名校
解题方法
1 . 设函数
的函数值表示不超过x的最大整数,则在同一个直角坐标系中,函数
的图象与圆
(
)的公共点个数可以是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1550a97c21c1d71c9e95dde569668be0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6bd56acbaed7465cc7b453fdccc0aef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0fff6e7e2b9f2b68b1647f6350b98dc8.png)
A.1个 | B.2个 | C.3个 | D.4个 |
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2024-05-25更新
|
2123次组卷
|
3卷引用:2024届广东省深圳市二模数学试题
2 . 平面直角坐标系中,圆M经过点
,
,
.
(1)求圆M的标准方程;
(2)设
,过点D作直线
,交圆M于PQ两点,PQ不在y轴上.
①过点D作与直线
垂直的直线
,交圆M于EF两点,记四边形
的面积为S,求S的最大值;
②设直线OP,BQ相交于点N,试证明点N在定直线上,求出该直线方程.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b42d63fab09f90a520a52c87ab3eb39.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab400b66fae56036c1e66c1e0a8fb899.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a9e157d75836912739cfa68013a48f07.png)
(1)求圆M的标准方程;
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d29d7fac4f150927e672507bdb26285.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e9b0f5f44abbc6544a2f672b025b013.png)
①过点D作与直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e9b0f5f44abbc6544a2f672b025b013.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f6f17bc385bafb37e8f964e5eb99cd0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ebcad0585886e2d7bac28a0e292a1d37.png)
②设直线OP,BQ相交于点N,试证明点N在定直线上,求出该直线方程.
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3 . 已知圆
过点
,圆心在直线
上,截
轴弦长为
.
(1)求圆
的方程;
(2)若圆
半径小于
,点
在该圆上运动,点
,记
为过
、
两点的弦的中点,求
的轨迹方程;
(3)在(2)的条件下,若直线
与直线
交于点
,证明:
恒为定值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0778813665f307942db9769077032f97.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ab466aedd6e176088d8dee7bc3e3aaa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b91d650c2fc1a741fabdb333b09aeb6.png)
(1)求圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)若圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d07ae0b4264da6a8812454ffd2f20d94.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1e5eea5f7f98ca8632358b7e49ceb25.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
(3)在(2)的条件下,若直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/33400b08941f4f9c0ed12b0e0cdff822.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/975d6f7561fc2bce5f4c94a33b3c0ccf.png)
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解题方法
4 . 若圆
与圆
外切,则
的取值范围是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a4f696f360654d17c0b26f89bfbd76ff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/852f8adbd4f6938db6f80880f31b70ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd34eda66afc688494af1f8ca5704667.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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5 . 如图,点C是以AB为直径的圆O上的一个动点,点Q是以AB为直径的圆O的下半个圆(包括A,B两点)上的一个动点,
,则
的最小值为___________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9a547a90936ff4635dd6d5f303e12986.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32979e540e6770ffc0f6417395ec5c74.png)
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解题方法
6 . 已知点
为圆
的两条切线,切点分别为
,则下列说法正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a827fd8990d58c1f979be7b84a438b9c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/248e009d63256e6a893b037fd67d2b78.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7789a500686c7a73770404ead6af0590.png)
A.圆![]() ![]() ![]() |
B.切线![]() |
C.直线![]() ![]() |
D.![]() |
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解题方法
7 . 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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名校
解题方法
8 . 某中学开展“劳动创造美好生活”的劳动主题教育活动,展示劳动实践成果并进行评比,某学生设计的一款如图所示的“心形”工艺品获得了“十佳创意奖”,该“心形”由上、下两部分组成,并用矩形框
虚线
进行镶嵌,上部分是两个半径都为
的半圆,
分别为其直径,且
,下部分是一个“半椭圆”,并把椭圆的离心率叫做“心形”的离心率.
,则当该矩形框面积最大时,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/258e40e3883271c3580c1d3c805dcac6.png)
__________ ;
(2)若
,图中阴影区域的面积为
,则该“心形”的离心率为__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fd995178601c2ad7b40f973d268c7bb7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04582116cd765fcc5a52f44279ad6c94.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11bc05f41215f9894e11d1df0465751a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2d578394cd8e4d7a705599269c512960.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ce072bcd423b56c827e40f0b731de85.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8da45c443af7994a26ffa9d8894e7262.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/258e40e3883271c3580c1d3c805dcac6.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7551011cfb75b26f35b07d6617c6a18b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df9eb8f0efaf6fe8a69150fcdf3c83fc.png)
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2024-03-03更新
|
282次组卷
|
3卷引用:北京市第二中学2023-2024学年高二上学期12月第二学段考试数学试卷
9 . 如图,一个酒杯的内壁的轴截面是抛物线的一部分,杯口宽
,杯深
,称为抛物线酒杯. 在杯内放入一个小的玻璃球,要使球触及酒杯底部,则玻璃球的半径的最大值为_____________
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4de7cb50f975c3d9c82d1e71ccf8020.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/07fb986e1a656aac2f6982351c4324ae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72026e70f45cea6a7c49881592e73a02.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/2/24/a5f11f9e-4462-4209-a262-891a86ec9c00.png?resizew=241)
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10 . 已知双曲线
的离心率为
,且左焦点
到渐近线的距离为
.过
作直线
分别交双曲线
于
和
,且线段
的中点分别为
,
.
(1)求双曲线
的标准方程;
(2)若直线
斜率的乘积为
,试探究:是否存在定圆
,使得直线
被圆
截得的弦长恒为4?若存在,请求出圆
的标准方程;若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2ef66f4832adc43902055a7e6d258037.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/839c7616cd0d90265f4b2c9c021254fe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7ffe8515ff6183c1c7775dc6f94bdb8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/44434b647ec546fe787e2164e0be6cd2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39acab3cfb59bfc9591371721ab01d93.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b5c4cd264c97c1f261229925cc5a6761.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
(1)求双曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
(2)若直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/44434b647ec546fe787e2164e0be6cd2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/feab93da829a3c07a98ed41b4707967e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
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