(1)先求出点D(-1,0),设点M(
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224056525403.png)
),根据动点
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224055558399.png)
到直线
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224055589396.png)
的距离是它到点
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224055621315.png)
的距离的2倍,建立关于x,y的方程,然后化简整理可得所求动点M的轨迹方程.
(2)按斜率存在和斜率不存在两种情况进行讨论.当直线EF的斜率不存在时,O、P、K三点共线,直线PK的斜率为0.然后再设EF的方程
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224056728599.png)
它与椭圆方程联立消y后得关于x的一元二次方程
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232240567591184.png)
,然后根据
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224056775585.png)
,K点坐标为(2,0)
可得
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224056884994.png)
,再借助直线方程和韦达定理建立m,b的方程,从而用m表示b,再代入直线方程可求出定点坐标.然后把KP的斜率表示成关于m的函数,利用函数的方法求其范围.
(1)依题意知,点C(-4,0),由
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224056993603.png)
得点D(-1,0)
设点M(
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224056525403.png)
),则:
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224057040927.png)
整理得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224057134728.png)
动点M的轨迹方程为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224056463706.png)
(2)当直线EF的斜率不存在时,由已知条件可知,O、P、K三点共线,直线PK的斜率为0.
当直线EF的斜率存在时,可设直线EF的方程为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224056728599.png)
代入
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224056463706.png)
,整理
得
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232240567591184.png)
设
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224057337873.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232240573992248.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224057415585.png)
,K点坐标为(2,0)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224057446991.png)
,代入整理得
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224057477875.png)
解得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224057493745.png)
当
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224057524510.png)
时,直线EF的方程为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224057602676.png)
恒过点
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224057617474.png)
,与已知矛盾,舍去.
当
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224057633621.png)
时,
设
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224057664809.png)
,由
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224056416697.png)
知
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232240578202343.png)
直线KP的斜率为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232240578361426.png)
当
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224057867425.png)
时,直线KP的斜率为0, 符合题意
当
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224057883450.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224057914769.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224057945833.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224058007639.png)
时取“=”)或
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224058023581.png)
≤-
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224058054767.png)
时取“=”)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224058070761.png)
或
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224058397680.png)
综合以上得直线KP斜率的取值范围是
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224056494687.png)
.