题目内容
如图,Rt△APC的顶点A,P在反比例函数![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022164706279939177/SYS201310221647062799391009_ST/0.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022164706279939177/SYS201310221647062799391009_ST/1.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022164706279939177/SYS201310221647062799391009_ST/2.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022164706279939177/SYS201310221647062799391009_ST/images3.png)
A.
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022164706279939177/SYS201310221647062799391009_ST/3.png)
B.2021054
C.2022060
D.
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022164706279939177/SYS201310221647062799391009_ST/4.png)
【答案】分析:设CP=m,由tanA=
=
得AC=mn,则A(1-m,1+mn),将A点坐标代入
中,得出an=1-m的表达式,寻找运算规律.
解答:
解:依题意设CP=m,
∵P点横坐标为1,则C点横坐标为1-m,
即an=1-m,
又∵tanA=
=
,
∴AC=mn,则A(1-m,1+mn),
将A点坐标代入
中,得(1-m)(1+mn)=1,
1-m+mn-m2n=1,
m(n-1-mn)=0,
则n-1-mn=0,
1-m=
,
则an=1-m=
,即
=n,
∴
=2+3+4+…+2010
=
=2021054.
故选B.
点评:本题主要考查反比例函数的图象和性质,关键是根据三角函数值设直角三角形的边长,表示A点坐标,根据A点在双曲线上,满足反比例函数解析式,从而得出一般规律.
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022164706279939177/SYS201310221647062799391009_DA/0.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022164706279939177/SYS201310221647062799391009_DA/1.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022164706279939177/SYS201310221647062799391009_DA/2.png)
解答:
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022164706279939177/SYS201310221647062799391009_DA/images3.png)
∵P点横坐标为1,则C点横坐标为1-m,
即an=1-m,
又∵tanA=
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022164706279939177/SYS201310221647062799391009_DA/3.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022164706279939177/SYS201310221647062799391009_DA/4.png)
∴AC=mn,则A(1-m,1+mn),
将A点坐标代入
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022164706279939177/SYS201310221647062799391009_DA/5.png)
1-m+mn-m2n=1,
m(n-1-mn)=0,
则n-1-mn=0,
1-m=
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022164706279939177/SYS201310221647062799391009_DA/6.png)
则an=1-m=
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022164706279939177/SYS201310221647062799391009_DA/7.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022164706279939177/SYS201310221647062799391009_DA/8.png)
∴
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022164706279939177/SYS201310221647062799391009_DA/9.png)
=
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022164706279939177/SYS201310221647062799391009_DA/10.png)
故选B.
点评:本题主要考查反比例函数的图象和性质,关键是根据三角函数值设直角三角形的边长,表示A点坐标,根据A点在双曲线上,满足反比例函数解析式,从而得出一般规律.
![](http://thumb2018.1010pic.com/images/loading.gif)
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a2 |
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a3 |
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| ||
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