Question

5£®Èçͼ£¬ÔÚƽÃæÖ±½Ç×ø±êϵxOyÖУ¬ÍÖÔ²$\frac{x^2}{a^2}$+$\frac{y^2}{b^2}$=1£¨a£¾b£¾0£©µÄÁ½½¹µã·Ö±ðΪF₁£¨-$\sqrt{3}$£¬0£©£¬F₂£¨$\sqrt{3}$£¬0£©£¬ÇÒ¾­¹ýµã£¨$\sqrt{3}$£¬$\frac{1}{2}$£©£®
£¨1£©ÇóÍÖÔ²µÄ·½³Ì¼°ÀëÐÄÂÊ£»
£¨2£©ÉèµãB£¬C£¬DÊÇÍÖÔ²Éϲ»Í¬ÓÚÍÖÔ²¶¥µãµÄÈýµã£¬µãBÓëµãD¹ØÓÚÔ­µãO¶Ô³Æ£®ÉèÖ±ÏßCD£¬CB£¬OB£¬OCµÄбÂÊ·Ö±ðΪk₁£¬k₂£¬k₃£¬k₄£¬ÇÒk₁k₂=k₃k₄£®
¢ÙÇók₁k₂µÄÖµ£»
¢ÚÇóOB²+OC²µÄÖµ£®

Answer 1

·ÖÎö £¨1£©ÒÀÌâÒ⣬c=$\sqrt{3}$£¬a²=b²+3£¬£¨$\sqrt{3}$£¬$\frac{1}{2}$£©´úÈëÍÖÔ²·½³Ì£¬Çó³öa£¬b£¬¼´¿ÉÇóÍÖÔ²µÄ·½³Ì¼°ÀëÐÄÂÊ£»
£¨2£©¢ÙÀûÓÃбÂʹ«Ê½£¬¼´¿ÉÇók₁k₂µÄÖµ£»
¢ÚÓÉ¢ÙÖª£¬k₃k₄=k₁k₂=$-\frac{1}{4}$£¬¹Êx₁x₂=-4y₁y₂£®ÀûÓÃOB²+OC²=$x_1^2+y_1^2+x_2^2+y_2^2$£¬ÇóOB²+OC²µÄÖµ£®

½â´ð ½â£º£¨1£©ÒÀÌâÒ⣬c=$\sqrt{3}$£¬a²=b²+3£¬¡­2·Ö
ÓÉ$\frac{3}{{{b^2}+3}}+\frac{{\frac{1}{4}}}{b^2}=1$£¬½âµÃb²=1£¨b²=$-\frac{3}{4}$£¬²»ºÏ£¬ÉáÈ¥£©£¬´Ó¶øa²=4£®
¹ÊËùÇóÍÖÔ²·½³ÌΪ£º$\frac{x^2}{4}+{y^2}=1$£¬ÀëÐÄÂÊe=$\frac{{\sqrt{3}}}{2}$£®¡­5·Ö
£¨2£©¢ÙÉèB£¨x₁£¬y₁£©£¬C£¨x₂£¬y₂£©£¬ÔòD£¨-x₁£¬-y₁£©£¬
ÓÚÊÇk₁k₂=$\frac{{{y_2}-{y_1}}}{{{x_2}-{x_1}}}•\frac{{{y_2}+{y_1}}}{{{x_2}+{x_1}}}$=$\frac{{£¨1-\frac{x_2^2}{4}£©-£¨1-\frac{x_1^2}{4}£©}}{x_2^2-x_1^2}$=$-\frac{1}{4}$£®¡­8·Ö
¢ÚÓÉ¢ÙÖª£¬k₃k₄=k₁k₂=$-\frac{1}{4}$£¬¹Êx₁x₂=-4y₁y₂£®
ËùÒÔ£¨x₁x₂£©²=£¨-4y₁y₂£©²£¬¼´£¨x₁x₂£©²=$16£¨1-\frac{x_1^2}{4}£©£¨1-\frac{x_2^2}{4}£©$=$16-4£¨x_1^2+x_2^2£©+x_1^2x_2^2$£¬
ËùÒÔ£¬$x_1^2+x_2^2$=4£®¡­11·Ö
ÓÖ2=$£¨\frac{x_1^2}{4}+y_1^2£©+£¨\frac{x_2^2}{4}+y_2^2£©$=$\frac{x_1^2+x_2^2}{4}+y_1^2+y_2^2$£¬¹Ê$y_1^2+y_2^2=1$£®
ËùÒÔ£¬OB²+OC²=$x_1^2+y_1^2+x_2^2+y_2^2$=5£®¡­14·Ö

µãÆÀ ±¾Ì⿼²éÍÖÔ²·½³ÌÓëÐÔÖÊ£¬¿¼²éбÂʹ«Ê½µÄÔËÓ㬿¼²éѧÉúµÄ¼ÆËãÄÜÁ¦£¬ÊôÓÚÖеµÌ⣮