Question

17£®Èçͼ£¬Å×ÎïÏßy=$\frac{2}{3}$x²+bx+c¾­¹ýµãB£¨3£¬0£©£¬C£¨0£¬-2£©£¬Ö±Ïßl£ºy=-$\frac{2}{3}$x-$\frac{2}{3}$½»yÖáÓÚµãE£¬ÇÒÓëÅ×ÎïÏß½»ÓÚA£¬DÁ½µã£¬PΪÅ×ÎïÏßÉÏÒ»¶¯µã£¨²»ÓëA£¬DÖغϣ©£®
£¨1£©ÇóÅ×ÎïÏߵĽâÎöʽ£»
£¨2£©µ±µãPÔÚÖ±ÏßlÏ·½Ê±£¬¹ýµãP×÷PM¡ÎxÖá½»lÓÚµãM£¬PN¡ÎyÖá½»lÓÚµãN£¬ÇóPM+PNµÄ×î´óÖµ£®
£¨3£©ÉèFΪֱÏßlÉϵĵ㣬ÒÔE£¬C£¬P£¬FΪ¶¥µãµÄËıßÐÎÄÜ·ñ¹¹³ÉƽÐÐËıßÐΣ¿ÈôÄÜ£¬Çó³öµãFµÄ×ø±ê£»Èô²»ÄÜ£¬Çë˵Ã÷ÀíÓÉ£®

Answer 1

·ÖÎö £¨1£©°ÑB£¨3£¬0£©£¬C£¨0£¬-2£©´úÈëy=$\frac{2}{3}$x²+bx+c½â·½³Ì×é¼´¿ÉµÃµ½½áÂÛ£»
£¨2£©ÉèP£¨m£¬$\frac{2}{3}$m²-$\frac{4}{3}$m-2£©£¬µÃµ½N£¨m£¬-$\frac{2}{3}$m-$\frac{2}{3}$£©£¬M£¨-m²+2m+2£¬$\frac{2}{3}$m²-$\frac{4}{3}$m-2£©£¬¸ù¾Ý¶þ´Îº¯ÊýµÄÐÔÖʼ´¿ÉµÃµ½½áÂÛ£»
£¨3£©ÇóµÃE£¨0£¬-$\frac{2}{3}$£©£¬µÃµ½CE=$\frac{4}{3}$£¬ÉèP£¨m£¬$\frac{2}{3}$m²-$\frac{4}{3}$m-2£©£¬¢ÙÒÔCEΪ±ß£¬¸ù¾ÝCE=PF£¬Áз½³ÌµÃµ½m=1£¬m=0£¨ÉáÈ¥£©£¬¢ÚÒÔCEΪ¶Ô½ÇÏߣ¬Á¬½ÓPF½»CEÓÚG£¬CG=GE£¬PG=FG£¬µÃµ½G£¨0£¬-$\frac{4}{3}$£©£¬ÉèP£¨m£¬$\frac{2}{3}$m²-$\frac{4}{3}$m-2£©£¬ÔòF£¨-m£¬$\frac{2}{3}$m-$\frac{2}{3}$£©£¬Áз½³ÌµÃµ½´Ë·½³ÌÎÞʵÊý¸ù£¬ÓÚÊǵõ½½áÂÛ£®

½â´ð ½â£º£¨1£©°ÑB£¨3£¬0£©£¬C£¨0£¬-2£©´úÈëy=$\frac{2}{3}$x²+bx+cµÃ£¬$\left\{\begin{array}{l}{\frac{2}{3}¡Á{3}^{2}+3b+c=0}\\{c=-2}\end{array}\right.$£¬
¡à$\left\{\begin{array}{l}{b=-\frac{4}{3}}\\{c=-2}\end{array}\right.$
¡àÅ×ÎïÏߵĽâÎöʽΪ£ºy=$\frac{2}{3}$x²-$\frac{4}{3}$x-2£»
£¨2£©ÉèP£¨m£¬$\frac{2}{3}$m²-$\frac{4}{3}$m-2£©£¬
¡ßPM¡ÎxÖᣬPN¡ÎyÖᣬM£¬NÔÚÖ±ÏßADÉÏ£¬
¡àN£¨m£¬-$\frac{2}{3}$m-$\frac{2}{3}$£©£¬M£¨-m²+2m+2£¬$\frac{2}{3}$m²-$\frac{4}{3}$m-2£©£¬
¡àPM+PN=-m²+2m+2-m-$\frac{2}{3}$m-$\frac{2}{3}$-$\frac{2}{3}$m²+$\frac{4}{3}$m+2=-$\frac{5}{3}$m²+$\frac{5}{3}$m+$\frac{10}{3}$=-$\frac{5}{3}$£¨m-$\frac{1}{2}$£©²+$\frac{15}{4}$£¬
¡àµ±m=$\frac{1}{2}$ʱ£¬PM+PNµÄ×î´óÖµÊÇ$\frac{15}{4}$£»
£¨3£©ÄÜ£¬
ÀíÓÉ£º¡ßy=-$\frac{2}{3}$x-$\frac{2}{3}$½»yÖáÓÚµãE£¬
¡àE£¨0£¬-$\frac{2}{3}$£©£¬
¡àCE=$\frac{4}{3}$£¬
ÉèP£¨m£¬$\frac{2}{3}$m²-$\frac{4}{3}$m-2£©£¬
ÈôÒÔE£¬C£¬P£¬FΪ¶¥µãµÄËıßÐÎÄܹ¹³ÉƽÐÐËıßÐΣ¬
¢ÙÒÔCEΪ±ß£¬¡àCE¡ÎPF£¬CE=PF£¬
¡àF£¨m£¬-$\frac{2}{3}$m-$\frac{2}{3}$£©£¬
¡à-$\frac{2}{3}$m-$\frac{2}{3}$-$\frac{2}{3}$m²+$\frac{4}{3}$m+2=$\frac{4}{3}$£¬»ò$\frac{2}{3}$m²-$\frac{4}{3}$m-2+$\frac{2}{3}$m+$\frac{2}{3}$=$\frac{4}{3}$£¬
¡àm₁=1£¬m₂=0£¨ÉáÈ¥£©£¬m₃=$\frac{1+\sqrt{17}}{2}$£¬m₄=$\frac{1-\sqrt{17}}{2}$£¬
¢ÚÒÔCEΪ¶Ô½ÇÏߣ¬Á¬½ÓPF½»CEÓÚG£¬
¡àCG=GE£¬PG=FG£¬
¡àG£¨0£¬-$\frac{4}{3}$£©£¬
ÉèP£¨m£¬$\frac{2}{3}$m²-$\frac{4}{3}$m-2£©£¬ÔòF£¨-m£¬$\frac{2}{3}$m-$\frac{2}{3}$£©£¬
¡à$\frac{1}{2}$¡Á£¨$\frac{2}{3}$m²-$\frac{4}{3}$m-2+$\frac{2}{3}$m-$\frac{2}{3}$£©=-$\frac{4}{3}$£¬
¡àm=1£¬m=0£¨ÉáÈ¥£©£¬
×ÛÉÏËùÊö£¬F£¨1£¬-$\frac{4}{3}$£©£¬£¨$\frac{1+\sqrt{17}}{2}$£¬-$\frac{\sqrt{17}}{3}$£©£¬£¨$\frac{1-\sqrt{17}}{2}$£¬$\frac{\sqrt{17}}{3}$£©£¬£¨-1£¬0£©ÒÔE£¬C£¬P£¬FΪ¶¥µãµÄËıßÐÎÄܹ¹³ÉƽÐÐËıßÐΣ®

µãÆÀ ±¾Ì⿼²éÁË´ý¶¨ÏµÊý·¨Çóº¯ÊýµÄ½âÎöʽ£¬Æ½ÐÐËıßÐεÄÐÔÖÊ£¬¶þ´Îº¯ÊýµÄÐÔÖÊ£¬ÕýÈ·µÄÀí½âÌâÒâÊǽâÌâµÄ¹Ø¼ü£®