解:(1)点轴上································································································· 1分

理由如下:连接,如图所示,在中,

由题意可知:

轴上,轴上.············································································ 2分

(2)过点轴于点

中,  在第一象限,

的坐标为····························································································· 4分

由(1)知,点轴的正半轴上

的坐标为

的坐标为······························································································· 5分

抛物线经过点, 

由题意,将代入中得

  解得

所求抛物线表达式为:·························································· 7分

(3)存在符合条件的点,点.·············································································· 8分

理由如下:矩形的面积

为顶点的平行四边形面积为

由题意可知为此平行四边形一边,  又

边上的高为2······································································································ 10分

依题意设点的坐标为

在抛物线

解得,

  为顶点的四边形是平行四边形,

当点的坐标为时,

的坐标分别为

当点的坐标为时,

的坐标分别为.·················································· 12分

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