设椭圆x2⼀a2+y2⼀b2=1(a>b>0)左右焦点分别为f1f2过点f1的直线与椭圆c相交于a,b两点如图∠f1f2b

(i)后的第三行应当不难理解。从A，B向x轴作垂线，垂足C, D; ACF2 与BDF相似，|CA| = 2|DB| (因为题中两个三角形的底相等).

F1(-c,0),F2(c,0)
s△F1F2A=2s△F1F2B
yA=-2yB,xA=3c-2xB
∠f1f2b=2π/3
k(AB)=tg(π/3)=√3
AB:y=√3*(x-c)
yB=√3*(xB-c)
C:x^2/a^2+y^2/b^2=1
xA^2/a^2+yA^2/b^2=1......(1)
xB^2/a^2+yB^2/b^2=1......(2)
(1)-(2)
b^2*(xA+xB)*(xA-xB)+a^2*(yA+yB)*(yA-yB)=0
b^2*(xA+xB) +a^2*(yA+yB)*(yA-yB)/(xA-xB)=0
b^2*(xA+xB)+√3*a^2*(yA+yB)=0
b^2*(3c-xB)+√3a^2*(-yB)=0
b^2*(3c-xB)-√3a^2*√3(xB-c)=0
b^2*(3c-xB)-3a^2(xB-c)=0......(3)
(xA-xB)^2+(yA-yB)^2=AB^2=(15/2)^2
1+(yA-yB)^2/(xA-xB)^2=56.25/(xA-xB)^2
1+3=56.25/(3c-3xB)^2
36=56.25/(xB-c)^2
xB-c=1.25
yB=1.25√3,yA=-2.5√3
xB=1.25+c,xA=3c-2xB=c-2.5代入(3),(2)
b^2*(2c-1.25)-3.75a^2=0
b^2*(2c-1.25)-3.75(b^2+c^2)=0
2cb^2-5b^2-3.75c^2=0......(4)
(1.25+c)^2/(b^2+c^2)+(1.25√3)^2/b^2=1......(5)
(4),(5):
b^2=,c^2=,a^2=
x^2/a^2+y^2/b^2=1