a=√2
F1M+F2M=2a=2√2
F1N+F2N=2a=2√2
所以MN=F1M+F1N=2√2+2√2-2√26/3=4√2-2√26/3
c²=a²-b²=1
F1(-1,0)
所以是y-0=k(x+1)
y=kx+k
代入x^2+2y^2=2
(2k^2+1)x^2+4k^2x+(2k^2-2)=0
x1+x2=-4k^2/(2k^2+1)
x1x2=(2k^2-2)/(2k^2+1)
(x1-x2)^2=(x1+x2)^2-4x1x2=(8k^2+8)/(2k^2+1)^2
y=kx+k
(y1-y2)^2=(kx1-kx2)^2=k^2(x1-x2)^2=(8k^4+8k^2)/(2k^2+1)^2
MN^2=(x1-x2)^2+(y1-y2)^2
=(8k^4+16k^2+8)/(2k^2+1)^2
=8(k^2+1)^2/(2k^2+1)^2=(4√2-2√26/3)^2
2√2(k^2+1)/(2k^2+1)=4√2-2√26/3
(k^2+1)/(2k^2+1)=2-√13/3
(2k^2+2)/(2k^2+1)=4-2√13/3
1+1/(2k^2+1)=4-2√13/3
2k^2+1=3/(9-2√13)
解出k即可