Question

1£®ÒÑÖªÅ×ÎïÏßC£ºy²=2px£¨p£¾0£©µÄ½¹µãFºÍÍÖÔ²$\frac{x^2}{4}$+$\frac{y^2}{3}$=1µÄÓÒ½¹µãÖغϣ¬Ö±Ïßl¹ýµãF½»Å×ÎïÏßÓÚA¡¢BÁ½µã£®
£¨1£©ÇóÅ×ÎïÏßCµÄ¹ý³Ì£»
£¨2£©ÈôÖ±Ïßl½»yÖáÓÚµãM£¬ÇÒ$\overrightarrow{MA}$=m$\overrightarrow{AF}$£¬$\overrightarrow{MB}$=n$\overrightarrow{BF}$£¬¶ÔÈÎÒâµÄÖ±Ïßl£¬m+nÊÇ·ñΪ¶¨Öµ£¿ÈôÊÇ£¬Çó³öm+nµÄÖµ£»·ñÔò£¬ËµÃ÷ÀíÓÉ£®

Answer 1

·ÖÎö £¨1£©ÓÉÍÖÔ²Óɽ¹µã×ø±êF£¨1£¬0£©£¬¼´$\frac{p}{2}$=1£¬¼´¿ÉÇóµÃÅ×ÎïÏßCµÄ¹ý³Ì£»
£¨2£©½«Ö±Ïß·½³Ì´úÈëÍÖÔ²·½³Ì£¬ÓÉ¡÷£¾0£¬¸ù¾ÝΤ´ï¶¨Àí¿ÉÖªx₁+x₂=$\frac{2{k}^{2}+4}{{k}^{2}}$£¬x₁•x₂=1£¬¸ù¾ÝÏòÁ¿µÄ×ø±ê±íʾ$\overrightarrow{MA}$=m$\overrightarrow{AF}$£¬$\overrightarrow{MB}$=n$\overrightarrow{BF}$£¬¼´¿ÉÇóµÃm=$\frac{{x}_{1}}{1-{x}_{1}}$£¬n=$\frac{{x}_{2}}{1-{x}_{2}}$£¬´úÈë¼´¿ÉÇóµÃm+n=-1£®

½â´ð ½â£º£¨1£©ÍÖÔ²µÄ$\frac{x^2}{4}$+$\frac{y^2}{3}$=1ÓÒ½¹µãF£¨1£¬0£©£¬
¡à$\frac{p}{2}$=1£¬p=2£¬
¡àÅ×ÎïÏßCµÄ·½³Ìy²=4x£»
£¨2£©ÓÉÒÑÖª¿ÉµÃ£ºÖ±ÏßlµÄбÂÊÒ»¶¨´æÔÚ£¬
¡àÉèl£ºy=k£¨x-1£©£¬lÓëyÖá½»ÓÚM£¨0£¬-k£©£¬
ÉèÖ±ÏßlÓëÅ×ÎïÏß½»ÓÚA£¨x₁£¬y₁£©B£¨x₂£¬y₂£©£¬
ÓÉ$\left\{\begin{array}{l}{y=k£¨x-1£©}\\{{y}^{2}=4x}\end{array}\right.$£¬Ôòk²x²-2£¨k²+2£©+k²=0£¬
¡÷=4£¨k²+2£©²-4k⁴=16£¨k²+1£©£¾0£¬
¡àx₁+x₂=$\frac{2{k}^{2}+4}{{k}^{2}}$£¬x₁•x₂=1£¬
$\overrightarrow{MA}$=m$\overrightarrow{AF}$£¬
¡à£¨x₁£¬y₁+k£©=m£¨1-x₁£¬-y₁£©£¬
x₁=m£¨1-x₁£©
¼´m=$\frac{{x}_{1}}{1-{x}_{1}}$£¬Í¬Àín=$\frac{{x}_{2}}{1-{x}_{2}}$£¬
m+n=$\frac{{x}_{1}}{1-{x}_{1}}$+$\frac{{x}_{2}}{1-{x}_{2}}$=$\frac{{x}_{1}+{x}_{2}-2{x}_{1}{x}_{2}}{1-£¨{x}_{1}+{x}_{2}£©+{x}_{1}•{x}_{2}}$=-1£¬
¹Ê¶ÔÈÎÒâµÄÖ±Ïßl£ºm+nµÄÖµ-1£®

µãÆÀ ±¾Ì⿼²éÅ×ÎïÏߵıê×¼·½³Ì¼°Æä¼òµ¥ÐÔÖÊ£¬¿¼²éÖ±ÏßÓëÅ×ÎïÏßµÄλÖùØϵ£¬Î¤´ï¶¨ÀíµÄÓ¦Ó㬿¼²é¼ÆËãÄÜÁ¦£¬ÊôÓÚÖеµÌ⣮