Question

2£®ÒÑÖªÍÖÔ²$C£º\frac{x^2}{a^2}+\frac{y^2}{b^2}=1£¨a£¾b£¾0£©$µÄ×óÓÒ½¹µãΪF₁£¬F₂£¬ÆäÀëÐÄÂÊΪ$\frac{{\sqrt{2}}}{2}$£¬ÓÖÅ×ÎïÏßx²=4yÔÚµãP£¨2£¬1£©´¦µÄÇÐÏßÇ¡ºÃ¹ýÍÖÔ²CµÄÒ»¸ö½¹µã£®
£¨1£©ÇóÍÖÔ²CµÄ·½³Ì£»
£¨2£©¹ýµãM£¨-4£¬0£©Ð±ÂÊΪk£¨k¡Ù0£©µÄÖ±Ïßl½»ÍÖÔ²CÓÚA£¬BÁ½µã£¬Ö±ÏßAF₁£¬BF₁µÄбÂÊ·Ö±ðΪk₁£¬k₂£¬ÊÇ·ñ´æÔÚ³£Êý¦Ë£¬Ê¹µÃk₁k+k₂k=¦Ëk₁k₂£¿Èô´æÔÚ£¬Çó³ö¦ËµÄÖµ£»Èô²»´æÔÚ£¬Çë˵Ã÷ÀíÓÉ£®

Answer 1

·ÖÎö £¨1£©ÍƵ¼³öÅ×ÎïÏß¹ýxÖáÉÏ£¨1£¬0£©µã£¬´Ó¶øc=1£¬ÔÙÓÉÀëÐÄÂÊÄÜÇó³ö$a=\sqrt{2}£¬b=1$£¬ÓÉ´ËÄÜÇó³öÍÖÔ²CµÄ·½³Ì£®
£¨2£©ÉèlµÄ·½³ÌΪy=k£¨x+4£©£¬ÁªÁ¢$\left\{{\begin{array}{l}{y=k£¨x+4£©}\\{{x^2}+2{y^2}=2}\end{array}}\right.⇒£¨1+2{k^2}£©{x^2}+16{k^2}x+32{k^2}-2=0$£¬ÓÉ´ËÀûÓøùµÄÅбðʽ¡¢Î¤´ï¶¨Àí£¬½áºÏÒÑÖªÌõ¼þÄÜÇó³ö³£Êý$¦Ë=\frac{2}{7}$£®

½â´ð £¨1£©¡ßÅ×ÎïÏßx²=4yÔÚµãP£¨2£¬1£©´¦µÄÇÐÏß·½³ÌΪy=x-1£¬
¡àËü¹ýxÖáÉÏ£¨1£¬0£©µã£¬
¡àÍÖÔ²CµÄÒ»¸ö½¹µãΪ£¨1£¬0£©¼´c=1
ÓÖ¡ß$e=\frac{c}{a}=\frac{{\sqrt{2}}}{2}$£¬
¡à$a=\sqrt{2}£¬b=1$£¬
¡àÍÖÔ²CµÄ·½³ÌΪ$\frac{x^2}{2}+{y^2}=1$
£¨2£©ÉèA£¨x₁£¬y₁£©£¬B£¨x₂£¬y₂£©£¬lµÄ·½³ÌΪy=k£¨x+4£©£¬
ÁªÁ¢$\left\{{\begin{array}{l}{y=k£¨x+4£©}\\{{x^2}+2{y^2}=2}\end{array}}\right.⇒£¨1+2{k^2}£©{x^2}+16{k^2}x+32{k^2}-2=0$£¬
¡à$\left\{{\begin{array}{l}{¡÷£¾0}\\{{x_1}+{x_2}=-\frac{{16{k^2}}}{{1+2{k^2}}}}\\{{x_1}{x_2}=\frac{{32{k^2}-2}}{{1+2{k^2}}}}\end{array}}\right.$£¬¡ß${F_1}£¨-1£¬0£©£¬{k_1}=\frac{y_1}{{{x_1}+1}}£¬{k_2}=\frac{y_2}{{{x_2}+1}}$£¬
¡à$\frac{1}{k_1}+\frac{1}{k_2}=\frac{{{x_1}+1}}{y_1}+\frac{{{x_2}+1}}{y_2}=\frac{1}{k}£¨\frac{{{x_1}+1}}{{{x_1}+4}}+\frac{{{x_2}+1}}{{{x_2}+4}}£©$£¬
¡à$\frac{k}{{k_1^{\;}}}+\frac{k}{k_2}=\frac{{2{x_1}{x_2}+5£¨{x_1}+{x_2}£©+8}}{{{x_1}{x_2}+4£¨{x_1}+{x_2}£©+16}}=\frac{2}{7}$£¬
¡à${k_1}k+{k_2}k=\frac{2}{7}{k_1}{k_2}$£¬
¡à´æÔÚ³£Êý$¦Ë=\frac{2}{7}$£®

µãÆÀ ±¾Ì⿼²éÍÖÔ²·½³ÌÇ󷨣¬¿¼²éÂú×ãÌõ¼þµÄʵÊýÖµµÄÇ󷨣¬¿¼²éÍÖÔ²¡¢Î¤´ï¶¨Àí¡¢¸ùµÄÅбðʽ¡¢Ö±Ïß·½³Ì¡¢ÏÒ³¤¹«Ê½µÈ»ù´¡ÖªÊ¶£¬¿¼²éÍÆÀíÂÛÖ¤ÄÜÁ¦¡¢ÔËËãÇó½âÄÜÁ¦£¬¿¼²é»¯¹éÓëת»¯Ë¼Ïë¡¢º¯ÊýÓë·½³Ì˼Ï룬ÊÇÖеµÌ⣮