【答案】
分析:(1)当切线斜率不存在时,直线与圆位置关系是相交,不合题意,所以设切线方程的斜率为k,根据P的坐标写出切线的方程,然后根据直线与圆相切时,圆心到直线的距离等于圆的半径,利用点到直线的距离公式表示出圆心到直线的距离d,让d等于半径r列出关于k的方程,求出方程的解即可得到k的值,根据求出的k的值和P的坐标写出切线方程即可;
(2)当切线斜率不存在时,直线与圆位置关系是外离,不合题意,所以设出切线方程的斜率为k,根据直线与圆相切时,圆心到直线的距离等于圆的半径,利用点到直线的距离公式表示出圆心到直线的距离d,让d等于圆的半径r列出关于k的方程,求出方程的解即可得到k的值,由k的值和Q的坐标写出切线方程即可;
(3)设出切点的坐标为(a,b),根据已知的斜率为-1,表示出切线的方程,然后利用点到直线的距离公式表示出圆心到所设直线的距离d,让d等于圆的半径r列出关于a与b的绝对值关系式,经讨论得到关于a与b的两关系式,分别记作①和②,把切点的坐标代入圆的方程,得到关于a与b的关系式,记作③,把①③联立,②③联立,分别求出两对a与b的值,得到切点的坐标有两个,根据求出的切点坐标和已知的切线的斜率写出切线方程即可.
解答:解:(1)经判断,得到点P在圆上,
当斜率k不存在时,直线与圆相交,不合题意,所以设切线方程的斜率为k,
则切线方程为:y-1=k(x-
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101221811021509996/SYS201311012218110215099012_DA/0.png)
),
所以圆心(0,0)到直线的距离d=
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101221811021509996/SYS201311012218110215099012_DA/1.png)
=r=2,
化简得:
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101221811021509996/SYS201311012218110215099012_DA/2.png)
=0,解得k=-
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101221811021509996/SYS201311012218110215099012_DA/3.png)
,
所以切线方程为:y=-
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101221811021509996/SYS201311012218110215099012_DA/4.png)
x+4;
(2)当直线斜率不存在时,直线与圆外离,不合题意,设过点Q的切线方程的斜率为k,
则切线方程为y=k(x-3),
所以圆心到直线的距离d=
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101221811021509996/SYS201311012218110215099012_DA/5.png)
=r=2,
化简得:k=±
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101221811021509996/SYS201311012218110215099012_DA/6.png)
,
所以切线方程为:y=
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101221811021509996/SYS201311012218110215099012_DA/7.png)
x-
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101221811021509996/SYS201311012218110215099012_DA/8.png)
或y=-
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101221811021509996/SYS201311012218110215099012_DA/9.png)
x+
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101221811021509996/SYS201311012218110215099012_DA/10.png)
;
(3)设切点坐标为(a,b),则切线方程为:y-a=-(x-b),即x+y-a-b=0,
所以圆心到直线的距离d=
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101221811021509996/SYS201311012218110215099012_DA/11.png)
=2,即a+b=2
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101221811021509996/SYS201311012218110215099012_DA/12.png)
①或a+b=-2
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101221811021509996/SYS201311012218110215099012_DA/13.png)
②,
又把切点坐标代入圆的方程得:a
2+b
2=4③,
由①得:a=2
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101221811021509996/SYS201311012218110215099012_DA/14.png)
-b,代入③得:a=b=
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101221811021509996/SYS201311012218110215099012_DA/15.png)
;由②得:a=-2
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101221811021509996/SYS201311012218110215099012_DA/16.png)
-b,代入③得:a=b=-
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101221811021509996/SYS201311012218110215099012_DA/17.png)
,
所以切点坐标分别为(
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101221811021509996/SYS201311012218110215099012_DA/18.png)
,
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101221811021509996/SYS201311012218110215099012_DA/19.png)
)或(-
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101221811021509996/SYS201311012218110215099012_DA/20.png)
,-
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101221811021509996/SYS201311012218110215099012_DA/21.png)
),
则切线方程为:y-
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101221811021509996/SYS201311012218110215099012_DA/22.png)
=-(x-
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101221811021509996/SYS201311012218110215099012_DA/23.png)
)或y+
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101221811021509996/SYS201311012218110215099012_DA/24.png)
=-(x+
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101221811021509996/SYS201311012218110215099012_DA/25.png)
),
即x+y-2
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101221811021509996/SYS201311012218110215099012_DA/26.png)
=0或x+y+2
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101221811021509996/SYS201311012218110215099012_DA/27.png)
=0.
点评:此题考查学生掌握直线与圆相切时圆心到直线的距离等于圆的半径,灵活运用点到直线的距离公式化简求值,是一道中档题.