聪明文档网
最新最全的文档下载山东省2020年高考数学 第21题优美解
2020年高考数学（山东）第21题（理）试题优美解
试题（山东、 理科21）
在平面直角坐标系xOy中，F是抛物线C：x2=2py（p＞0）的焦点，M是抛物线C上位于第一象限内的任意一点，过M，F，O三点的圆的圆心为Q，点Q到抛物线C的准线的距离为![](https://wkrtcs.bdimg.com/rtcs/image?w=16.00&md5sum=9f8290b07f91a7454cc95288609b1502&sign=8ac78f20a1&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=250&ipr={"t":"img","r":"r_3","w":"16.00","h":"41.33","dataType":"gif","c":"word/media/image1.gif"})
。
（Ⅰ）求抛物线C的方程；
（Ⅱ）是否存在点M，使得直线MQ与抛物线C相切于点M？若存在，求出点M的坐标；若不存在，说明理由；
（Ⅲ）若点M的横坐标为![](https://wkrtcs.bdimg.com/rtcs/image?w=25.33&md5sum=9f8290b07f91a7454cc95288609b1502&sign=8ac78f20a1&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=250&ipr={"t":"img","r":"r_3","w":"25.33","h":"22.67","dataType":"gif","c":"word/media/image2.gif"})
，直线l：y=kx+与抛物线C有两个不同的交点A，B，l与圆Q有两个不同的交点D，E，求当≤k≤2时，的最小值。
解法
（Ⅰ）F抛物线C：x2=2py（p＞0）的焦点F![](https://wkrtcs.bdimg.com/rtcs/image?w=42.67&md5sum=9f8290b07f91a7454cc95288609b1502&sign=8ac78f20a1&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=250&ipr={"t":"img","r":"r_3","w":"42.67","h":"41.33","dataType":"gif","c":"word/media/image6.gif"})
，设M，，由题意可知，则点Q到抛物线C的准线的距离为，解得，于是抛物线C的方程为.
（Ⅱ）假设存在点M，使得直线MQ与抛物线C相切于点M，
而![](https://wkrtcs.bdimg.com/rtcs/image?w=178.67&md5sum=9f8290b07f91a7454cc95288609b1502&sign=8ac78f20a1&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=250&ipr={"t":"img","r":"r_3","w":"178.67","h":"45.33","dataType":"gif","c":"word/media/image13.gif"})
，，，
![](https://wkrtcs.bdimg.com/rtcs/image?w=212.00&md5sum=9f8290b07f91a7454cc95288609b1502&sign=8ac78f20a1&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=250&ipr={"t":"img","r":"r_3","w":"212.00","h":"45.33","dataType":"gif","c":"word/media/image16.gif"})
，，
由![](https://wkrtcs.bdimg.com/rtcs/image?w=56.00&md5sum=9f8290b07f91a7454cc95288609b1502&sign=8ac78f20a1&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=250&ipr={"t":"img","r":"r_3","w":"56.00","h":"24.00","dataType":"gif","c":"word/media/image12.gif"})
可得，，则，
即![](https://wkrtcs.bdimg.com/rtcs/image?w=110.67&md5sum=9f8290b07f91a7454cc95288609b1502&sign=8ac78f20a1&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=250&ipr={"t":"img","r":"r_3","w":"110.67","h":"26.67","dataType":"gif","c":"word/media/image21.gif"})
，解得，点M的坐标为.
（Ⅲ）若点M的横坐标为，则点M，。
由![](https://wkrtcs.bdimg.com/rtcs/image?w=80.00&md5sum=9f8290b07f91a7454cc95288609b1502&sign=8ac78f20a1&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=250&ipr={"t":"img","r":"r_3","w":"80.00","h":"64.00","dataType":"gif","c":"word/media/image26.gif"})
可得，设，
![](https://wkrtcs.bdimg.com/rtcs/image?w=228.00&md5sum=9f8290b07f91a7454cc95288609b1502&sign=8ac78f20a1&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=250&ipr={"t":"img","r":"r_2","w":"228.00","h":"29.33","dataType":"gif","c":"word/media/image29.gif"})
圆![](https://wkrtcs.bdimg.com/rtcs/image?w=264.00&md5sum=9f8290b07f91a7454cc95288609b1502&sign=8ac78f20a1&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=250&ipr={"t":"img","r":"r_3","w":"264.00","h":"45.33","dataType":"gif","c":"word/media/image31.gif"})
，
![](https://wkrtcs.bdimg.com/rtcs/image?w=249.33&md5sum=9f8290b07f91a7454cc95288609b1502&sign=8ac78f20a1&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=250&ipr={"t":"img","r":"r_3","w":"249.33","h":"48.00","dataType":"gif","c":"word/media/image33.gif"})
，
于是![](https://wkrtcs.bdimg.com/rtcs/image?w=286.67&md5sum=9f8290b07f91a7454cc95288609b1502&sign=8ac78f20a1&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=250&ipr={"t":"img","r":"r_3","w":"286.67","h":"48.00","dataType":"gif","c":"word/media/image34.gif"})
，令
![](https://wkrtcs.bdimg.com/rtcs/image?w=534.67&md5sum=9f8290b07f91a7454cc95288609b1502&sign=8ac78f20a1&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=250&ipr={"t":"img","r":"r_3","w":"534.67","h":"48.00","dataType":"gif","c":"word/media/image36.gif"})
，
设![](https://wkrtcs.bdimg.com/rtcs/image?w=153.33&md5sum=9f8290b07f91a7454cc95288609b1502&sign=8ac78f20a1&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=250&ipr={"t":"img","r":"r_3","w":"153.33","h":"41.33","dataType":"gif","c":"word/media/image37.gif"})
，，
当![](https://wkrtcs.bdimg.com/rtcs/image?w=58.67&md5sum=9f8290b07f91a7454cc95288609b1502&sign=8ac78f20a1&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=250&ipr={"t":"img","r":"r_3","w":"58.67","h":"41.33","dataType":"gif","c":"word/media/image39.gif"})
时，，
即当![](https://wkrtcs.bdimg.com/rtcs/image?w=81.33&md5sum=9f8290b07f91a7454cc95288609b1502&sign=8ac78f20a1&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=250&ipr={"t":"img","r":"r_3","w":"81.33","h":"41.33","dataType":"gif","c":"word/media/image41.gif"})
时.
故当![](https://wkrtcs.bdimg.com/rtcs/image?w=41.33&md5sum=9f8290b07f91a7454cc95288609b1502&sign=8ac78f20a1&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=250&ipr={"t":"img","r":"r_3","w":"41.33","h":"41.33","dataType":"gif","c":"word/media/image43.gif"})
时，.
试题欣赏
理科第(21)题采用了开放性的设问方式和对新定义的阅读和理解以及应用，创新意识的考查，注重对未来继续学习的能力考查。
Word文档免费下载：
山东省2020年高考数学 第21题优美解
-
1个月 推荐
￥29.8
￥45每天只需1.0元 -
1天
￥9.9
￥15 -
3个月
￥59.8
￥90
选择支付方式
-
微信付款
