(1)S=1/2(2√2+√6)*h=1+√3/2
h=√2/2
以AB中点为原点O,OA为x轴建立坐标系,CD位于AB上方
则A(-√2,0),B(√2,0),C(√6/2,√2/2),D(-√6/2,√2/2)
设椭圆方程为x^2/a^2+y^2/(a^2-2)=1
将C点坐标代入得
a^2=3或a^2=1(舍)
所以椭圆的标准方程为x^2/3+y^2=1
(2)设M(x1,y1),N(x2,y2),则由OM·ON=0
得x1x2+y1y2=0
即x1^2*x2^2=y1^2*y2^2=(1-x1^2/3)(1-x2^2/3)
8x1^2*x2^2-3x1^2-3x2^2-9=0
x2^2=(3x1^2+9)/(8x1^2-3)
|MN|=√[(x1-x2)^2+(y1-y2)^2]
=√[x1^2+x2^2+y1^2+y2^2]
=√[2x1^2/3+2x^2/3+2]=√[2/3(x1^2+x2^2)+2]
=√[2/3(x1^2+(3x1^2+9)/(8x1^2-3))+2]
=√[2/3(x1^2+3/8+(81/8)/(8x1^2-3))+2]
=√[2/3(1/8(8x1^2-3)+(81/8)/(8x1^2-3)+3/4)+2]
≥√[2/3(2*9/8+3/4)+2]=2
当x1=√6/2时等号成立
所以|MN|min=2