点到直线距离公式的另一种推导方法
高中《解析几何》中，《点到直线的距离》一节，用“分类讨论”的办法，得到了点到直线的距离公式，并说“用求交点，再用两点之间的距离公式求点到直线的距离运算很繁”。其实，若采用“设而不求”的办法，这种方法的运算量也不是很大，而且“设而不求”的方法，在后面第二章中将会常常用到，故在此给出，供大家参考。
问题：求点P![](https://wkrtcs.bdimg.com/rtcs/image?w=52&md5sum=5163ed6d136077d756cd83b7d972fd0c&sign=7dd35e32f2&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=96&ipr={"t":"img","w":"52","h":"24","dataType":"gif","c":"word/media/image1.png"})
到直线L：的距离。
解：过P作直线L的垂线PQ交直线L于Q，设Q![](https://wkrtcs.bdimg.com/rtcs/image?w=36&md5sum=5163ed6d136077d756cd83b7d972fd0c&sign=7dd35e32f2&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=96&ipr={"t":"img","w":"36","h":"20","dataType":"gif","c":"word/media/image3.png"})
，PQ的方程为：，因为 PPQ，
所以 ![](https://wkrtcs.bdimg.com/rtcs/image?w=35&md5sum=5163ed6d136077d756cd83b7d972fd0c&sign=7dd35e32f2&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=96&ipr={"t":"img","w":"35","h":"23","dataType":"gif","c":"word/media/image6.png"})
，PQ：，
即 PQ：![](https://wkrtcs.bdimg.com/rtcs/image?w=168&md5sum=5163ed6d136077d756cd83b7d972fd0c&sign=7dd35e32f2&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=96&ipr={"t":"img","w":"168","h":"24","dataType":"gif","c":"word/media/image9.png"})
，
由
![](https://wkrtcs.bdimg.com/rtcs/image?w=108&md5sum=5163ed6d136077d756cd83b7d972fd0c&sign=7dd35e32f2&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=96&ipr={"t":"img","w":"108","h":"20","dataType":"gif","c":"word/media/image2.png"})
得
![](https://wkrtcs.bdimg.com/rtcs/image?w=287&md5sum=5163ed6d136077d756cd83b7d972fd0c&sign=7dd35e32f2&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=96&ipr={"t":"img","w":"287","h":"24","dataType":"gif","c":"word/media/image10.png"})
有 ![](https://wkrtcs.bdimg.com/rtcs/image?w=56&md5sum=5163ed6d136077d756cd83b7d972fd0c&sign=7dd35e32f2&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=96&ipr={"t":"img","w":"56","h":"24","dataType":"gif","c":"word/media/image11.png"})
![](https://wkrtcs.bdimg.com/rtcs/image?w=59&md5sum=5163ed6d136077d756cd83b7d972fd0c&sign=7dd35e32f2&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=96&ipr={"t":"img","w":"59","h":"24","dataType":"gif","c":"word/media/image13.png"})
，
故![](https://wkrtcs.bdimg.com/rtcs/image?w=32&md5sum=5163ed6d136077d756cd83b7d972fd0c&sign=7dd35e32f2&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=96&ipr={"t":"img","w":"32","h":"27","dataType":"gif","c":"word/media/image15.png"})
＝
![](https://wkrtcs.bdimg.com/rtcs/image?w=312&md5sum=5163ed6d136077d756cd83b7d972fd0c&sign=7dd35e32f2&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=96&ipr={"t":"img","w":"312","h":"52","dataType":"gif","c":"word/media/image17.png"})
![](https://wkrtcs.bdimg.com/rtcs/image?w=116&md5sum=5163ed6d136077d756cd83b7d972fd0c&sign=7dd35e32f2&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=96&ipr={"t":"img","w":"116","h":"51","dataType":"gif","c":"word/media/image18.png"})
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