X 2/9+Y 2 =1,and the equation passing through the straight line MN with the left focus slope of k is y=k(x+2√2), which is sorted out by substituting it into the elliptic equation, (1/9+k 2) x 2+4 √ 2× k 2x.
x 1+X2 =-4√2×k2/(k+ 1/9)。 ①，x 1x 2 =(8k 2- 1)/(k+ 1/9)。
| Mn | = √ (k+1) ×| x2-x1| = √ (k+1) √ [(x1+x2) 2-4x1x2] = short axis.
Substitute ① ② into K = √ 3/3, that is, when K = √ 3/3, the value of |MN| is equal to the length of the short axis.