名校
解题方法
1 . 已知直线
与直线
.
(1)若两条直线垂直,求实数a的值;
(2)若两条直线的夹角为
,求实数a的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6229fbdb2c3609beb75bf79952bcdc1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4de869ae6b6dc5b79fcae3de540b30bf.png)
(1)若两条直线垂直,求实数a的值;
(2)若两条直线的夹角为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/beb214b260f8cd16ece2898eecef2866.png)
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名校
解题方法
2 . 已知直线l的方程为
.根据以下条件,求直线m的方程.
(1)若直线m过点
,且直线m与直线l的夹角为
,求直线m的方程;
(2)若直线m的倾斜角为
,将直线m绕其上一点P逆时针旋转α后得到直线n,直线n与y轴交于点
,将直线n绕点P逆时针旋转
后得到直线l,求直线m的方程.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3aa3adcb154f6144903d456289ecb0f.png)
(1)若直线m过点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fad20e2bc6576fc461419f8f138d26e7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5f700dfadb2d40a13c4797c368acc72d.png)
(2)若直线m的倾斜角为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2f9ac7f5c2a744c676e60413cca43236.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1803dc3c76fd2b51696647aa18602412.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d6d5d1dac32b2b62e28fcc24e92d0d00.png)
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3 . 已知点
和非零实数
,若两条不同的直线
均过点
,且斜率之积为
,则称直线
是一组“
共轭线对”,如直线
是一组“
共轭线对”,其中O是坐标原点.规定相交直线所成的锐角或直角为两条相交直线的夹角.
(1)已知
是一组“
共轭线对”,求
的夹角的最小值;
(2)已知点
,直线
是“
共轭线对”,当
的斜率变化时,求原点O到直线
的距离之积的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/44434b647ec546fe787e2164e0be6cd2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/44434b647ec546fe787e2164e0be6cd2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/280f1e7d3e287061e928c064f2197e11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a2ccb7065696491da7cb857d0f2eb2e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a5b0c1277c5cfe00b24d32ec4ec1630.png)
(1)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/44434b647ec546fe787e2164e0be6cd2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c903f3bfb8ecbbf0cd87e20c928f732.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/44434b647ec546fe787e2164e0be6cd2.png)
(2)已知点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8385ba2fd0cc7cf87e5e7ff302718e13.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/44434b647ec546fe787e2164e0be6cd2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6224cdba8faffb7445f0a483969e6e37.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e9b0f5f44abbc6544a2f672b025b013.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/44434b647ec546fe787e2164e0be6cd2.png)
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23-24高二上·上海·课后作业
4 . 根据下列方程,判定直线
与
的位置关系,若相交,求出夹角.
(1)
,
;
(2)
,
;
(3)
,
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e9b0f5f44abbc6544a2f672b025b013.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f6f17bc385bafb37e8f964e5eb99cd0.png)
(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c15d419ac2b9832ef6fb704d98790487.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2db191a381bcc131aad1216844cfe0fc.png)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b00b8eb5fee88a104b321d1c2d5c5b59.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0031a6f0727e81ec9966c95fb2a5eb12.png)
(3)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83342b9d257e062198077c2897e77dc9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5810390abd0a156828293ad5885442d8.png)
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23-24高二上·上海·课后作业
5 . 求直线
与
的夹角.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a537e5c7208f0a82d242a0943d0fda.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ee15ce4a6000b524447bbe8d0829cc35.png)
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23-24高二上·上海·课后作业
解题方法
6 . 若直线
与直线
的夹角为
,求
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a72347f2c8fb45e1ba2c1462c2005ce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e8a3365e99f926b1dafa901ab232152.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15615de1a6df206dbd081251f676578e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
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23-24高二上·上海·课后作业
7 . 给定直线
与
.求证:
(1)
;
(2)如果直线
与
不互相垂直,那么它们的夹角
满足
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fa4211831377655be51acaf182280085.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/266e3689cc366d5f0a7718ce9c20e818.png)
(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/44b29b23cdf9eb410bf9360ad48aeebd.png)
(2)如果直线
![](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/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e3a755d4f2c8d8eb39b023462e3838c0.png)
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解题方法
8 . 已知直线
经过点
且与
和
分别交于
两点,若
,求直线
的方程.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c3c9708ef0dc6d6f5dcf6596d3e4f6e5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c78e1c4c7e0534809d2190e9df630448.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ad8df4ddf82673e99b4ec811eceffe3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ff1c07d3ab5f594be5fffe13ebaaccb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
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名校
9 . 已知拋物线
和
,其中
.
与
在第一象限内的交点为
.
与
在点
处的切线分别为
和
,定义
和
的夹角为曲线
的夹角.
(1)若
的夹角为
,
,求
的值;
(2)若直线
既是
也是
的切线,切点分别为
,当
为直角三角形时,求出相应
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39eebc6a2c5bab554e6fa43c15d60728.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d1138c04fc3a0e1c217db0d432e4aff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94440d3e4c073f94f2b266ff99d50e74.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/dad2a36927223bd70f426ba06aea4b45.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/dad2a36927223bd70f426ba06aea4b45.png)
![](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/2e9b0f5f44abbc6544a2f672b025b013.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f6f17bc385bafb37e8f964e5eb99cd0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/21e9feabc99f62ee569b460e61526e2e.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/21e9feabc99f62ee569b460e61526e2e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/950ebd2fdaf9e6f461c3ac713f3fea52.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(2)若直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c9fce9427c9b17e4d3cda0c3ff3e2e14.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/d8ca00309261a540934d9b3ed9ba05b6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c28abb154f41e1ca9816c9c9c2433ca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
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名校
解题方法
10 . 已知直线
经过两条直线![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d71187e41a7278c6f1893852944bb782.png)
与![](https://staticzujuan.xkw.com/quesimg/Upload/formula/56d4b99f920ce922c4709077a6662446.png)
的交点
且与直线
的夹角为
,求直线
的方程.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d71187e41a7278c6f1893852944bb782.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0985b973395bcd371cd1e26d3fcd1c36.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/56d4b99f920ce922c4709077a6662446.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a1fd72f36c81a55cafa92739ee87f743.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/86ca0ea5118346386bd9eb489cb8dda2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/037fb348109dc2063a268b10eb925a57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
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