设点
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823215951706302.png)
是抛物线
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823215951721292.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823215951752604.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823215951768509.png)
的焦点,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823215951846674.png)
是抛物线
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823215951862280.png)
上的
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823215951971297.png)
个不同的点(
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823215951986443.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823215952049503.png)
).
(1) 当
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823215952096421.png)
时,试写出抛物线
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823215952314281.png)
上的三个定点
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823215952392315.png)
、
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823215952439348.png)
、
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823215952454352.png)
的坐标,从而使得
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823215952501798.png)
;
(2)当
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823215952688421.png)
时,若
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823215952720907.png)
,
求证:
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232159527351053.png)
;
(3) 当
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823215952688421.png)
时,某同学对(2)的逆命题,即:
“若
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232159527351053.png)
,则
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823215952720907.png)
.”
开展了研究并发现其为假命题.
请你就此从以下三个研究方向中任选一个开展研究:
① 试构造一个说明该逆命题确实是假命题的反例(本研究方向最高得4分);
② 对任意给定的大于3的正整数
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823215951971297.png)
,试构造该假命题反例的一般形式,并说明你的理由(本研究方向最高得8分);
③ 如果补充一个条件后能使该逆命题为真,请写出你认为需要补充的一个条件,并说明加上该条件后,能使该逆命题为真命题的理由(本研究方向最高得10分).
【评分说明】本小题若填空不止一个研究方向,则以实得分最高的一个研究方向的得分作为本小题的最终得分.