【答案】
分析:所求式子看做经过(2,1)点直线的斜率,根据题意画出图形,找出P与C,D重合时直线的斜率,即为求出所求式子的最大值与最小值.
解答:![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131103175014044366253/SYS201311031750140443662006_DA/images0.png)
解:根据题意画出图形,当P与C(或D)重合时,直线BC(BD)与圆A相切,
设直线BC解析式为y-1=k(x-2),即kx-y-2k+1=0,
∴圆心(0,1)到直线BC的距离d=r,即
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131103175014044366253/SYS201311031750140443662006_DA/0.png)
=1,
解得:k=±
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131103175014044366253/SYS201311031750140443662006_DA/1.png)
,
∴-
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131103175014044366253/SYS201311031750140443662006_DA/2.png)
≤k≤
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131103175014044366253/SYS201311031750140443662006_DA/3.png)
,即-
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131103175014044366253/SYS201311031750140443662006_DA/4.png)
≤
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131103175014044366253/SYS201311031750140443662006_DA/5.png)
≤
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131103175014044366253/SYS201311031750140443662006_DA/6.png)
,
则
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131103175014044366253/SYS201311031750140443662006_DA/7.png)
的最大值与最小值分别为
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131103175014044366253/SYS201311031750140443662006_DA/8.png)
,-
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131103175014044366253/SYS201311031750140443662006_DA/9.png)
.
故选B
点评:此题考查了直线与圆的位置关系,涉及的知识有:圆的标准方程,直线的点斜式方程,点到直线的距离公式,利用了数形结合的思想,解题的关键是所求式子看做为经过(2,1)点直线的斜率.