解题方法
1 . 已知双曲线C:
,B是右顶点,F是右焦点,点A在x轴的正半轴上,且
、
和
成等比数列,过点F作双曲线C在第一、三象限的渐近线的垂线l,垂足为点P.
(1)求证:
.
(2)若l与双曲线C的左右两支分别相交于点D、E,求双曲线的离心率e的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3040b6c904477030ecf8ba20b2b18759.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e578668ad5d0ad2b240ac6906c66f2c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4303745ad1b7549611d2e86ec3ea8e20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df576259f315a81398ce80af7989dddd.png)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1f4b58e605fb54f91dfab690996955a.png)
(2)若l与双曲线C的左右两支分别相交于点D、E,求双曲线的离心率e的取值范围.
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解题方法
2 . 平面向量
,若存在整数
使
,且
,试求
的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22a4521f0658e021ad28318d8db27426.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09737e117adb481fd3c4affdf38ff45f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/245d4ac7e6c91f3ea4028bdeb2fb1a9b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8f92ffd4c85efa2d4a1692459a06cbbb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/44bdb686910f213670f2c6f9e46c1f62.png)
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3 . 已知
、
是椭圆
上两动点,
为原点,定点
,向量
,
在向量
方向上的投影分别为
,
,且
,动点
满足
.
(1)求点
的轨迹
的方程;
(2)记点
,
,求证:无论动点
在轨迹
上如何运动,
恒为一个常数.
![](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/f578c87d528cc32b2acb0e913391c26e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/33a62c9695f5a1691c5fe8724fa764b7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a4f605ec0729ce6d72237ad662a06862.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6fc9656d8286c4d6fa309d6ae347c89e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/21909dd065ccc349a2cbfd4c3cf4976b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca70897450a4208d95018c8fac6138ba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/98094053649f93909ac555de3694ad52.png)
(1)求点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
(2)记点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ae140e4db2c5563e5f902fcbebaac262.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c95963e8e4dcc511f0d86b1853ddcdc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b894af0328f96959e0ef1c19ff50cdd.png)
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4 . 在直角坐标系xOy上有两点
、
,给定三个条件:①
,②
,③
.请从上述三个条件中选出两个分别填在下列空白处(只填代号),使其构成一个真命题:当且仅当___________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3539590ba3e226d4d6480f8460a6247d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8f12e628b7b0bd5c387c4c46cef47c17.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1f52965de7291b547f4c325d3626bd9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5dc261508f42e75323971f0b8282b0ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d8e8c4db99a1f1ef1032002a070d65a0.png)
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解题方法
5 . 已知
,
,且存在实数
和
,使得
,
,且
,求
的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f9798750055cc3be811b36accdec975d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/28e135eb6709b5d964f89ce0e391062a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4749a22a7ae411cf646c64fdbf08e1ba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6f4b8e9bf365f34a58866b4fdf2096e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/654c8d7389fbd7fffd7e00071f814016.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/69a17fbeeb7bc8ff39e2b0403042f026.png)
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解题方法
6 . 已知直线
与圆
交于
两点,则
平行于向量( ).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a1d1e97d1268ad01b4a91ff0e68c6a5a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f240cccaf24af8a796abb95cb42be52e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/277cf6fdff3911e963cd83cfbb4b8a76.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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解题方法
7 . 已知点集
,其中
,点
为
与
轴的公共点,等差数列
的公差为1.
(1)求数列
的通项公式;
(2)若
,数列
的前
项和
满足
对任意的
都成立,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/95d8ce02203ac6e6acf2349e18856c96.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/604ba8370e3c48e55cc0ac1cf6acbdd0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/243ed164b1170d010d2c9300a4ef470e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c88d9142df6ba8e43c1a93bd04a1362.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(1)求数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/62e567d7e9761951a266953c8d5042ac.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ff55ecab1e81c7c604bc43eda6e74366.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/67ef8539a7a09303a95b4e79fb9949fc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/641c660da352120c5705d25632acd5c3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a37a59558292ad6b3d0978bfd7484990.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
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解题方法
8 . 已知单位向量
,则函数
的最大值为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/976db62a864859d5659e17b7e1bc5687.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ea5a7cac866891616317966e65427988.png)
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9 . 已知函数
,其中![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61228bf9a9e5bfa26e7e3eaeb48cf909.png)
.
(1)若函数
的图象过点
、
,求函数
的解析式;
(2)如图,点M、N是函数
的图象在
轴两侧与
轴的两个相邻交点,函数图象上一点
满足
,求函数
的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/064b3e9a9a7551f282b530877228cb8d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61228bf9a9e5bfa26e7e3eaeb48cf909.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b61df454f1485f9d448402586945b868.png)
(1)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/023c93917c96d00ea4ba399fa441a685.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51b97fc3360cdfc250e7eb1bf24bcb64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
(2)如图,点M、N是函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9efe6112e30f859a68339fb6793bded8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dc6aa5d475a3da2ee13c1a407bc6e0b3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
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10 . 在椭圆
中,A为长轴的一个端点,B为短轴的一个端点,
为两个焦点.若
,求
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b454cdb97c408300b50d945f002c2cb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d2a97987f71835f519b462f5b8f5957.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad36e15240bd3499dc70befc4a8f253c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ccca27ddae404d05ab937afe24ff415.png)
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