1 . 设抛物线
与
相切,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/380bbacf854e30e2e747fc286d2b9997.png)
______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/344ccbf79da6ad7e3709d6fa72efb756.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c0d45d146c1df586e69bad35394a85bd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/380bbacf854e30e2e747fc286d2b9997.png)
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解题方法
2 . 求函数
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c33d8c876f71b58fa25a3c4f20bae4c.png)
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3 . 材料:对抛物线
,定义:点
叫做该抛物线的焦点,直线
叫做该抛物线的准线,且该抛物线上任意一点到焦点的距离与它到准线的距离相等.运用上述材料解决如下问题:
如图,已知抛物线
的图象与
轴交于
两点,且过点
,
的解析式和点A的坐标;
(2)若将抛物线
的图象向左平移2个单位,再向上平移1个单位得抛物线
的图象.
①设
为抛物线
位于第一象限内图象上的任意一点,
轴于点
,求
的最小值;
②若过抛物线
的焦点
作直线
,与抛物线
交于
两点,再过
两点分别作抛物线的切线,两条切线交于点
,求
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5fa0e961df94bc4cc1c8407ed65f2557.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a415f2567685ccf84257ac443f779a5c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/423f5e7781b94f96ff4aac23cba2964d.png)
如图,已知抛物线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f61d802cc79e61fc37a79d4bae4e98c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f9cda46e575acf64941ea0964b89ee99.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/980240503eeea96e431bb5540ba60879.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)若将抛物线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4c8a9c4957431681ddfc77895a88508.png)
①设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4c8a9c4957431681ddfc77895a88508.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/52cbeb9b1c1d637b903cf3e5c7f730f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f25f0610430d94cc16ca340e3ac5757.png)
②若过抛物线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4c8a9c4957431681ddfc77895a88508.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4c8a9c4957431681ddfc77895a88508.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/44091f07aab1b00f3c5abb14d98fed25.png)
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4 . 已知抛物线
经过点
.设点
,请在抛物线的对称轴上确定一点D,使得
的值最大,则D点的坐标为_______
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c44f12ece8a44b5943afbec664e7d40e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e95e84f5c91c910aaafc5e74dbfbdf59.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f8389dc56f2a2e736b50307d04d641b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/004dcc4b1ac504ce1c70d11d4b7bce04.png)
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5 . 如图,过抛物线
上一点P作切线l,过O点作l的垂线交
于点Q,
,则
的面积是_________ .
![](https://img.xkw.com/dksih/QBM/2023/2/7/3169696117899264/3169822596808704/STEM/803c4c244ccb41debe2c503c5c4c2f06.png?resizew=335)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b072ff6d1b83232bebd7d4709ffba4ef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fb26d84907c923278ac4626a9d58947.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/592c77c484e727c95cc104c2c291153e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf2e3b2eb83eb2fe1119e78f4e6e8eaa.png)
![](https://img.xkw.com/dksih/QBM/2023/2/7/3169696117899264/3169822596808704/STEM/803c4c244ccb41debe2c503c5c4c2f06.png?resizew=335)
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6 . 已知椭圆
的焦点坐标为
,若直线l与椭圆
相切,点
到直线l的距离分别为
.证明:
(1)
.
(2)![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cfe6c4c8fffd4c5b21f1d11b26963fcf.png)
(3)
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c0c118b51ab426bc1c1b56179094f146.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/475a3fdc7b27d5ffa8c864dc478903d4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b94469fd19f40116e2dec334919d6586.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d2a97987f71835f519b462f5b8f5957.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf2f1c1409a06278e847e6b573cef254.png)
(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58fc0e502b29162d63e46d46220cce8e.png)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cfe6c4c8fffd4c5b21f1d11b26963fcf.png)
(3)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8acbfb924b9d207b057fc6593ffeb9d9.png)
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7 . 已知函数
在定义域
上严格单调递增.
(1)证明:函数
至多存在一个零点.
(2)若函数
存在零点
,证明:存在
,使得
对于任意
恒成立的充分必要条件是
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b7a19bbab2270fc8e694527e801556cf.png)
(1)证明:函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
(2)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b752f0f189e5d8666daea73e145dff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22dd8b3dc4c609bab82d356a5cc2208d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/07cbf6c4e062852a2bdee01b9992713b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2ad65ff0ab0a64844ab78062df2e6ce7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13747fa9a42164caebe2c9b7c5d06d3a.png)
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8 . 已知函数
,其中
.
(1)证明:
有唯一零点.
(2)设
为函数
的零点,证明:
①
;
②
.
参考数据:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/de7573ad83c7fdbdd21e7ad648dabcff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9cb4dfdd6761f67b2586ebd408bc15f6.png)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b752f0f189e5d8666daea73e145dff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/450efeff8a78e650d17cfca8d411c4f7.png)
②
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c04c89a680ff7556cee31f24e9b77c09.png)
参考数据:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5361b129a7ae086d392b48ec52417bc0.png)
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解题方法
9 . 如图,已知抛物线
,点A在抛物线上,且在第一象限,以点A为切点作抛物线的切线l交x轴于点B,过点B作垂直于l的直线
交抛物线于C,D两点,其中点C在第一象限,设
与y轴交于点K.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/3/22/34e3d803-c747-4660-a65a-42f14b74a55c.png?resizew=248)
(1)若点A的横坐标为2,求切线l的方程.
(2)连结
,记
的面积分别为
,求
的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e42102c1c07562853219ca5918803a27.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13dea1bd3d0dd84b8b6f6ff634c5600c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13dea1bd3d0dd84b8b6f6ff634c5600c.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/3/22/34e3d803-c747-4660-a65a-42f14b74a55c.png?resizew=248)
(1)若点A的横坐标为2,求切线l的方程.
(2)连结
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb1bf0ea1c9191c3c88acf28bff2c4e4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/27c68e39785ea98cde06005ff449b3a7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fcf4e20ea341827ce5f9552daee39462.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/958608935f2e6d69c15cb1f130428245.png)
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10 . 已知双曲线
左右焦点分别为
,过点
作与一条渐近线垂直的直线l,且l与双曲线的左右两支分别交于M,N两点,若
,则该双曲线的渐近线方程为__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/528a243eb93f63c1e126409be1fb3fb5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d2a97987f71835f519b462f5b8f5957.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5076289823db419f94e9c0c8f4aafd9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/859c116c6d7695fd354a6d6df63c6800.png)
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2023-02-07更新
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562次组卷
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3卷引用:2021年清华大学文科营暨工科营(冬令营)数学试题