Question

1£®Èçͼ£¬Ö±Ïßy=$\frac{\sqrt{3}}{3}$x-$\sqrt{3}$ÓëxÖá½»ÓÚµãB£¬ÓëyÖá½»ÓÚµãC£¬Å×ÎïÏßy=ax²+bx-$\sqrt{3}$¾­¹ýµãA¡¢B¡¢C£¬ÇÒµãA×ø±êÊÇ£¨-1£¬0£©£¬µãDÊÇÖ±ÏßBCÏ·½Å×ÎïÏßÉϵÄÒ»¶¯µã£®
£¨1£©ÇóÅ×ÎïÏߵĽâÎöʽ£»
£¨2£©µ±ËıßÐÎABDCÃæ»ý×î´óʱ£¬ÇëÇó³öµãDµÄ×ø±êºÍËıßÐÎABDCÃæ»ýµÄ×î´óÖµ£¿
£¨3£©ÉèÅ×ÎïÏߵĶԳÆÖáÓëxÖáÏཻÓÚµãE£¬ÔÚÉäÏßCEÉÏÊÇ·ñ´æÔÚµãP£¬Ê¹µÃ¡÷ABPÊÇÖ±½ÇÈý½ÇÐΣ¿Èç¹û´æÔÚ£¬ÇëÖ±½Óд³öAPµÄ³¤¶È£»Èç¹û²»´æÔÚ£¬Çë˵Ã÷ÀíÓÉ£®

Answer 1

·ÖÎö £¨1£©¸ù¾Ý×Ô±äÁ¿Ó뺯ÊýÖµµÄ¶ÔÓ¦¹Øϵ£¬¿ÉµÃBµã×ø±ê£¬¸ù¾Ý´ý¶¨ÏµÊý·¨£¬¿ÉµÃº¯Êý½âÎöʽ£»
£¨2£©¸ù¾ÝƽÐÐÓÚyÖáÖ±ÏßÉÏÁ½µã¼äµÄ¾àÀëÊǽϴóµÄ×Ý×ø±ê¼ä½ÏСµÄ×Ý×ø±ê£¬¿ÉµÃDFµÄ³¤£¬¸ù¾ÝÃæ»ýµÄºÍ²î£¬¿ÉµÃ¶þ´Îº¯Êý£¬¸ù¾Ý¶þ´Îº¯ÊýµÄÐÔÖÊ£¬¿ÉµÃ´ð°¸£»
£¨3£©¸ù¾Ý¹´¹É¶¨Àí£¬¿ÉµÃ¹ØÓÚnµÄ·½³Ì£¬¸ù¾Ý½â·½³Ì£¬¿ÉµÃnµÄÖµ£¬ÔÙ¸ù¾Ý¹´¹É¶¨Àí£¬¿ÉµÃ´ð°¸£®

½â´ð ½â£º£¨1£©ÓÉy=$\frac{\sqrt{3}}{3}$x-$\sqrt{3}$£¬µÃy=0ʱ£¬x=3£¬Bµã×ø±êΪ£¨3£¬0£©£¬
½«A¡¢Bµã×ø±ê´úÈ뺯Êý½âÎöʽ£¬µÃ
$\left\{\begin{array}{l}{a-b-\sqrt{3}=0}\\{9a+3b-\sqrt{3}=0}\end{array}\right.$£¬½âµÃ$\left\{\begin{array}{l}{a=\frac{\sqrt{3}}{3}}\\{b=-\frac{2\sqrt{3}}{3}}\end{array}\right.$£¬
y=$\frac{\sqrt{3}}{3}$x²-$\frac{2\sqrt{3}}{3}$x-$\sqrt{3}$£»
£¨2£©Èçͼ1£¬¹ýµãD×÷DE¡ÍxÖáÓÚµãE½»BCÓÚFµã£¬
ÉèDµãµÄ×ø±êΪ£¨m£¬$\frac{\sqrt{3}}{3}$m²-$\frac{2\sqrt{3}}{3}$m-$\sqrt{3}$£©µãFµÄ×ø±êΪ£¨m£¬$\frac{\sqrt{3}}{3}$m-$\sqrt{3}$£©
DF=£¨$\frac{\sqrt{3}}{3}$m-$\sqrt{3}$£©-£¨$\frac{\sqrt{3}}{3}$m²-$\frac{2\sqrt{3}}{3}$m-$\sqrt{3}$£©=-$\frac{\sqrt{3}}{3}$m²+$\sqrt{3}$m£¬
S_{ËıßÐÎABDC}=S_¡÷DFB+S_¡÷DFC+S_¡÷ABC
=$\frac{1}{2}$DF•OB+$\frac{1}{2}$¡Á4¡Á$\sqrt{3}$
=$\frac{1}{2}$£¨-$\frac{\sqrt{3}}{3}$m²+$\sqrt{3}$m£©¡Á3+2$\sqrt{3}$
=-$\frac{\sqrt{3}}{2}$£¨m-$\frac{3}{2}$£©²+$\frac{25\sqrt{3}}{8}$£¨0£¼m£¼3£©£¬
¡àµ±m=$\frac{3}{2}$ʱ£¬ËıßÐÎABDCµÄÃæ»ýÈ¡µÃ×î´óÖµ$\frac{25\sqrt{3}}{8}$£¬´ËʱµãDµÄ×ø±êΪ£¨$\frac{3}{2}$£¬-$\frac{5\sqrt{3}}{4}$£©£»
£¨3£©y=$\frac{\sqrt{3}}{3}$x²-$\frac{2\sqrt{3}}{3}$x-$\sqrt{3}$=$\frac{\sqrt{3}}{3}$£¨x-1£©²-$\frac{4\sqrt{3}}{3}$£¬
¶Ô³ÆÖáÓÚxÖáµÄ½»µãEΪ£¨1£¬0£©£¬
CEµÄ½âÎöʽΪy=$\sqrt{3}$x-$\sqrt{3}$£¬
ÉèPµã×ø±êΪ£¨n£¬$\sqrt{3}$n-$\sqrt{3}$£©£¬PÔÚCEµÄÉäÏßÉÏ£¬¡à¡ÏPAB£¼90¡ã£¬
AP²=£¨n+1£©²+£¨$\sqrt{3}n-\sqrt{3}$£©²£¬PB²=£¨n-3£©²+£¨$\sqrt{3}$n-$\sqrt{3}$£©²
¢Ùµ±¡ÏAPB=90¡ãʱ£¬AP²+PB²=AB²£¬
¼´£¨n+1£©²+£¨$\sqrt{3}$n-$\sqrt{3}$£©²+£¨n-3£©²+£¨$\sqrt{3}$n-$\sqrt{3}$£©²=4²£¬
»¯¼ò£¬µÃ8n²-16n=0£¬
½âµÃn₁=0£¬n₂=2£¬
µ±n=0ʱ£¬AP=$\sqrt{£¨0+1£©^{2}+£¨-\sqrt{3}£©^{2}}$=2£¬
µ±n=2ʱ£¬AP=$\sqrt{£¨2+1£©^{2}+£¨2\sqrt{3}-\sqrt{3}£©^{2}}$=2$\sqrt{3}$£»
¢Úµ±¡ÏABP=90¡ãʱ£¬BP¡ÍABÓÚB£¬µÃ
n=3£¬
AP=$\sqrt{£¨3+1£©^{2}+£¨3\sqrt{3}-\sqrt{3}£©^{2}}$=2$\sqrt{7}$£¬
×ÛÉÏËùÊö£ºAPµÄ³¤Îª2£¬2$\sqrt{3}$£¬2$\sqrt{7}$£®

µãÆÀ ±¾Ì⿼²éÁ˶þ´Îº¯Êý×ÛºÏÌ⣬½â£¨1£©µÄ¹Ø¼üÊÇ´ý¶¨ÏµÊý·¨£¬½â£¨2£©µÄ¹Ø¼üÊÇÀûÓÃÃæ»ýµÄºÜ²îµÃ³ö¶þ´Îº¯Êý£¬ÓÖÀûÓÃÁ˶þ´Îº¯ÊýµÄÐÔÖÊ£»½â£¨3£©µÄ¹Ø¼üÊǹ´¹É¶¨ÀíµÃ³ö¹ØÓÚnµÄ·½³Ì£¬Òª·ÖÀàÌÖÂÛ£¬ÒÔ·ÀÒÅ©£®