设直线BC的方程为x=my-1,B(x1,y1),C(x2,y2)
将x=my-1代入x²+3y²=4,得(m²+3)y²-2my-3=0
则y1+y2=2m/(m²+3),y1y2=-3/(m²+3),
因为A(-2,0),所以向量AB=(x1+2,y1)=(my1+1,y1),向量AC=(my2+1,y2),
向量AB*向量AC=(my1+1)(my2+1)+y1y2
=(m²+1)y1y2+m(y1+y2)+1
=-3(m²+1)/(m²+3)+2m²/(m²+3)+1=0
所以AB⊥AC.
因为|AT|=1
所以三角形ABC的面积S=(1/2)|AT||y1-y2|=(1/2)|y1-y2|,
(y1-y2)²=(y1+y2)²-4y1y2=(16m²+36)/(m²+3)²,
设t=m²,f(t)=(y1-y2)²=(16t+36)/(t+3)²(t≥0)
因为f'(t)=-(16t+24)/(t+3)³
童茂松
关键词: 椭圆 x2 3y2 顶点 定点