Question

20£®Èçͼ£¬ÒÑÖªÅ×ÎïÏß$y=-\frac{1}{3}{x^2}+\frac{2}{3}x+5$ÓëxÖá½»ÓÚA¡¢BÁ½µã£¨µãAÔÚµãBµÄÓҲࣩ£¬ÓëyÖá½»ÓÚµãC£¬ÓÐÒ»¿í¶ÈΪ1µÄ¿Ì¶È³ßÑØxÖá·½ÏòƽÒÆ£¬ÓëyÖáƽÐеÄÒ»×é¶Ô±ß½»Å×ÎïÏßÓÚµãPºÍµãQ£¬½»Ö±ÏßACÓÚµãMºÍµãN£¬½»xÖáÓÚµãEºÍµãF£®

£¨1£©ÇóµãA¡¢B¡¢CµÄ×ø±ê£»
£¨2£©µ±µãMºÍµãN¶¼ÔÚÏ߶ÎACÉÏʱ£¬Á¬½ÓEN£¬Èç¹ûµãEµÄ×ø±êΪ£¨4£¬0£©£¬Çósin¡ÏANEµÄÖµ£»
£¨3£©Ôڿ̶ȳßƽÒƹý³ÌÖУ¬µ±ÒÔµãP¡¢Q¡¢N¡¢MΪ¶¥µãµÄËıßÐÎÊÇƽÐÐËıßÐÎʱ£¬ÇóµãNµÄ×ø±ê£®

Answer 1

·ÖÎö £¨1£©ÀûÓÃ×ø±êÖáÉϵãµÄ×ø±êÌØÕ÷¼´¿É½áÂÛ£»
£¨2£©ÏÈÈ·¶¨³öAF=FN=2£¬GE=$\frac{\sqrt{2}}{2}$£¬ÔÙÀûÓù´¹É¶¨ÀíÇó³öNE=$\sqrt{5}$£¬¼´¿ÉµÃ³ö½áÂÛ£»
£¨3£©ÏÈÈ·¶¨³öÖ±ÏßACµÄº¯Êý±í´ïʽΪy=-x+5£®ÔÙ·ÖMNΪ±ßºÍ¶Ô½ÇÏßÁ½ÖÖÇé¿ö£¬½¨Á¢·½³ÌÇó½â¼´¿ÉµÃ³ö½áÂÛ£®

½â´ð ½â£º£¨1£©Áîy=0µÃ£º$-\frac{1}{3}{x^2}+\frac{2}{3}x+5$=0£¬½âµÃx=5»òx=-3£®
¡ßµãAÔÚµãBµÄÓҲ࣬
¡àµãA¡¢BµÄ×ø±ê·ÖΪ£¨5£¬0£©¡¢£¨-3£¬0£©£®
µ±x=0ʱ£¬y=5£¬
¡àµãCµÄ×ø±êΪ£¨0£¬5£©£®

£¨2£©Èçͼ1£¬×÷EG¡ÍAC£¬´¹×ãΪµãG£®

¡ßµãEµÄ×ø±êΪ£¨4£¬0£©£¬
¡àOE=4£®
¡ßOA=OC=5£¬
¡àAE=1£¬¡ÏOAC=45¡ã£®
¡àAF=FN=2£¬GE=AE•sin45¡ã=$\frac{\sqrt{2}}{2}$
ÔÚRt¡÷EFNÖУ¬ÒÀ¾Ý¹´¹É¶¨Àí¿ÉÖªNE=$\sqrt{E{F}^{2}+F{N}^{2}}$=$\sqrt{5}$£¬
¡àsin¡ÏANE=$\frac{GE}{EN}$=$\frac{\frac{\sqrt{2}}{2}}{\sqrt{5}}$=$\frac{\sqrt{10}}{10}$£¬
£¨3£©ÉèÖ±ÏßACµÄº¯Êý±í´ïʽΪy=kx+b£®
½«µãAºÍµãCµÄ×ø±ê´úÈëµÃ£º$\left\{\begin{array}{l}5k+b=0\\ b=5\end{array}\right.$£¬
½âµÃk=-1£¬b=5£®
¡àÖ±ÏßACµÄº¯Êý±í´ïʽΪy=-x+5£®
¢Ùµ±MNΪ±ßʱ£¬Èçͼ2Ëùʾ£º

ÉèµãQ£¨n£¬$-\frac{1}{3}{n^2}+\frac{2}{3}n+5$£©£¬
ÔòµãP£¨n+1£¬$-\frac{1}{3}{n^2}+\frac{16}{3}$£©£¬µãN£¨n£¬-n+5£©M£¨n+1£¬-n+4£©£®
¡ßQN=PM
¡à$£¨-\frac{1}{3}{n^2}+\frac{2}{3}n+5£©-£¨-n+5£©=£¨-\frac{1}{3}{n^2}+\frac{16}{3}£©-£¨-n+4£©$£¬½âµÃn=2£®
¡àµãNµÄ×ø±êΪ£¨2£¬3£©£®
¢Úµ±MNÊÇƽÐÐËıßÐεĶԽÇÏßʱ£¬Èçͼ3Ëùʾ£º

ÉèµãFµÄ×ø±êΪ£¨m£¬0£©£¬
ÔòN£¨m£¬-m+5£©£¬M£¨m+1£¬-m+4£©£¬
Q£¨m£¬$-\frac{1}{3}{m^2}+\frac{2}{3}m+5$£©£¬P£¨m+1£¬$-\frac{1}{3}{m^2}+\frac{16}{3}$£©£®
¡ßQN=PM£¬
¡à$£¨-m+5£©-£¨-\frac{1}{3}{m^2}+\frac{2}{3}m+5£©=£¨-\frac{1}{3}{m^2}+\frac{16}{3}£©-£¨-m+4£©$£¬½âµÃm=2¡À$\sqrt{6}$£®
¡àµãNµÄ×ø±êΪ£¨2+$\sqrt{6}$£¬3-$\sqrt{6}$£©»ò£¨2-$\sqrt{6}$£¬3+$\sqrt{6}$£©£®
×ÛÉÏËùÊö£¬ÒÔµãP¡¢Q¡¢N¡¢MΪ¶¥µãµÄËıßÐÎÊÇƽÐÐËıßÐÎʱ£¬µãNµÄ×ø±êΪ£¨2£¬3£©
»ò£¨2+$\sqrt{6}$£¬3-$\sqrt{6}$£©»ò£¨2-$\sqrt{6}$£¬3+$\sqrt{6}$£©£®

µãÆÀ ´ËÌâÊǶþ´Îº¯Êý×ÛºÏÌ⣬Ö÷Òª¿¼²éÁË´ý¶¨ÏµÊý·¨£¬¹´¹É¶¨Àí£¬Æ½ÐÐËıßÐεÄÐÔÖÊ£¬½â£¨2£©µÄ¹Ø¼üÊÇÇó³öNEµÄ³¤£¬½â£¨3£©µÄ¹Ø¼üÊÇ·ÖMNΪƽÐÐËıßÐεıߺͶԽÇÏßÁ½ÖÖÇé¿ö£¬Ó÷½³ÌµÄ˼Ïë½â¾öÎÊÌ⣬ÊÇÒ»µÀÖп¼³£¿¼Ì⣮