题目内容
阅读理解:对于任意正实数a、b,∵(![](http://thumb.1010pic.com/pic1/imagenew2/czsx/1/15321.png)
![](http://thumb.1010pic.com/pic1/imagenew2/czsx/2/15322.png)
![](http://thumb.1010pic.com/pic1/imagenew2/czsx/3/15323.png)
![](http://thumb.1010pic.com/pic1/imagenew2/czsx/3/15323.png)
结论:在a+b≥2
![](http://thumb.1010pic.com/pic1/imagenew2/czsx/3/15323.png)
![](http://thumb.1010pic.com/pic1/imagenew2/czsx/6/15326.png)
![](http://thumb.1010pic.com/pic1/imagenew2/czsx/6/15326.png)
(1)若m>0,只有当m= 时,m+
![](http://thumb.1010pic.com/pic1/imagenew2/czsx/8/15328.png)
若m>0,只有当m= 时,2m+
![](http://thumb.1010pic.com/pic1/imagenew2/czsx/9/15329.png)
(2)如图,已知直线L1:y=
![](http://thumb.1010pic.com/pic1/imagenew2/czsx/10/15330.png)
![](http://thumb.1010pic.com/pic1/imagenew2/czsx/11/15331.png)
(x>0)相交于点B(2,m),求直线L2的解析式.
![](http://thumb.1010pic.com/pic1/imagenew2/czsx/12/15332.png)
(3)在(2)的条件下,若点C为双曲线上任意一点,作CD∥y轴交直线L1于点D,试
求当线段CD最短时,点A、B、C、D围成的四边形面积.
(1)当
时,
有最小值为2;当
时,
有最小值为8
(2)
(3)23解析:
解:(1)∵m>0,只有当
时,
有最小值;
m>0,只有当
时,
有最小值.
∴m>0,只有当
时,
有最小值为2;
m>0,只有当
时,
有最小值为8
(2)对于
,令y=0,得:x=-2,
∴A(-2,0)
又点B(2,m)在
上,
∴![](http://thumb.1010pic.com/pic1/imagenew2/czsx/8/15348.png)
设直线
的解析式为:
,
则有,![](http://thumb.1010pic.com/pic1/imagenew2/czsx/11/15351.png)
解得:![](http://thumb.1010pic.com/pic1/imagenew2/czsx/12/15352.png)
∴直线
的解析式为:
;
(3)设
,则:
,
∴CD=
,
∴CD最短为5,
此时
,n=4,C(4,-2),D(4,3)
过点B作BE∥y轴交AD于点E,则B(2,-4),E(2,2),BE=6,
∴S四边形ABCD=S△ABE+S四边形BEDC
![](http://thumb.1010pic.com/pic1/imagenew2/czsx/13/15333.png)
![](http://thumb.1010pic.com/pic1/imagenew2/czsx/14/15334.png)
![](http://thumb.1010pic.com/pic1/imagenew2/czsx/15/15335.png)
![](http://thumb.1010pic.com/pic1/imagenew2/czsx/16/15336.png)
(2)
![](http://thumb.1010pic.com/pic1/imagenew2/czsx/17/15337.png)
解:(1)∵m>0,只有当
![](http://thumb.1010pic.com/pic1/imagenew2/czsx/18/15338.png)
![](http://thumb.1010pic.com/pic1/imagenew2/czsx/14/15334.png)
m>0,只有当
![](http://thumb.1010pic.com/pic1/imagenew2/czsx/0/15340.png)
![](http://thumb.1010pic.com/pic1/imagenew2/czsx/16/15336.png)
∴m>0,只有当
![](http://thumb.1010pic.com/pic1/imagenew2/czsx/13/15333.png)
![](http://thumb.1010pic.com/pic1/imagenew2/czsx/14/15334.png)
m>0,只有当
![](http://thumb.1010pic.com/pic1/imagenew2/czsx/15/15335.png)
![](http://thumb.1010pic.com/pic1/imagenew2/czsx/16/15336.png)
(2)对于
![](http://thumb.1010pic.com/pic1/imagenew2/czsx/6/15346.png)
∴A(-2,0)
又点B(2,m)在
![](http://thumb.1010pic.com/pic1/imagenew2/czsx/7/15347.png)
∴
![](http://thumb.1010pic.com/pic1/imagenew2/czsx/8/15348.png)
设直线
![](http://thumb.1010pic.com/pic1/imagenew2/czsx/9/15349.png)
![](http://thumb.1010pic.com/pic1/imagenew2/czsx/10/15350.png)
则有,
![](http://thumb.1010pic.com/pic1/imagenew2/czsx/11/15351.png)
解得:
![](http://thumb.1010pic.com/pic1/imagenew2/czsx/12/15352.png)
∴直线
![](http://thumb.1010pic.com/pic1/imagenew2/czsx/9/15349.png)
![](http://thumb.1010pic.com/pic1/imagenew2/czsx/17/15337.png)
(3)设
![](http://thumb.1010pic.com/pic1/imagenew2/czsx/15/15355.png)
![](http://thumb.1010pic.com/pic1/imagenew2/czsx/16/15356.png)
∴CD=
![](http://thumb.1010pic.com/pic1/imagenew2/czsx/17/15357.png)
∴CD最短为5,
此时
![](http://thumb.1010pic.com/pic1/imagenew2/czsx/18/15358.png)
过点B作BE∥y轴交AD于点E,则B(2,-4),E(2,2),BE=6,
∴S四边形ABCD=S△ABE+S四边形BEDC
![](http://thumb.1010pic.com/pic1/imagenew2/czsx/19/15359.png)
![](http://thumb2018.1010pic.com/images/loading.gif)
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