Question

2£®Èçͼ£¬ÔÚƽÃæÖ±½Ç×ø±êϵxOyÖУ¬ÍÖÔ²E£º$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1£¨a£¾b£¾0£©$µÄ×󶥵ãΪA£¬ÓëxÖáƽÐеÄÖ±ÏßÓëÍÖÔ²E½»ÓÚB¡¢CÁ½µã£¬¹ýB¡¢CÁ½µãÇÒ·Ö±ðÓëÖ±ÏßAB¡¢AC´¹Ö±µÄÖ±ÏßÏཻÓÚµãD£®ÒÑÖªÍÖÔ²EµÄÀëÐÄÂÊΪ$\frac{{\sqrt{5}}}{3}$£¬ÓÒ½¹µãµ½ÓÒ×¼ÏߵľàÀëΪ$\frac{{4\sqrt{5}}}{5}$£®
£¨1£©ÇóÍÖÔ²EµÄ±ê×¼·½³Ì£»
£¨2£©Ö¤Ã÷µãDÔÚÒ»Ìõ¶¨Ö±ÏßÉÏÔ˶¯£¬²¢Çó³ö¸ÃÖ±Ïߵķ½³Ì£»
£¨3£©Çó¡÷BCDÃæ»ýµÄ×î´óÖµ£®

Answer 1

·ÖÎö £¨1£©ÀûÓÃ$\frac{c}{a}=\frac{{\sqrt{5}}}{3}$£¬$\frac{a^2}{c}-c=\frac{{4\sqrt{5}}}{5}$£¬¼ÆËã¼´¿É£»
£¨2£©Í¨¹ýÉèB¡¢Cµã×ø±ê¡¢Ð´³öÖ±ÏßAB¡¢AC¡¢BD¡¢CDµÄбÂÊ£¬ÁªÁ¢Ö±ÏßBD¡¢CDµÄ·½³Ì£¬¼ÆËã¼´¿É£»
£¨3£©Í¨¹ý¼ÆËã¿ÉµÃµãDµÄ×Ý×ø±ê£¬½ø¶ø¿ÉµÃµãDµ½Ö±ÏßBCµÄ¾àÀ룬ÀûÓÃÈý½ÇÐεÄÃæ»ý¹«Ê½¼°»ù±¾²»µÈʽ¼´µÃ½áÂÛ£®

½â´ð £¨1£©½â£ºÓÉÌâÒâµÃ$\frac{c}{a}=\frac{{\sqrt{5}}}{3}$£¬$\frac{a^2}{c}-c=\frac{{4\sqrt{5}}}{5}$£¬
½âµÃ$a=3£¬c=\sqrt{5}$£¬
¡àb²=a²-c²=4£¬
¡àÍÖÔ²EµÄ±ê×¼·½³ÌΪ$\frac{x^2}{9}+\frac{y^2}{4}=1$£®
£¨2£©Ö¤Ã÷£ºÉèB£¨x₀£¬y₀£©£¬C£¨-x₀£¬y₀£©£¬ÏÔȻֱÏßAB£¬AC£¬BD£¬CDµÄбÂʶ¼´æÔÚ£¬
ÉèΪk₁£¬k₂£¬k₃£¬k₄£¬Ôò${k_1}=\frac{y_0}{{{x_0}+3}}£¬{k_2}=\frac{y_0}{{-{x_0}+3}}$£¬${k_3}=-\frac{{{x_0}+3}}{y_0}£¬{k_4}=\frac{{{x_0}-3}}{y_0}$£¬
¡àÖ±ÏßBD£¬CDµÄ·½³ÌΪ£º$y=-\frac{{{x_0}+3}}{y_0}£¨x-{x_0}£©+{y_0}£¬y=\frac{{{x_0}-3}}{y_0}£¨x+{x_0}£©+{y_0}$£¬
ÏûÈ¥yµÃ£º$-\frac{{{x_0}+3}}{y_0}£¨x-{x_0}£©+{y_0}=\frac{{{x_0}-3}}{y_0}£¨x+{x_0}£©+{y_0}$£¬
»¯¼òµÃx=3£¬¹ÊµãDÔÚ¶¨Ö±Ïßx=3ÉÏÔ˶¯£®
£¨3£©½â£ºÓÉ£¨2£©µÃµãDµÄ×Ý×ø±êΪ${y_D}=\frac{{{x_0}-3}}{y_0}£¨3+{x_0}£©+{y_0}=\frac{x_0^2-9}{y_0}+{y_0}$£¬
ÓÖ¡ß$\frac{x_0^2}{9}+\frac{y_0^2}{4}=1$£¬¡à$x_0^2-9=-\frac{9y_0^2}{4}$£¬
Ôò${y_D}=\frac{{{x_0}-3}}{y_0}£¨3+{x_0}£©+{y_0}=\frac{{-\frac{9}{4}y_0^2}}{y_0}+{y_0}=-\frac{5}{4}{y_0}$£¬
¡àµãDµ½Ö±ÏßBCµÄ¾àÀëh=$|{{y_D}-{y_0}}|=|{-\frac{5}{4}{y_0}-{y_0}}|=\frac{9}{4}|{y_0}|$£¬
½«y=y₀´úÈë$\frac{x^2}{9}+\frac{y^2}{4}=1$£¬µÃ$x=¡À3\sqrt{1-\frac{y_0^2}{4}}$£¬
¡à¡÷BCDÃæ»ý${S_{¡÷ABC}}=\frac{1}{2}BC•h=\frac{1}{2}¡Á6\sqrt{1-\frac{y_0^2}{4}}•\frac{9}{4}|{y_0}|$
=$\frac{27}{2}\sqrt{1-\frac{y_0^2}{4}}•\frac{1}{2}|{y_0}|¡Ü\frac{27}{2}•\frac{{1-\frac{y_0^2}{4}+\frac{y_0^2}{4}}}{2}=\frac{27}{4}$£¬
µ±ÇÒ½öµ±$1-\frac{y_0^2}{4}=\frac{y_0^2}{4}$£¬¼´${y_0}=¡À\sqrt{2}$ʱµÈºÅ³ÉÁ¢£¬
¹Ê${y_0}=¡À\sqrt{2}$ʱ£¬¡÷BCDÃæ»ýµÄ×î´óֵΪ$\frac{27}{4}$£®

µãÆÀ ±¾Ì⿼²éÍÖÔ²µÄ¶¨Òå¼°Æä±ê×¼·½³Ì¡¢Ö±ÏßÓëÍÖÔ²µÄλÖùØϵ¡¢Èý½ÇÐεÄÃæ»ý¼ÆËãµÈ»ù´¡ÖªÊ¶£¬¿¼²éÔËËãÇó½âÄÜÁ¦£¬×¢Òâ½âÌâ·½·¨µÄ»ýÀÛ£¬ÊôÓÚÖеµÌ⣮