题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215352-81032.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215353-33857.png)
(1)求圆
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215353-96871.png)
(2)若直线
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215353-13506.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215353-56819.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215353-96871.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215353-64443.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215353-13506.png)
(3)是否存在斜率是1的直线
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215354-94619.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215354-94619.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215353-96871.png)
原点?若存在,试求出直线
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215354-94619.png)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215354-42463.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215354-23371.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215354-85374.png)
解析
试题分析:(1)用两点的距离公式求出圆的半径,就可写出圆的标准方程;(2)法一:由圆的弦长可求得圆心到直线的距离,再用点斜式设出所求直线的方程,应用待定系数法:由点到直线的距离公式,就可求出所求直线的斜率,从而就可求得所求的直线方程,只是一定要注意:斜率不存在情形的讨论;法二:设出直线的斜率,写出直线方程,与圆方程联立,消去y得到关于x的一元二次方程,应用韦达定理及弦长公式,就可用斜率的代数式将弦长表示出来,从而获得关于斜率的方程解之即得;一样也需考虑斜率不存在情形;(3)法一:假设所求直线存在,先用斜截式设出其方程
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215355-73324.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215355-71814.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215355-71814.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215355-46038.png)
试题解析:(1)圆
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215353-96871.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215355-51677.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215355-12065.png)
∴圆
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215353-96871.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215354-42463.png)
(2)方法一 如图所示,设直线
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215353-13506.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215353-96871.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215356-40590.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215356-34723.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215356-21395.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215357-15817.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215357-13669.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215357-30563.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215357-87391.png)
∵圆
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215353-96871.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215357-25384.png)
∴在
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215358-96546.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215358-43989.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215353-96871.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215353-13506.png)
(i)当所求直线
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215353-13506.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215358-27616.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215358-24316.png)
由点到直线的距离公式得:
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215359-81176.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215359-11304.png)
∴此时直线
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215359-74800.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215354-23371.png)
(ii)当直线
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215359-74800.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215359-74800.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215354-85374.png)
将
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215354-85374.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215354-42463.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215359-19928.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215359-82336.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215400-98657.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215400-16799.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215400-85843.png)
∴方程为
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215354-85374.png)
∴所求直线
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215353-13506.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215354-23371.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215354-85374.png)
方法二:当所求直线
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215353-13506.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215358-27616.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215358-24316.png)
联立直线与圆
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215353-96871.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215400-55477.png)
消去
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215401-71243.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215401-80462.png)
设方程①的两根为
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215401-87940.png)
由根与系数的关系得
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215402-84614.png)
由弦长公式得
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215402-65836.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215402-61265.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215402-56429.png)
将②式代入③,并解得
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215359-11304.png)
此时直线
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215353-13506.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215354-23371.png)
当直线
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215359-74800.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215359-74800.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215354-85374.png)
仿方法一验算得方程为
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215354-85374.png)
∴所求直线
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215353-13506.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215354-23371.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215354-85374.png)
(3)方法一:假设存在直线
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215354-94619.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215354-94619.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215355-73324.png)
则
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215402-11875.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215403-12942.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215355-73324.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215403-58494.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215403-42465.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215403-37665.png)
∵以
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215402-11875.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215355-71814.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215404-69889.png)
又∵
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215404-48283.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215404-41335.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215404-90453.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215404-35674.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215405-56719.png)
∵方程
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215405-56719.png)
∴不存在满足题设条件的直线
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215354-94619.png)
方法二: 假设存在直线
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215354-94619.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215354-94619.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215355-73324.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215405-87000.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215405-69663.png)
联立直线与圆
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215353-96871.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215406-74336.png)
消去
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215401-71243.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215406-11316.png)
由根与系数的关系得
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215406-50840.png)
∵以
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215402-11875.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215355-71814.png)
若
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215407-75640.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215407-76117.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215407-60768.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215408-57977.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215353-96871.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215408-74775.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215408-57977.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215408-67568.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215408-43820.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215409-30252.png)
化简得:
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215409-38811.png)
以④代入并化简得
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215409-24283.png)
∵方程
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215409-24283.png)
∴不存在满足题设条件的直线
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215354-94619.png)
核心考点
试题【已知圆心为的圆经过点.(1)求圆的标准方程;(2)若直线过点且被圆截得的线段长为,求直线的方程;(3)是否存在斜率是1的直线,使得以被圆所截得的弦EF为直径的圆】;主要考察你对点到直线的距离等知识点的理解。[详细]
举一反三
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215342-34497.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215342-93012.png)
(1)过点M向圆O引切线,求切线的方程;
(2)求以点M为圆心,且被直线
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215342-66620.png)
(3)设P为(2)中圆M上任意一点,过点P向圆O引切线,切点为Q,试探究:平面内是否存在一定点R,使得
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215342-97599.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215332-36688.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215315-33784.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215315-33360.png)
(1)求满足条件a+b≥9的概率;
(2)求直线ax+by+5=0与x2+y2=1相切的概率
(3)将a,b,5的值分别作为三条线段的长,求这三条线段能围成等腰三角形的概率。
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215245-11232.png)
(1)求直线
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022215245-57828.png)
(2)过点G(1,3)作两条与圆C相切的直线,切点分别为M,N求直线MN的方程
(3)若与直线l1垂直的直线l与圆C交于不同的两点P,Q,若∠POQ为钝角,求直线l纵截距的取值范围.
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