(1)c=√3,a=2,b=1
x^2+y^2/4=1
(2)设A(x1,y1),B(x2,y2)
直线AB:y=kx+1与椭圆y^2/4+x^2=1联立
(k^2+4)x^2+2kx-3=0
x1+x2=-2k/(4+k^2),x1*x2=-3/(4+k^2)
向量OA⊥向量OB,x1x2+y1y2=0
x1x2+y1y2=x1x2+(kx1+1)(kx2+1)
=(1+k^2)x1x2+k(x1+x2)+1
=(1-4k^2)/(4+k^2)=0
k=1/2 或k=-1/2
I向量ABI=|AB|=√(1+k^2)|x1-x2|
=√(1+k^2)*√[(x1+x2)^2-4x1x2]
=√(1+k^2)*√[16(k^2+3)/(k^2+4)^2]
=√(5/4)*√[52/(17/4)^2]
=4√65/17