ÌâÄ¿ÄÚÈÝ
19£®ÒÑÖªÍÖÔ²C£º$\frac{{x}^{2}}{{a}^{2}}$+$\frac{{y}^{2}}{{b}^{2}}$=1£¨a£¾b£¾0£©¹ýµã£¨1£¬$\frac{\sqrt{3}}{2}$£©£¬ÀëÐÄÂÊΪ$\frac{\sqrt{3}}{2}$£¬¹ýÍÖÔ²ÓÒ¶¥µãAµÄÁ½ÌõбÂʳ˻ýΪ-$\frac{1}{4}$µÄÖ±Ïß·Ö±ð½»ÍÖÔ²CÓÚM£¬NÁ½µã£®£¨I£©ÇóÍÖÔ²CµÄ±ê×¼·½³Ì£»
£¨¢ò£©Ö±ÏßMNÊÇ·ñ¹ý¶¨µãD£¿Èô¹ý¶¨µãD£¬Çó³öµãDµÄ×ø±ê£»Èô²»¹ý£¬Çë˵Ã÷ÀíÓÉ£®
·ÖÎö £¨I£©ÓÉÍÖÔ²C£º$\frac{{x}^{2}}{{a}^{2}}$+$\frac{{y}^{2}}{{b}^{2}}$=1£¨a£¾b£¾0£©¹ýµã£¨1£¬$\frac{\sqrt{3}}{2}$£©£¬ÀëÐÄÂÊΪ$\frac{\sqrt{3}}{2}$£¬½¨Á¢·½³Ì£¬Çó³öa£¬b£¬¼´¿ÉÇóÍÖÔ²CµÄ±ê×¼·½³Ì£»
£¨¢ò£©ÉèAM¡¢ANµÄ·½³Ì£¬´úÈëÍÖÔ²·½³Ì£¬Çó³öM£¬NµÄ×ø±ê£¬½ø¶ø¿ÉµÃMNµÄ·½³Ì£¬¼´¿ÉµÃ³ö½áÂÛ£®
½â´ð ½â£º£¨I£©ÓÉÒÑÖª$\left\{\begin{array}{l}{\frac{c}{a}=\frac{\sqrt{3}}{2}}\\{\frac{1}{{a}^{2}}+\frac{3}{4{b}^{2}}=1}\end{array}\right.$£¬¡àa=2£¬b=1£¬
¡àÍÖÔ²CµÄ±ê×¼·½³ÌΪ$\frac{{x}^{2}}{4}+{y}^{2}=1$£»
£¨¢ò£©Ö±ÏßMN¹ý¶¨µãD£¨0£¬0£©£®
Ö¤Ã÷ÈçÏ£ºÓÉÌâÒ⣬A£¨2£¬0£©£¬Ö±ÏßAMºÍÖ±ÏßANµÄбÂÊ´æÔÚÇÒ²»Îª0£¬
ÉèAMµÄ·½³ÌΪy=k£¨x-2£©£¬´úÈëÍÖÔ²·½³ÌµÃ£¨1+4k2£©x2-16k2x+16k2-4=0
¡à2xM=$\frac{16{k}^{2}-4}{1+4{k}^{2}}$£¬
¡àxM=$\frac{8{k}^{2}-2}{1+4{k}^{2}}$£¬
¡àyM=k£¨xM-2£©=$\frac{-4k}{1+4{k}^{2}}$£¬
¡àM£¨$\frac{8{k}^{2}-2}{1+4{k}^{2}}$£¬$\frac{-4k}{1+4{k}^{2}}$£©£¬
¡ßÍÖÔ²ÓÒ¶¥µãAµÄÁ½ÌõбÂʳ˻ýΪ-$\frac{1}{4}$µÄÖ±Ïß·Ö±ð½»ÍÖÔ²CÓÚM£¬NÁ½µã£¬
¡àÉèÖ±ÏßANµÄ·½³ÌΪy=-$\frac{1}{4k}$£¨x-2£©£¬
ͬÀí¿ÉµÃN£¨$\frac{2-8{k}^{2}}{1+4{k}^{2}}$£¬$\frac{4k}{1+4{k}^{2}}$£©£¬
xM¡ÙxN£¬¼´k$¡Ù¡À\frac{1}{2}$ʱ£¬kMN=$\frac{2k}{1-4{k}^{2}}$£¬
¡àÖ±ÏßMNµÄ·½³ÌΪy-$\frac{4k}{1+4{k}^{2}}$=$\frac{2k}{1-4{k}^{2}}$£¨x-$\frac{2-8{k}^{2}}{1+4{k}^{2}}$£©£¬¼´y=$\frac{2k}{1-4{k}^{2}}$x£¬
¡àÖ±ÏßMN¹ý¶¨µãD£¨0£¬0£©£®
xM=xN£¬¼´k=$\frac{1}{2}$ʱ£¬Ö±ÏßMN¹ý¶¨µãD£¨0£¬0£©£®
×ÛÉÏËùÊö£¬Ö±ÏßMN¹ý¶¨µãD£¨0£¬0£©£®
µãÆÀ ±¾Ì⿼²éÍÖÔ²·½³Ì£¬¿¼²éÖ±ÏßÓëÍÖÔ²µÄλÖùØϵ£¬¿¼²éÖ±Ïß¹ý¶¨µã£¬¿¼²éѧÉú·ÖÎö½â¾öÎÊÌâµÄÄÜÁ¦£¬ÊôÓÚÖеµÌ⣮