1 . 已知
为椭圆
上的点,对椭圆
上的任意两点P、Q,用如下办法定义它们的“和”
:过点S作一条平行于
(若点P与Q重合,则直线
表示椭圆
在P处的切线)的直线l与椭圆
交于不同于S的另一点,记作
(若l与椭圆
相切,则规定S为
).并规定
.
(1)若点
,求
、
以及
的坐标.
(2)在椭圆
上是否存在不同于S的点P,满足
?若存在,求出所有满足条件的点P的坐标;若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cacd4c2119a2381d627a45de68e2d263.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/459208d3062060f65fa347e6368e1457.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b94469fd19f40116e2dec334919d6586.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f460a3d908ee5647b7ba52c19740ef08.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a5f1641947153c80b987320885a2b57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a5f1641947153c80b987320885a2b57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b94469fd19f40116e2dec334919d6586.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b94469fd19f40116e2dec334919d6586.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f460a3d908ee5647b7ba52c19740ef08.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f460a3d908ee5647b7ba52c19740ef08.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c9eb1745e3e06d9a875107480c658b54.png)
(1)若点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/abc301000af55006ce42a131e6d56097.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f460a3d908ee5647b7ba52c19740ef08.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1624d5e6bace3bfe3f7b69ddbd774a9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ded53338ed7b2c20fd5e96522debbe4.png)
(2)在椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b94469fd19f40116e2dec334919d6586.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/75cb0ff741c5a50624b7fa1a0ddfc3b1.png)
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2 . 已知动点
在椭圆
的内部,过
作椭圆的弦
.证明:
、
中必有一个不超过
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c3c9708ef0dc6d6f5dcf6596d3e4f6e5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/441f8aa7df5c6dee72692429f8e01db4.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/d063ec7f9dbeba72fabf4437f9400e07.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e3ce8eed2660a33465a7f764e76cdba8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c222550cd67b33e01e72b7f1331dd2b7.png)
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3 . 已知单位圆
上三个点
,
,
满足
.则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f2499f2c837c79b4a3234304e7274b82.png)
__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f240cccaf24af8a796abb95cb42be52e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12a3efb79f35db8448f3391252ab7d4e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8df332f01628130c084fd46aaca0a4b7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c16dac1e9bf5804c8907cbc59014d04.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e79718083af8cb21877878f9170142e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f2499f2c837c79b4a3234304e7274b82.png)
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4 . 如图,已知椭圆
,
、
为椭圆与
轴的交点,
,
,且
,
.动点
在
轴上方的
上移动.则
的最小值是_____ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/271e595c257e4c0ade90a9bbbf0e6b0d.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/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9536a2be7b84612f45cc875a00c5a5d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5bd2e101f851bb77cfa793f4038015cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e54efb07b75e9842fd51a807b086806.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e30fb8caf1df96127bfbfbde82a930a8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1010846eeec6c9da29640f5aa3f8738.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e03cab2c9f78f55c14738123d22f3278.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/9/6bb4c3a0-4ea8-43e3-831a-e048a2e3e425.png?resizew=135)
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5 . 椭圆
的左焦点为
,直线
与椭圆在第一象限的焦点为
,过
作椭圆的切线交
轴于点
.则
的面积是.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6cae00bdc6f8b564b6b15b32572c848b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/45cc81cfaccc00aa4b7139de5a35a102.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.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/c84c1f9cf5571338256afc8479182a54.png)
A.![]() | B.![]() |
C.![]() | D.![]() |
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6 . 已知椭圆
的上顶点为
,右焦点为
,直线
与椭圆交于
、
两点.试判断是否存在直线
,使得点
是
的(1)重心;(2)垂心.若存在,求出对应的直线
的方程;若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7895aece9dd5a90424879ea9184037a2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.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/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d4a95e72206b23ff0b477d1ea9033a93.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
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7 . 复平面上动点
的轨迹为( ).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22580fce96922c7b5b41391fa0222168.png)
A.圆 | B.椭圆 | C.双曲线 | D.抛物线 |
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8 . 设经过定点
的直线
与抛物线
相交于
、
两点,若
为常数,则
的值为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d811e3371b79f5bd7a816acfa3aa4d0e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/745de5ef1fd897d16e37464172d5e8c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/537a720ac4508b5ba43279509bb02176.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
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9 . 若动点P(x,y)以等角速度
在单位圆上逆时针运动,则点
的运动方程是( ).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/074c228ffc7b1e306f8410afe7bc4b5c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ae582bd27ce0354d68ea35974d757b66.png)
A.以角速度![]() | B.以角速度![]() |
C.以角速度2![]() | D.以角速度2![]() |
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10 . 称联结椭圆上两不同点的线段为椭圆的一条弦.则椭圆
上长为
的弦的中点的轨迹方程为_______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c82e7d9f4f7ace849e09e9adcb786b7f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2967337e3fcb228dded64ab0c41a17e0.png)
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