解题方法
1 . 数学中有许多形状优美的曲线.例如曲线
:
,当
时,是我们熟知的圆;当
时,是形状如“四角星”的曲线,称为星形线,则下列关于曲线
的结论正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/21b436687ddf3870afec4dc85b792b1c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc2d3df37e73a8abea815f37dbb3fff5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/262129c4595a1e460f6be9ba0e0ed1a0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
A.对任意正实数![]() ![]() |
B.存在无数个正实数![]() ![]() |
C.星形线围成的封闭图形的面积大于2 |
D.星形线与圆![]() |
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解题方法
2 . 已知离心率为
的双曲线
:
过椭圆
:
的左,右顶点A,B.
(1)求双曲线
的方程;
(2)
是双曲线
上一点,直线AP,BP与椭圆
分别交于D,E,设直线DE与x轴交于
,且
,记
与
的外接圆的面积分别为
,
,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/047616f1d1d39bf6c3cd07cf63ef5b80.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3040b6c904477030ecf8ba20b2b18759.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6cae00bdc6f8b564b6b15b32572c848b.png)
(1)求双曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/89cddda9100666fa42033cc4f2688fb9.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/16862304173044aef35e32375095ff5a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8245026524d605717d6a5fbe30abd95a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df7f9f7952c4be18cb73393047c2ad75.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab2a2834d80ff574e79eae8ca8d4e94f.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/142b9d242ef0c6b807d1257f2638b37b.png)
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2024-03-01更新
|
1365次组卷
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4卷引用:浙江省名校协作体2023-2024学年高二下学期2月月考数学试题
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解题方法
3 . 设F为双曲线
的右焦点,O为坐标原点.若圆
交C的右支于A,B两点,则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e2872b941d9e4748a823d37f14335fd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f8cb0e4f2335a997c29a3e578a21f0b.png)
A.C的焦距为![]() | B.![]() |
C.![]() | D.![]() |
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2024-02-28更新
|
682次组卷
|
3卷引用:重庆市第一中学校2023-2024学年高二下学期开学考试数学试题
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4 . 已知等轴双曲线
过定点
,直线
与双曲线
交于
两点,记
,且
.
(1)求等轴双曲线
的标准方程;
(2)证明:直线
过定点.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79238e14cbbc951421c74471f9df8692.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/f6bce3d91ca23b86d8c6625f2632e437.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c21fdf33b6370afbdd5e1b3fee34cc1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/445bf1af0740c861e152d076f20f1ab2.png)
(1)求等轴双曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)证明:直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
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5 . 过点
作直线l与双曲线C:
交于A,B两点,P是双曲线C的左顶点,直线
与y轴分别交于
.
(1)求直线l斜率的取值范围;
(2)求证:线段
的中点M为定点,并求出点M的坐标.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ef7883f9b72effd33bda22f803789ff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b2724acdc1cbf15077b0b0295ab21a3d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/790ef3382b1c731f2885eecfd92c2a86.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d8ca00309261a540934d9b3ed9ba05b6.png)
(1)求直线l斜率的取值范围;
(2)求证:线段
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5d7b816eca15d4b7d060013df53edd53.png)
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6 . 已知椭圆
:
(
).
(1)若椭圆
的焦距为6,求
的值;
(2)设
,若椭圆
上两点M,N满足
,求点N横坐标取最大值时
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb6e9badea4fda62bdbfa1ae44107fd2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3be362dec96173f246ff747264007817.png)
(1)若椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22840186db0afc0e2b2e8915ce79b998.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b2d5450b9ec0e906919efbb87e5a9e8d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
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7 . 已知点
,直线
上有且仅有一点
满足
,则
可能是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df7e26bd228679011198cf33e7b8bcb5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a646ac0df52f1d728c8fe4659e6675c0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e1bd05b71aa0f9f979b0d4cb543ae3c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
A.0 | B.-1 | C.![]() | D.![]() |
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8 . 已知双曲线
,直线
与C交于A,B两点,点P是C上异于A,B的一点,则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/82c4ed5a210e7f97bf194e37fd38ad8f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c3016baf1a9ce777f16ea9ce469b2510.png)
A.C的焦点到其渐近线的距离为![]() |
B.直线![]() ![]() |
C.过C的一个焦点作弦长为4的直线只有1条 |
D.点P到两条渐近线的距离之积为![]() |
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9 . 曲线
上动点
与
构成
,若
,则实数
的取值范围为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b67d93773ed34cfe79fd47025c9dac1c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51296b06e4f0ac74bea949ce2d9faa6a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1ed4c4e8edbd179f3fc38a6653f18c1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03ac378e1dd219184c4a0d722acb10f3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
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2024-02-23更新
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974次组卷
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2卷引用:浙江省舟山市2023-2024学年高二上学期期末检测数学试题
10 . 已知
为平面直角坐标系
上的动点,记其轨迹为曲线
.
(1)请从以下两个条件中选择一个,求对应曲线
的方程.
①已知点
,直线
,动点
到点
的距离与到直线
的距离之比为
:
②已知点
是圆
上的任意一点,点
为圆
的圆心,点
与点
关于原点对称,线段
的垂直平分线与线段
交于点
;
注:如果选择多个条件分别作答,则按第一个解答计分.
(2)延长
至
,使
,点
的轨迹为曲线
,过点
的直线
交曲线
于
两点,求
面积的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ee31829d0d4d5f779a957d7df8058ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(1)请从以下两个条件中选择一个,求对应曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
①已知点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/53e4a3aea4b4b0d87c014030a6ff9027.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6261e2ccfee2e4df6e1f2021de5da27e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f89eef3148f2d4d09379767b4af69132.png)
②已知点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b012d1310aab427e9deb746015a30d0c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f8e55590555905eb4f57889bbd16e39a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cad0a19415e796564f30906f2e7dbf76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6aa2b5e09f8ec785c59900a529390a02.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
注:如果选择多个条件分别作答,则按第一个解答计分.
(2)延长
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/abd13974aebe38eb2a1d744a01ea5aa5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c56b32ddc6969b161fa2a1b3ae73d33e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.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/7789a500686c7a73770404ead6af0590.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ed283a253b61df01f2a1cdc0cd8003f3.png)
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