Question

11£®ÒÑÖªÍÖÔ²C£º$\frac{{x}^{2}}{{a}^{2}}$+$\frac{{y}^{2}}{{b}^{2}}$=1£¨a£¾b£¾0£©µÄÀëÐÄÂÊΪ$\frac{2\sqrt{2}}{3}$£¬ÇÒÍÖÔ²ÉÏÒ»µãÓëÍÖÔ²µÄÁ½¸ö½¹µã¹¹³ÉµÄÈý½ÇÐεÄÖܳ¤Îª6+4$\sqrt{2}$£®
£¨¢ñ£©ÇóÍÖÔ²CµÄ·½³Ì£»
£¨¢ò£©ÉèÖ±Ïßl£ºx=ky+mÓëÍÖÔ²C½»ÊÖA¡¢BÁ½µã£¬ÈôÒÔABΪֱ¾¶µÄÔ²¾­¹ýÍÖÔ²µÄÓÒ¶¥µãD£¬Çó¡÷ABDÃæ»ýµÄ×î´óÖµ£®

Answer 1

·ÖÎö £¨I£©¸ù¾ÝÍÖÔ²µÄ¶¨Òå·½³Ì£¬µÃ³ö$\frac{c}{a}$=$\frac{2\sqrt{2}}{3}$£¬2a$+2c=6+4\sqrt{2}$£¬Çó½â¼´¿ÉµÃ³ö·½³Ì£®
£¨II£©²»·ÁÉèABµÄ·½³ÌΪx=ky+m£¬ÓëÍÖÔ²·½³ÌÁªÁ¢£¬¸ù¾ÝÒÔABΪֱ¾¶µÄÔ²¹ýµãD£¬¿ÉµÃ $\overrightarrow{DA}$•$\overrightarrow{DB}$=0£¬´Ó¶ø¿ÉÇómµÄÖµ£¬½ø¶ø¿É±íʾÈý½ÇÐεÄÃæ»ý£¬»»Ôª£¬¼´¿ÉÇóµÃABCÃæ»ýµÄ×î´óÖµ

½â´ð ½â£º£¨I£©¡ßÍÖÔ²C£º$\frac{{x}^{2}}{{a}^{2}}$+$\frac{{y}^{2}}{{b}^{2}}$=1£¨a£¾b£¾0£©µÄÀëÐÄÂÊΪ$\frac{2\sqrt{2}}{3}$£¬ÇÒÍÖÔ²ÉÏÒ»µãÓëÍÖÔ²µÄÁ½¸ö½¹µã¹¹³ÉµÄÈý½ÇÐεÄÖܳ¤Îª6+4$\sqrt{2}$£¬
¡à$\frac{c}{a}$=$\frac{2\sqrt{2}}{3}$£¬2a$+2c=6+4\sqrt{2}$£¬
¡àa=3£¬c=2$\sqrt{2}$£¬b=1£¬
¼´ÍÖÔ²CµÄ·½³ÌΪ$\frac{{x}^{2}}{9}$$+\frac{{y}^{2}}{1}$=1£¬
£¨II£©²»·ÁÉèABµÄ·½³ÌΪx=ky+m£®
ÓÉ$\left\{\begin{array}{l}{x=ky+m}\\{\frac{{x}^{2}}{9}+{y}^{2}=1}\end{array}\right.$ÏûÈ¥xµÃ£¨k²+9£©y²+2kmy+m²-9=0£¬
ÉèA£¨x₁£¬y₁£©£¬B£¨x₂£¬y₂£©£¬ÔòÓÐy₁+y₂=-$\frac{2km}{{k}^{2}+9}$£¬y₁y₂=$\frac{{m}^{2}-9}{{k}^{2}+9}$¢Ù
ÒòΪÒÔABΪֱ¾¶µÄÔ²¹ýµãD£¬ËùÒÔ $\overrightarrow{DA}$•$\overrightarrow{DB}$=0
ÓÉ $\overrightarrow{DA}$=£¨x₁-3£¬y₁£©£¬$\overrightarrow{DB}$=£¨x₂-3£¬y₂£©£¬µÃ£¨x₁-3£©£¨x₂-3£©+y₁y₂=0£®
½«x₁=ky₁+m£¬x₂=ky₂+m´úÈëÉÏʽ£¬
µÃ£¨k²+1£©y₁y₂+k£¨m-3£©£¨y₁+y₂£©+£¨m-3£©²=0£®
½«¢Ù´úÈëÉÏʽ£¬½âµÃm=$\frac{12}{5}$»òm=3£¨Éᣩ£®
ËùÒÔm=$\frac{12}{5}$£¨´ËʱֱÏßAB¾­¹ý¶¨µãD£¨$\frac{12}{5}$£¬0£©£¬ÓëÍÖÔ²ÓÐÁ½¸ö½»µã£©£®
ËùÒÔS_¡÷ABC=$\frac{1}{2}$S¡÷ABC=$\frac{1}{2}$|AB||y₁-y₂|=$\frac{1}{2}¡Á$ $\frac{3}{5}$$\sqrt{£¨{y}_{1}+{y}_{2}£©^{2}-4{y}_{1}{y}_{2}}$=$\frac{9}{5}$£®
Éèt=$\frac{1}{{k}^{2}+9}$£¬0$£¼t¡Ü\frac{1}{9}$£¬ÔòS_¡÷ABC=$\frac{9}{5}$£®
ËùÒÔµ±t=$\frac{25}{288}$ʱ£¬S_¡÷ABCÈ¡µÃ×î´óÖµ$\frac{3}{8}$£®

µãÆÀ ±¾Ì⿼²éÍÖÔ²µÄ±ê×¼·½³Ì£¬¿¼²éÈý½ÇÐÎÃæ»ýµÄ¼ÆË㣬¿¼²éÖ±ÏßÓëÍÖÔ²·½³ÌµÄÁªÁ¢£¬ÕýÈ·±íʾÈý½ÇÐεÄÃæ»ýÊǹؼü