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 . 设锐角
满足
.令
,且令
,
、
为任意正实数.证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a44c8eb08b253c1c92163ae04b7f45f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f2adc8b386a6de97e6d04c7057c13803.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9cb7d1d8d3e710a664de5d312b733bae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f3e798612e0276022aaedf44826c2481.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10b4f24f8545981f4fda173d6a90caf2.png)
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3 . 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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4 . 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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5 . 已知同心圆
与椭圆
有公共点.求半径R的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7d3311792d8021cce363cea15edae9fd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4a5f4b244b3b0799cfb1994364036eb1.png)
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6 . 设椭圆
,
,过原点
引射线分别与椭圆
、
交于点
、
,
为线段
上一点.
(1)求证:
、
、
成等比数列的充要条件点
的轨迹方程为
.
(2)试利用合情推理,将(1)的结论类比到双曲线得出相应的正确结论(不要求证明).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/99cd361ce118bca96a731b241a9c587d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b0b48878a72078d715357cd5301d5c54.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.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/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c388f1f7160d3397610633dfde015fa0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f645d4b09fba53f971172cd2602c691.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85a82596d3e0eea2ea14448e3ede29c2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e918b255600da996c1ea631f4df7759.png)
(2)试利用合情推理,将(1)的结论类比到双曲线得出相应的正确结论(不要求证明).
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7 . 已知椭圆
,过中心
作两条互相垂直的弦
和
,设点
、
的离心角分别为
、
.求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7dd54b9df3402ad91e2d34c40efe0c7a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](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/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/25f64fa38725c136504f723019a18dc5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e93fa313adc4ac7608ba9449fd755212.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/992e1c215af2ab79ebf756538c89e2d8.png)
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8 . 已知椭圆
与y轴正向的交点为B,求以点B为直角顶点的椭圆内接等腰直角三角形的个数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7dd54b9df3402ad91e2d34c40efe0c7a.png)
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9 . 已知椭圆
,过中心
作互相垂直的两条弦
、
.设点
、
的离心角分别为
、
,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b5c9bebea391a1f9956dfcca98d9d1f1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](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/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/25f64fa38725c136504f723019a18dc5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e93fa313adc4ac7608ba9449fd755212.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1f7917aa62af366b02d642ee4f00139.png)
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10 . 以
的边
为实轴的双曲线交此三角形的另两边
、
的延长线于
、
.过点
、
分别作该双曲线的切线,两切线交于点
.求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0faed94a64b2dcfc6801b4fca0f16675.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.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/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bfbad7ad1465d1c4c177e3321e6ed12a.png)
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