解析,
a>b>0
e=c/a=√6/3,
短轴一个端点到右焦点的距离为√3,即是a=√3
那么,c=√2,b=1
椭圆的方程为：x²+3y²-3=0
设直线的方程为mx+ny=1
原点到直线的距离为√3/2,
即是,1/√(m²+n²)=√3/2,
整理得,m²+n²=4/3.
当m=0时,即AB垂直y轴,此时S(△ABC)=3/4.
当n=0时,即AB垂直x轴,此时S(△ABC)=3/4.
当m,n≠0时,
设A(x1,y1),B(x2,y2)
x1-x2=n(y2-y1)/m,
S(△ABC)=1/2*√3/2*√[(y1-y2)²+(x1-x2)²]
=√3/4*√[(y1-y2)²+(x1-x2)²]
=√3/4*|y1-y2|*√[(m²+n²)/m²]
=1/2*|y1-y2|/|m|.
mx+ny=1,x²+3y²-3=0联立,
整理得,(3m²+n²)y²-2ny+1-3m²=0,
y1+y2=2n/(3m²+n²),y1*y2=(1-3m²)/(3m²+n²)
|y1-y2|=√[(y1+y2)²-4y1*y2]=2√3*|m|*√[(3m²+n²-1)/(3m²+n²)²]
因此,
S(△ABC)=1/2*|y1-y2|/|m|
=√3*√[(3m²+n²-1)/(3m²+n²)²]
又,m²+n²=4/3代入
原式=√3*√[(2m²+1/3)/(2m²+4/3)²]【配方法】
=√3/√[(2m²+1/3)+1/(2m²+1/3)+2]
≦√3/√(2+2)=√3/2
当且仅当2m²+1/3=1/(2m²+1/3),即是m²=1/3时取等号.
又,√3/2>3/4
S(△ABC)(max)=√3/2.