847

椭圆 ${\displaystyle C:{\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)}}$ 的左右焦点为 ${\displaystyle F_1}$ 和 ${\displaystyle F_2}$, ${\displaystyle O}$ 为椭圆的中心，过 ${\displaystyle F_1}$ 作直线 ${\displaystyle l_1,l_2}$, 分别交椭圆 ${\displaystyle C}$ 于 ${\displaystyle A,B}$ 和 ${\displaystyle C,D}$, 且 ${\displaystyle |AB|}$ 的最大值为 ${\displaystyle 2\sqrt{6}}$, ${\displaystyle |CD|}$ 的最小值为 ${\displaystyle {\displaystyle \frac{2\sqrt{6}}{3}}}$.

1. 求椭圆 ${\displaystyle C}$ 的方程;
2. 设线段 ${\displaystyle AB,CD}$ 的中点分别为 ${\displaystyle M,N}$, 记 ${\displaystyle \mathrm{\Delta }OMN}$ 的面积为 ${\displaystyle S_1}$, ${\displaystyle \mathrm{\Delta }{F}_{2}MN}$ 的面积为 ${\displaystyle S_2}$. 若直线 ${\displaystyle l_1,l_2}$ 的斜率为 ${\displaystyle k_1,k_2}$, 且 ${\displaystyle k_1{k}_2={\displaystyle \frac{1}{2}}}$, 求证: ${\displaystyle {\displaystyle \frac{S_1}{S_2}}}$ 为定值，并求出这个定值.

1. 椭圆的焦点弦在长轴取到最长，在通径取到最短，因此 ${\displaystyle {\displaystyle 2a=2\sqrt{6},\frac{2b^2}{a}=\frac{2\sqrt{6}}{3}\Rightarrow C:\frac{x^2}{6}+\frac{y^2}{2}=1}}$.
2. 设 ${\displaystyle l_1:x=t_{1}y-2,l_2:x=t_{2}y-2}$, 这里 ${\displaystyle t_i={\displaystyle \frac{1}{k_i},i=1,2}}$, 且由于 ${\displaystyle k_1{k}_2={\displaystyle \frac{1}{2}}}$, 故 ${\displaystyle k_1,k_2\ne 0}$, ${\displaystyle t_1,t_2}$ 存在且 ${\displaystyle t_1{t}_2=2}$. 联立 ${\displaystyle l_1,C}$: $$\begin{array}{r}(t_1^2+3)y^2-4t_{1}y-2=0\Rightarrow y_M=\frac{y_A+y_B}{2}=\frac{2t_1}{t_1^2+3},\end{array}$$
   从而${\displaystyle x_M=t_1{y}_M-2=-{\displaystyle \frac{6}{t_1^2+3}}}$. 同理${\displaystyle {\displaystyle N(-\frac{6}{t_2^2+3},\frac{2t_2}{t_2^2+3})}}$.
   设 ${\displaystyle MN}$ 与 ${\displaystyle x}$ 轴交于 ${\displaystyle P}$, 则 $$\begin{array}{r}{x}_P=-\frac{x_M{y}_N-x_N{y}_M}{y_N-y_M}=\frac{12(t_1-t_2)}{6(t_1-t_2)-2t_1{t}_2(t_1-t_2)}=-6,\end{array}$$
   如图，作${\displaystyle F_2,O}$在${\displaystyle MN}$上的垂足${\displaystyle F^{\prime },O^{\prime }}$, 则$${\displaystyle \frac{S_1}{S_2}=\frac{O{O}^{\prime }}{F_2{F}^{\prime }}=\frac{OQ}{F_{2}Q}=\frac{3}{4}.}$$