题干

在平面直角坐标系xOy中,已知椭圆E:
x
2
a
2
+
y
2
b
2
=1
(a>b>0),圆O:x2+y2=r2(0<r<b),若圆O的一条切线l:y=kx+m与椭圆E相交于A,B两点.

(Ⅰ)当k=﹣
1
2
,r=1时,若点A,B都在坐标轴的正半轴上,求椭圆E的方程;
(Ⅱ)若以AB为直径的圆经过坐标原点O,探究a,b,r之间的等量关系,并说明理由.
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答案(点此获取答案解析)

解:(Ⅰ)依题意原点O到切线l:y=﹣
1
2
x+m的距离为半径1,∴ m1+

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