名校
解题方法
1 . 在平面直角坐标系中,过直线
上任一点
作该直线的垂线
,
,线段
的中垂线与直线
交于点
.
(1)当
在直线
上运动时,求点
的轨迹
的方程;
(2)过
向圆
引两条切线,与轨迹
的另一个交点分别为
,
.
(i)证明:直线
与圆
也相切;
(ii)求
周长的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/236a066876764d090523afe0ea734a21.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/892909e49156f7dcc0650fcd65243877.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/50154acc6ad77c6c777fffe3a08afb59.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2273ae1ee99cec9c1304323bc9ebf75f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/892909e49156f7dcc0650fcd65243877.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)过
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f817fa2d2a8c4e38191900ed7730c879.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
(i)证明:直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
(ii)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2205cffebf8c4d5f81d15ed7b85c8936.png)
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2 . 已知双曲线
的上、下顶点分别为
.
(1)若直线
与
交于
两点,记直线
与
的斜率分别为
,求
的值;
(2)过
上一点
作抛物线
的切线
和
,切点分别为
,证明:直线
与圆
相切.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a9ff07459dc1549f2a66429eca9829e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
(1)若直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94a8d6991873e79b298984a95b8954b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7789a500686c7a73770404ead6af0590.png)
![](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/90963760acac7bfad3ae03088c6c80b0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/67351fe10fcfc3f9072eec4c60bfaaa5.png)
(2)过
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38fda21944581898ccb13c7d4641b7f9.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/f6bce3d91ca23b86d8c6625f2632e437.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a5f1641947153c80b987320885a2b57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f240cccaf24af8a796abb95cb42be52e.png)
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2024高三·全国·专题练习
3 . 已知O为坐标原点,P,Q是双曲线
上的两个动点.
(1)若点P,Q在双曲线E的右支上且直线PQ的斜率为2,点T在双曲线E的左支上且
,
,求双曲线E的渐近线方程;
(2)若
,
,
成等比数列,
,证明直线PQ与定圆相切.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96c4088276acdbede4781b2ebc466366.png)
(1)若点P,Q在双曲线E的右支上且直线PQ的斜率为2,点T在双曲线E的左支上且
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23be75b7327cdbd25f37f91146b2217e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/84f3cf1c5892c448e6e617288d7a0454.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13502d46b8563c54c09b29b20b3006a4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18f0281e6bbdbe08beeccb55adf84536.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20d6fc9b90f370fbb27552876b650f8f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc7df99fe6438442a9453fc0c57fb703.png)
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4 . 已知抛物线
的准线
交
轴于
,过
作斜率为
的直线
交
于
,过
作斜率为
的直线
交
于
.
(1)若抛物线的焦点
,判断直线
与以
为直径的圆的位置关系,并证明;
(2)若
三点共线,
①证明:
为定值;
②求直线
与
夹角
的余弦值的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f32e577ae1f4449efbd64c1199efe7a3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a436db19eb954d31075d5398f1b92ecd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6defc43285a40f7ccb74c1cc04265eba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e9b0f5f44abbc6544a2f672b025b013.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39acab3cfb59bfc9591371721ab01d93.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f96c1f9f575ea0fd1487c9f4bb7745af.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/423b7ae39db552e60ee8b1d27312306f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f6f17bc385bafb37e8f964e5eb99cd0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/52041559f8fee18bfa3e2e2ac07c3bfa.png)
(1)若抛物线的焦点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b0d4e207a27fc83c2d7bf247841a788.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31e55e398e8520d8a36fb5a625a085b8.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/939658de45d1e76de0a85e1c50d29e12.png)
①证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/29212e455dd6df5379ae67f379a32158.png)
②求直线
![](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/c24095e409b025db711f14be783a406c.png)
您最近一年使用:0次
2023-12-22更新
|
575次组卷
|
2卷引用:重庆市沙坪坝区重庆一中2024届高三上学期12月月考数学试题
名校
5 . 已知点
和直线
,则点
到直线
的距离证明可用公式
计算.
例如:求点
到直线
的距离.
解:
直线
,其中
,
.
点
到直线
的距离为:
.
根据以上材料,解答下列问题:
(1)求点
到直线
的距离;
(2)已知⊙
的圆心
坐标为
,半径
为
,判断⊙
与直线
的位置关系,并说明理由:
(3)已知直线
与
平行,求这两条直线之间的距离.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7775aa57ca0e62216f3039ed88dceed0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c15fb18163df0690365a0d2e7ee88f5a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c15fb18163df0690365a0d2e7ee88f5a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0682ce7c7d01d65347c659227e6c3e15.png)
例如:求点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a42451bdbef6c82dbaf8e06f0614794.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/960c22d0509ff3a0d4620afe187b196a.png)
解:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16f3d198e76391779fa3badc848c8ac8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/960c22d0509ff3a0d4620afe187b196a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/367e788c32187ae2cc97aaa24da1d40d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ab1c19b66cda3fb899f06d9a25e973c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2de0d10ef8b748d4531250c37c5d3f9e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a42451bdbef6c82dbaf8e06f0614794.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/960c22d0509ff3a0d4620afe187b196a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/46ed48c24e5697d14fe19abf3586fa6f.png)
根据以上材料,解答下列问题:
(1)求点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8c2f21b1baf0624482fd41d7ba390341.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e235d7dd12f948f5ffb2e5afddc95612.png)
(2)已知⊙
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fb0e705301752424a492f6277ed7774e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11bc05f41215f9894e11d1df0465751a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ca7d1107389675d32b56ec097464c14.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5627cec233ab4cd6ea8a864e220a6946.png)
(3)已知直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1d7df623642896d720d6956ed1f0ff6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a515853c22f0145b36c512079134dd5.png)
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名校
6 . 下列说法或运算正确的是( )
A.![]() |
B.用反证法证明“一个三角形至少有两个锐角”时需设“一个三角形没有锐角” |
C.“![]() ![]() ![]() ![]() |
D.直线![]() ![]() |
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7 . 圆
,动圆
.
(1)求证:圆
、圆
相交于两个定点;
(2)设点
是圆
上的点,过点
作圆
的一条切线,切点为
,过点
作圆
的一条切线,切点为
,问:是否存在点
,使无穷多个圆
,满足
?如果存在,求出所有这样的点
;如果不存在,说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d08e615aa342c43c9d7990169d534996.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/98ee0624db0ec93e626206e064f4e945.png)
(1)求证:圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
(2)设点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f240cccaf24af8a796abb95cb42be52e.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/4e9a724b59c890095baa5cb73e267c44.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9275bd8ce17fcc4a786510b008414ab0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/875ff780d7ae008dcc9f73d3c28d9286.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
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8 . 在平面直角坐标系中,已知圆
,直线
.
(1)求证:直线
与圆
总有两个不同的交点;
(2)在①
,②
最小,③过A,B两点分别作圆
的切线,切线交于点
,这三个条件中任选一个,补充在下面问题中并求解;
设圆
的圆心为
,直线
与圆
交于A,B两点,当__________时,求直线
的方程.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/321933346d4fed890863be4cb25edb63.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/21ae92459c4682762669063b425ba963.png)
(1)求证:直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)在①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5e5aceeb2d9c0fa9e7ff8d1df4b86b4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf2743c965e63b856f45c9e581fcf719.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7bd4e5049fa304e4d352bfe6dee455d.png)
设圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
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2021-11-05更新
|
677次组卷
|
5卷引用:重庆市西南大学附属中学2021-2022学年高二上学期期中数学试题
解题方法
9 . 已知椭圆
.设O为原点.若点A在椭圆C上,点B在直线
上,且
,试判断直线AB与圆
的位置关系,并证明你的结论.综合性问题,对于平面内定点F(1,0)与定直线
,设d为点P到直线l的距离.若
,你知道此时动点P的轨迹是什么样的曲线吗?为什么?
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/689e1011525feb927bef1c8c81961c0b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/107babba45f110012183dc4dc54490f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3825ccc273ef9a672a606432d165b866.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d61985901c2bc698d72ac88f4e1eb65.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4180dae966f648d368a10edf3b7e3c3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5d5a7572674db39a70a6d9a72c56a103.png)
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