc87d4f280ee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2678f7eb2dd77cf46af0a1c08e1801f1.png)
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解题方法
6 . 已知椭圆C:
的离心率为
,点![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c84d316b10430afac5b0c065f31e1b2c.png)
分别是椭圆
的左,右焦点,以原点为圆心,椭圆
的短半轴为半径的圆与直线
相切.
(1)求椭圆
的方程;
(2)若过点
的直线l与椭圆
相交于
两点,求使
面积最大时直线l的方程.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7dd54b9df3402ad91e2d34c40efe0c7a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f89eef3148f2d4d09379767b4af69132.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c84d316b10430afac5b0c065f31e1b2c.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/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d38f9106cef0b638e62c90afc1c523cd.png)
(1)求椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)若过点
![](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/7789a500686c7a73770404ead6af0590.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76ea1c3fe8431260ecb8dffcdae8d570.png)
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解题方法
7 . 已知椭圆
的焦点在
轴上,它的一个顶点恰好是抛物线
的焦点,离心率
.
(1)求椭圆的标准方程;
(2)过椭圆
的右焦点
作直线
交椭圆
于
两点,交
轴于点
,若
,那么
为定值吗?证明你的结论.
![](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/e48f28aeccf369df5980ac787e9e313f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e5bb267505c2a5fc6f9abc02bd43120.png)
(1)求椭圆的标准方程;
(2)过椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.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/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad150fcb95a93b39a16af8cced105f0d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32705e629d8b9187b53efeee6605af15.png)
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解题方法
8 . 直线与
轴交于点
,与
轴交于点
,且直线与椭圆
相切,则
的最小值为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d7a6fd6d651ae341154c2e40928d628.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03eb5be422dc9f0f01736b0f2b08a3a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48da128547c4cf9745e8e4b99988a3db.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f4dfec890cdfdda355e19463f3be813.png)
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9 . 在
中,
分别是内角
的对边,
成等差数列,且
,
.
(1)求顶点
的轨迹
的方程;
(2)是否存在直线
,使得
过点
且与曲线
交于不同的两点
,且
?若存在,求出直线
的方程,若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f7f5573b30734d65648f61c0a94c98de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38335830b93ac4d99c28a8e209eecb3f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01d019e58114a7008512af5e72ad288a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c98312826bb039bac770fc7c7380ebdf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/395bbdcf87796c48d3a95fc923edb115.png)
(1)求顶点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
(2)是否存在直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.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/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a5b1b15a4605fce993cb13aefbf40360.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c2ccdaf90efc0fb7fd0ac132c344f603.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
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10 . 已知抛物线
,过其焦点
作两条相互垂直且不平行于
轴的直线,分别交抛物线
于点
和点
,线段
的中点分别记为
.
(1)求
面积的最小值;
(2)求线段
的中点
满足的方程.
![](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/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5425108c557f0f21474c045334f97d9c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1361a10d0d1973969bec5e537c518a3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5f67918697e8bcec994c392d39ee87b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/237c115d5b39d761e1cbcae031070b70.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4f1ce4c9091a7d1ba57d4bd99d1b08ca.png)
(2)求线段
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3dedd84baa5219a2af415be51947c301.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
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