名校
解题方法
1 . 已知椭圆
:
(
)的左、右焦点分别为![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5076289823db419f94e9c0c8f4aafd9.png)
,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3fb78c5f885034612c0e030b920143d.png)
,点P是
上一点,直线l:
(
).
(1)当
时,已知直线l恰经过
的右顶点A,求m的值;
(2)当
时,若P同时是l上一点且
,求a的值;
(3)设直线
交l于点Q,对每一个给定的
,任意满足
的实数a,都有
成立.则当m变化时,求
的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d7aea48c44781a844b5c19191f70f61.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a0c4c098615c6bc7e6dcf72e5b5201a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5076289823db419f94e9c0c8f4aafd9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/78f9b6b54e434819f50ae1349f534abd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3fb78c5f885034612c0e030b920143d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/44b7fcfde0f8949461eb9ea7af43ede8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ff99ce2491f3b979cbbfbc5e9f8d9b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5d670dcd0ce51abe372bc51a88ba1a7.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5af039cad52ca4e1f1e322277bc81afd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94a71a658362bd9faa329c3d9f6e6d2c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/377923225cb490323c85e4e5a12643e5.png)
(3)设直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0739793f234f8e86adc6177801ae7295.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5d670dcd0ce51abe372bc51a88ba1a7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/697cf7097b0133b2598b697fa8470d9b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ee83676fa76e3f845ab3d47ad678a8f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b213e4e1164ff957df10996756912a7c.png)
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2 . 函数
是我们最熟悉的函数之一,它是奇函数,且y轴和直线
是它的渐近线,在第一象限和第三象限存在图象,其图象实质是圆锥曲线中的双曲线.
的图象不仅是中心对称图形,而且还是轴对称图形,求其对称轴l的方程;
(2)若保持原点不动,长度单位不变,只改变坐标轴的方向的坐标系的变换,叫坐标系的旋转,简称转轴.
(i)请采用适当的变换方法,求函数
变换后所对应的双曲线标准方程;
(ii)已知函数
图象上任一点到平面内定点
的距离差的绝对值为定值,以线段
为直径的圆与
的图象一个交点为
,求
的面积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1305b9abebd7bef3171486df157286b3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d77f5191798242b7b9b88a75e17e4425.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1305b9abebd7bef3171486df157286b3.png)
(2)若保持原点不动,长度单位不变,只改变坐标轴的方向的坐标系的变换,叫坐标系的旋转,简称转轴.
(i)请采用适当的变换方法,求函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1305b9abebd7bef3171486df157286b3.png)
(ii)已知函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1305b9abebd7bef3171486df157286b3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e52586ca2a3b783bc8092415e2d4bf6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1305b9abebd7bef3171486df157286b3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2205cffebf8c4d5f81d15ed7b85c8936.png)
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|
55次组卷
|
2卷引用:山西省晋城市第一中学校2023-2024学年高二下学期第四次调研考试(5月)数学试题
3 . 已知定点
,动点N在直线
上,过点N作l的垂线,该垂线与NF的垂直平分线交于点T,记点T的轨迹为曲线C.
(1)求曲线C的方程;
(2)已知点P、A、B是曲线C上的点,且
.
(i)若点P的坐标为
,则动直线AB是否过定点?如果过定点,请求出定点坐标,反之,请说明理由;
(ii)若
,求
面积的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5421a28dc3675ae20190d6090793246e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d42502a5730e1930d77d7100d1e34707.png)
(1)求曲线C的方程;
(2)已知点P、A、B是曲线C上的点,且
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83640592853a53872d7af69c0cffc1bb.png)
(i)若点P的坐标为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fc7334402b0f5f606a82d027208e3e2c.png)
(ii)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d81529452eb2cb8578fd164db9e957ae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2205cffebf8c4d5f81d15ed7b85c8936.png)
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4 . 在棱长为2的正方体
中,点
满足
,则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/167c13c894f6d448cda166ac5f2a81e7.png)
A.当![]() ![]() ![]() ![]() |
B.任意![]() ![]() |
C.存在![]() ![]() ![]() ![]() |
D.当![]() ![]() ![]() |
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5 . 在棱长均为1的三棱柱
中,
,点
满足
,其中
,则下列说法一定正确的有( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42d3a82b8e587ee890467835bc4e854c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b11a79894447b31c2ca1163bc304871.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5d051546c15ec453fc53a6de2774b414.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b1ccdad989ebe7c7087be45cd0683a3.png)
A.当点![]() ![]() ![]() |
B.当![]() ![]() ![]() |
C.当点![]() ![]() ![]() |
D.当![]() ![]() ![]() ![]() |
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解题方法
6 . 已知椭圆
经过
和
,
分别为椭圆的左顶点、右顶点、上顶点.
(1)求椭圆
的标准方程;
(2)过
轴上点
(点
在椭圆
长轴上)作直线交椭圆
两点,且
,若
,求
点的坐标;
(3)过点
作直线交椭圆
于
点,交直线
于
,直线
于
轴相交于
,求证:
为定值,并求此定值.(其中
分别为直线
和直线l,
的斜率).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ed1bbd2face8220cbd14191212588aa4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2314fc922502d8fc6d13e4b9f2775b3a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6bd07fb0b22450681d23d5f0513d8e26.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce7af7c5df749c6fa9bbe87faa72c66d.png)
(1)求椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
(2)过
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3cf9b288c48c73463a2f214f02b6952a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d94e87c1ed4c9c0cd0f9e1f10f11a35f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/21791b26a0b406302709f9776dd9f28f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
(3)过点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63a253c7fdf589ee3dece13d5b5b5732.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/c4849d0599bbeaccc05eb0cece91ec97.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1eacfcd9c21fffe2820a00dd4f09ef25.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ce2790947716b1cfa9c5e7a65db4093.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf702adb116c1e46569eb7050d029f49.png)
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解题方法
7 . 已知双曲线E:
的左,右焦点分别为
,离心率为2,点B为
,直线
与圆
相切.
(1)求双曲线E方程;
(2)过
的直线l与双曲线E交于M,N两点,
①若
,求
的面积取值范围:
②若直线l的斜率为k,是否存在双曲线E上一点Q以及x轴上一点P,使四边形PMQN为菱形?若存在,求出
;若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3040b6c904477030ecf8ba20b2b18759.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d2a97987f71835f519b462f5b8f5957.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ad29801f799532ee7dda9658c30e373.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c3656055f5256cd06e636ea96e9f89c2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cbb7edd919d58c85c4813f43729eb5af.png)
(1)求双曲线E方程;
(2)过
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3fb78c5f885034612c0e030b920143d.png)
①若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/096a1ef5efc86957efc9fbf5a24c50a0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d4f02028a3847c4807c2d3cf0ea7efb8.png)
②若直线l的斜率为k,是否存在双曲线E上一点Q以及x轴上一点P,使四边形PMQN为菱形?若存在,求出
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b986e3613290b456532843d5ad4c6e67.png)
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8 . 已知动点M到点
的距离与到直线
的距离之比为
.
(1)求动点M的轨迹C的方程;
(2)过点
的直线与轨迹C交于P,Q两点,点P关于x轴对称的点为R,求
面积的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be92f0e0012a7696c78e3e00513edefd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e2977eea43a781e06d93e04a395a309.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1174142f3bba761585b6bc2653009b36.png)
(1)求动点M的轨迹C的方程;
(2)过点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e95e84f5c91c910aaafc5e74dbfbdf59.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2776dedca657ea11d05a420399ac123.png)
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9 . 在椭圆(双曲线)中,任意两条互相垂直的切线的交点都在同一个圆上,该圆的圆心是椭圆(双曲线)的中心,半径等于椭圆(双曲线)长半轴(实半轴)与短半轴(虚半轴)平方和(差)的算术平方根,则这个圆叫蒙日圆.已知椭圆
的蒙日圆的面积为
,该椭圆的上顶点和下顶点分别为
,且
,设过点
的直线
与椭圆
交于
两点(不与
两点重合)且直线
.
(1)求椭圆
的标准方程;
(2)证明:
的交点
的纵坐标为定值;
(3)求直线
围成的三角形面积的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9a200ca2c4af794f4d1c6a5443830b5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e28ff244e7fd9fcfec74c38151dafdf0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5425108c557f0f21474c045334f97d9c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4107b67c55c036c61bceacdf98b8e899.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/055d3ab1ca889ea898296dfc1abf725b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e9b0f5f44abbc6544a2f672b025b013.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/5425108c557f0f21474c045334f97d9c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c566bff96884d9e4214257a87ed0ef2f.png)
(1)求椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
(2)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a046cc6f537f0c967df678f4d5d10e74.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
(3)求直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac3e028a8b796c5b986ea00b297ef992.png)
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解题方法
10 . 已知点P在圆
上,过点P作x轴的垂线段
,D为垂足,Q为线段
的中点,当点P在圆上运动时,点Q的轨迹为Γ.
(1)求Γ的方程;
(2)设
,
,过点
作直线与Γ交于不同的两点M,N(异于A,B),直线
,
的交点为G.
(ⅰ)证明:点G在一条平行于x轴的直线上;
(ⅱ)设直线
,
交点为H,试问:
与
的面积之积是否为定值?若是,求出该定值;若不是,说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a5f5d967ad135991b6075ee45df55643.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0629ce42392a7fe9be21d25c39c3e64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0629ce42392a7fe9be21d25c39c3e64.png)
(1)求Γ的方程;
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/383f12cb70ca55eba4ff012771dbfa9d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3544997cc034ed882c0d0a3bdbf5f957.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c0e1ba8ef888dfe9a639dddd38d6d603.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e69d2b798744645af88a4fa411344a83.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f50b3ae183997b707d16eb4e7f6712fa.png)
(ⅰ)证明:点G在一条平行于x轴的直线上;
(ⅱ)设直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d50703c46b6153945d718b198f03b4b5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7785afeeaf274892253d04b4f693b367.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2b3b785ebbf5889849e872f461669f71.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f584dfa75ec20e4cba4216998b454dd.png)
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