(I)先建系,然后根据
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823230824091660.png)
为定值,可确定点M的轨迹是双曲线,
然后按照求双曲线标准方程的方法求解即可.
(II) 先设直线l的方程为y=kx+2,代入双曲线C的方程并整理得(1-k
2)x
2-4kx-6=0.
根据条件可知
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232308245441428.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823230824559256.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823230824590910.png)
,从而得到k的取值范围.
再利用弦长公式和韦达定理用k表示出|EF|,再利用点到直线的距离公式求出原点O到直线l的距离,从而表示出三角形的面积,这样三角形的面积就表示成了关于k的函数,
再根据
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823230824715527.png)
,得到关于k的不等式,从而解出k的取值范围,再与前面k的取值范围求交集即可.
(Ⅰ)解法1:以O为原点,AB、OD所在直线分别为x轴、y轴,建立平面直角坐标系,则A(-2,0),B(2,0),D(0,2),P(
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823230824746398.png)
),依题意得
|MA|-|MB|=|PA|-|PB|=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232308247621097.png)
<|AB|=4.
∴曲线C是以原点为中心,A、B为焦点的双曲线.
设实平轴长为a,虚半轴长为b,半焦距为c,
则c=2,2a=2
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823230824466336.png)
,∴a
2=2,b
2=c
2-a
2=2.∴曲线C的方程为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823230824809687.png)
.
解法2:同解法1建立平面直角坐标系,则依题意可得|MA|-|MB|=|PA|-|PB|<
|AB|=4.
∴曲线C是以原点为中心,A、B为焦点的双曲线.
设双曲线的方程为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823230824824834.png)
>0,b>0).
则由
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232308249021167.png)
解得a
2=b
2=2,∴曲线C的方程为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823230824450698.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/2014082323082502710265.jpg)
(Ⅱ)解法1:依题意,可设直线l的方程为y=kx+2,代入双曲线C的方程并整理得(1-k
2)x
2-4kx-6=0.
∵直线l与双曲线C相交于不同的两点E、F,
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232308245441428.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823230824559256.png)
∴k∈(-
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823230825292337.png)
,-1)∪(-1,1)∪(1,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823230825292337.png)
).
设E(x,y),F(x
2,y
2),则由①式得x
1+x
2=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823230825324903.png)
,于是
|EF|=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232308253551354.png)
=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232308253862044.png)
而原点O到直线l的距离d=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823230825402598.png)
,
∴S
△DEF=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232308254332547.png)
若△OEF面积不小于2
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823230824466336.png)
,即S
△OEF![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823230825480438.png)
,则有
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232308256822083.png)
③
综合②、③知,直线l的斜率的取值范围为[-
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823230824466336.png)
,-1]∪(1-,1) ∪(1,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823230824466336.png)
).
解法2:依题意,可设直线l的方程为y=kx+2,代入双曲线C的方程并整理,
得(1-k
2)x
2-4kx-6=0.
∵直线l与双曲线C相交于不同的两点E、F,
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232308245441428.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823230824559256.png)
∴k∈(-
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823230825292337.png)
,-1)∪(-1,1)∪(1,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823230825292337.png)
).
设E(x
1,y
1),F(x
2,y
2),则由①式得
|x
1-x
2|=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232308261971884.png)
③
当E、F在同一去上时(如图1所示),
S
△OEF=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232308262281815.png)
当E、F在不同支上时(如图2所示).
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823230826244676.png)
S
△ODE=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232308262751213.png)
综上得S
△OEF=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823230826291793.png)
于是
由|OD|=2及③式,得S
△OEF=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823230826322827.png)
若△OEF面积不小于2
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823230826338940.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232308263531992.png)
④
综合②、④知,直线l的斜率的取值范围为[-
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823230824466336.png)
,-1]∪(-1, 1)∪(1,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823230824466336.png)
).