切线方程

ゝ一纸荒年。 2023-06-14 13:58 28阅读 0赞

切线方程是研究[切线][Link 1]以及切线的斜率方程，涉及[几何][Link 2]、[代数][Link 3]、物理[向量][Link 4]、量子力学等内容。是关于几何图形的切线坐标向量关系的研究。分析方法有向量法和解析法。

## 定义 ##

切线方程是研究[切线][Link 5]以及切线的斜率方程。

## 方程的证明 ##

### 向量法 ###

设[圆上][Link 6]一点A为 ![format_png][] ,则该点与圆心O的向量![format_png 1][]

因为过该点的切线与该方向半径垂直，则有切线方向上的单位向量与向量OA的点积为0.

设直线上任意点B为(x,y)   则对于直线方向上的向量 ![format_png 2][]

有向量AB与OA的点积

![format_png 3][]

![format_png 4][]

![format_png 5][]

故有

![format_png 6][]

### 分析-解析法 ###

设圆上一点A为 ![format_png][] ,  则有： ![format_png 7][]

对[隐函数][Link 7][求导][Link 8]，则有： ![format_png 8][]

![format_png 9][]    （隐函数求导法亦可证明椭圆的切线方程，方法相同）

或直接

![format_png 10][]    （k1为与切线垂直的半径斜率。）

得

![format_png 11][]    （以上处理是假设斜率存在，在后面讨论[斜率][Link 9]不存在的情况）

所以切线方程可写为：  ![format_png 12][]

将点 ![format_png][]   ,可求出    ![format_png 13][]

所以:

![format_png 14][]

![format_png 15][]

![format_png 16][]

![format_png 17][]

当斜率不存在时，[切点][Link 10]为与x轴平行的直线过圆心与圆的交点。

此类切点有2个，不妨设为  ![format_png 18][]

![format_png 19][]

将2点带入上式，亦成立。

故得证。 \[1\]

## 常见切线方程的证明 ##

### 圆 ###

若点M ![format_png][]  在圆  ![format_png 20][]  上,

则过点M的切线方程为    ![format_png 21][]

或表述为： 若点M ![format_png][] 在圆  ![format_png 22][]  上，

则过点M的切线方程为  ![format_png 19][]

若已知点M![format_png][]  在圆 ![format_png 22][] 外，

则切点AB的直线方程也为  ![format_png 19][]

[![过圆外一点的2条切线][2]][2_2]过圆外一点的2条切线

### 椭圆 ###

若椭圆的方程为  ![format_png 23][] ,点P![format_png][] 在椭圆上，则过点P椭圆的切线方程为  ![format_png 24][]

证明：椭圆为     ![format_png 23][] ,[切点][Link 10]为 ![format_png][] ,则   ![format_png 25][]

...(1)

对椭圆求导得   ![format_png 26][]  , 即切线斜率    ![format_png 27][] ,

故切线方程是  ![format_png 28][]  ,将(1)代入并[化简][Link 11]得切线方程为   ![format_png 24][]  。

### 双曲线 ###

若双曲线的方程为    ![format_png 29][]  ,点P![format_png][] 在双曲线上，

则过点P双曲线的切线方程为   ![format_png 30][]

此命题的[证明方法][Link 12]与椭圆的类似。

### 抛物线 ###

若[抛物线][Link 13]的方程为  ![format_png 31][] , 点P![format_png][] 在抛物线上，则过点P的抛物线的切线方程为

![format_png 32][]

此命题的[证明方法][Link 12]亦与椭圆的类似，可设切线方程为

![format_png 33][]

联立[切线][Link 5]与抛物线,则  ![format_png 34][]  ,整理得

![format_png 35][]

因为相切，所以△=0

则

![format_png 36][]

可求得  ![format_png 37][]  ,代回

![format_png 33][]

![format_png 38][]

曲线的切线方程也可以用[导数][Link 14]求解。更为简便的计算方法：

设切线方程为   ![format_png 39][]  ，联立切线与抛物线

![format_png 40][]

△=0，

![format_png 41][] ，解得    ![format_png 42][]

切线方程：  ![format_png 43][]    ,化简得  ![format_png 44][]

[微积分][Link 15]方法：

在M(a,b)点斜率为求导：

2yy'=2p

代入点（a，b）则

![format_png 45][]

所以切线为：  ![format_png 46][]

[Link 1]: https://baike.baidu.com/item/%E5%88%87%E7%BA%BF/674562
[Link 2]: https://baike.baidu.com/item/%E5%87%A0%E4%BD%95/303227
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[Link 5]: https://baike.baidu.com/item/%E5%88%87%E7%BA%BF
[Link 6]: https://baike.baidu.com/item/%E5%9C%86%E4%B8%8A
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[Link 10]: https://baike.baidu.com/item/%E5%88%87%E7%82%B9
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[Link 14]: https://baike.baidu.com/item/%E5%AF%BC%E6%95%B0
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[Link 15]: https://baike.baidu.com/item/%E5%BE%AE%E7%A7%AF%E5%88%86/6065
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