Question

13£®£¨1£©Èçͼ1£¬ÒÑÖª¡÷ABCÈý¸ö¶¥µãµÄ×ø±ê·Ö±ðΪA£¨1£¬4£©¡¢B£¨4£¬1£©¡¢C£¨4£¬4£©£¬ÈôË«ÇúÏßy=$\frac{k}{x}$£¨x£¾0£©Óë¡÷ABCÓй«¹²µã£¬ÔòkµÄÈ¡Öµ·¶Î§ÊÇ4¡Ük¡Ü16£»
£¨2£©°Ñͼ1Öеġ÷ABCÑØÖ±ÏßAB·­ÕÛºóµÃµ½¡÷ABC₁£¬ÈôË«ÇúÏßy=$\frac{m}{x}$£¨x£¾0£©Óë¡÷ABC₁Óй«¹²µã£¬ÇómµÄÈ¡Öµ·¶Î§£»
СÃ÷½èÖúÒ»Ôª¶þ´Î·½³Ì¸ùµÄÅжÏʽԲÂúµØ½â¾öÁËÕâ¸öÎÊÌ⣬С·¼½èÖú¶þ´Îº¯ÊýÄ£ÐÍÒ²Ô²ÂúµØ½â¾öÁËÕâ¸öÎÊÌ⣮ÇëÄãÏÈÔÚͼ2Öл­³ö¡÷ABC₁£¬ÔÙд³ö×Ô¼ºµÄ½â´ð¹ý³Ì£®
£¨3£©Èçͼ3£¬ÒÑÖªµãAΪ£¨1£¬2£©£¬µãBΪ£¨4£¬1£©£¬ÈôË«ÇúÏßy=$\frac{n}{x}$£¨x£¾0£©ÓëÏ߶ÎABÓй«¹²µã£¬ÔònµÄÈ¡Öµ·¶Î§ÊÇ2¡Ün¡Ü$\frac{49}{12}$£®

Answer 1

·ÖÎö £¨1£©·Ö±ðÇóµÃË«ÇúÏßy=$\frac{k}{x}$£¨x£¾0£©Óë±ßAC£¬±ßBCÓй«¹²µãµÄkµÄÈ¡Öµ·¶Î§£¬È»ºóÇóµÃÖ±ÏßABµÄ½âÎöʽ£¬ÀûÓÃÒ»Ôª¶þ´Î·½³Ì¸ùµÄÅбðʽÇóµÃÓë±ßABÓй«¹²µãµÄkµÄÈ¡Öµ·¶Î§£¬Ôò¿ÉÇóµÃ´ð°¸£»
£¨2£©Ê×ÏÈÇóµÃµãCµÄ×ø±ê£¬¼´¿ÉµÃË«ÇúÏßy=$\frac{m}{x}$£¨x£¾0£©Óë±ßAC£¬±ßBCÓй«¹²µãµÄkµÄÈ¡Öµ·¶Î§£¬ÓÉ£¨1£©¿ÉÇóµÃÓë±ßABµÄÈ¡Öµ·¶Î§£¬¼Ì¶øÇóµÃ´ð°¸£»
£¨3£©Ê×ÏÈÇóµÃÏ߶ÎABµÄ½âÎöʽÒÔ¼°×Ô±äÁ¿µÄÈ¡Öµ·¶Î§£¬ÔÙÀûÓÃÒ»Ôª¶þ´Î·½³Ì¸ùµÄÅбðʽÇóµÃ´ð°¸£®

½â´ð ½â£º£¨1£©ÉèÖ±ÏßABµÄ½âÎöʽΪ£ºy=kx+b£¬
¡ßA£¨1£¬4£©¡¢B£¨4£¬1£©£¬
¡à$\left\{\begin{array}{l}{k+b=4}\\{4k+b=1}\end{array}\right.$£¬
½âµÃ£º$\left\{\begin{array}{l}{k=-1}\\{b=5}\end{array}\right.$£¬
¡àÖ±ÏßABµÄ½âÎöʽΪ£ºy=-x+5£¬
¡ß¡÷ABCÈý¸ö¶¥µãµÄ×ø±ê·Ö±ðΪA£¨1£¬4£©¡¢B£¨4£¬1£©¡¢C£¨4£¬4£©£¬
¡àÈôË«ÇúÏßy=$\frac{k}{x}$£¨x£¾0£©Óë±ßACÓй«¹²µã£¬Ôò4¡Ük¡Ü16£¬
ÈôË«ÇúÏßy=$\frac{k}{x}$£¨x£¾0£©Óë±ßBCÓй«¹²µã£¬Ôò4¡Ük¡Ü16£¬
ÈôË«ÇúÏßy=$\frac{k}{x}$£¨x£¾0£©Óë±ßABÓй«¹²µã£¬Ôò$\frac{k}{x}$=-x+5£¨1¡Üx¡Ü4£©£¬
¼´x²-5x+k=0£¬
¡à¡÷=25-4k¡Ý0£¬
½âµÃ£ºk¡Ü$\frac{25}{4}$£¬
¡àÈôË«ÇúÏßy=$\frac{k}{x}$£¨x£¾0£©Óë±ßABÓй«¹²µã£¬Ôò4¡Ük¡Ü$\frac{25}{4}$£»
×ÛÉϿɵãºÈôË«ÇúÏßy=$\frac{k}{x}$£¨x£¾0£©Óë¡÷ABCÓй«¹²µã£¬ÔòkµÄÈ¡Öµ·¶Î§ÊÇ£º4¡Ük¡Ü16£»
¹Ê´ð°¸Îª£º4¡Ük¡Ü16£»

£¨2£©Èçͼ£¬ÔòµãC₁£¨1£¬1£©£¬
ÓÉ£¨1£©µÃ£ºÈôË«ÇúÏßy=$\frac{m}{x}$£¨x£¾0£©Óë±ßABÓй«¹²µã£¬Ôò4¡Üm¡Ü$\frac{25}{4}$£¬
¡ßÈôË«ÇúÏßy=$\frac{m}{x}$£¨x£¾0£©Óë±ßACÓй«¹²µã£¬Ôò1¡Ük¡Ü4£¬
ÈôË«ÇúÏßy=$\frac{m}{x}$£¨x£¾0£©Óë±ßBCÓй«¹²µã£¬Ôò1¡Ük¡Ü4£¬
×ÛÉϿɵãºÈôË«ÇúÏßy=$\frac{m}{x}$£¨x£¾0£©Óë¡÷ABCÓй«¹²µã£¬ÔòkµÄÈ¡Öµ·¶Î§ÊÇ£º1¡Ük¡Ü$\frac{25}{4}$£»

£¨3£©Èçͼ3£¬ÉèÖ±ÏßABµÄ½âÎöʽΪ£ºy=mx+n£¬
¡ßµãAΪ£¨1£¬2£©£¬µãBΪ£¨4£¬1£©£¬
¡à$\left\{\begin{array}{l}{m+n=2}\\{4m+n=1}\end{array}\right.$£¬
½âµÃ£º$\left\{\begin{array}{l}{m=-\frac{1}{3}}\\{n=\frac{7}{3}}\end{array}\right.$£¬
¡àÖ±ÏßABµÄ½âÎöʽΪ£ºy=-$\frac{1}{3}$x+$\frac{7}{3}$£¨1¡Üx¡Ü4£©£¬
ÈôË«ÇúÏßy=$\frac{n}{x}$£¨x£¾0£©ÓëÏ߶ÎABÓй«¹²µã£¬
Ôò$\frac{n}{x}$=-$\frac{1}{3}$x+$\frac{7}{3}$£¬
ÕûÀíµÃ£ºx²-7x+3n=0£¬
¡à¡÷=49-12n¡Ý0£¬
¡àn¡Ü$\frac{49}{12}$£¬
¡ß1¡Üx¡Ü4£¬
¡àn¡Ý2£¬
¡àÈôË«ÇúÏßy=$\frac{n}{x}$£¨x£¾0£©ÓëÏ߶ÎABÓй«¹²µã£¬ÔònµÄÈ¡Öµ·¶Î§ÊÇ£º2¡Ün¡Ü$\frac{49}{12}$£®
¹Ê´ð°¸Îª£º2¡Ün¡Ü$\frac{49}{12}$£®

µãÆÀ ´ËÌâÊôÓÚ·´±ÈÀýº¯Êý×ÛºÏÌ⣬¿¼²éÁËÖ±ÏßÓë·´±ÈÀýº¯ÊýµÄ½»µãÎÊÌâÒÔ¼°Ò»Ôª¶þ´Î·½³ÌµÄ¸ùµÄÅбðʽ£®×¢ÒâÄܽèÖúÒ»Ôª¶þ´Î·½³Ì¸ùµÄÅжÏʽÇó½âÊǹؼü£®