ÌâÄ¿ÄÚÈÝ
(Àí)Éè¹ØÓÚxµÄ·½³Ìx2-mx-1=0ÓÐÁ½¸öʵ¸ù¦Á¡¢¦Â,ÇÒ¦Á£¼¦Â.¶¨Ò庯Êýf(x)=
(1)Çó¦Áf(¦Á)+¦Âf(¦Â)µÄÖµ;
(2)ÅжÏf(x)ÔÚÇø¼ä(¦Á,¦Â)Éϵĵ¥µ÷ÐÔ,²¢¼ÓÒÔÖ¤Ã÷;
(3)Èô¦Ë¡¢¦ÌΪÕýʵÊý,Ö¤Ã÷²»µÈʽ:|f()-f(
)|£¼|¦Á-¦Â|.
(ÎÄ)Èçͼ,ÔÚƽÃæÖ±½Ç×ø±êϵÖÐ,ÒÑÖª¶¯µãP(x,y),PM¡ÍyÖá,´¹×ãΪM,µãNÓëµãP¹ØÓÚxÖá¶Ô³Æ,ÇÒ=4.
(1)Ç󶯵ãPµÄ¹ì¼£WµÄ·½³Ì;
(2)ÈôµãQµÄ×ø±êΪ(2,0),A¡¢BΪWÉϵÄÁ½¸ö¶¯µã,ÇÒÂú×ãQA¡ÍQB,µãQµ½Ö±ÏßABµÄ¾àÀëΪd,ÇódµÄ×î´óÖµ.
´ð°¸£º(Àí)½â£º(1)¡ß¦Á¡¢¦ÂÊÇ·½³Ìx2-mx-1=0µÄÁ½¸öʵ¸ù,
¡à¡àf(¦Á)=
.
ͬÀí,f(¦Â)=.¡à¦Áf(¦Á)+¦Âf(¦Â)=2.
(2)¡ßf(x)=,¡àf¡ä(x)=
=
.
µ±x¡Ê(¦Á,¦Â)ʱ,x2-mx-1=(x-¦Á)(x-¦Â)£¼0,
¡àf¡ä(x)£¾0.¡àf(x)ÔÚ(¦Á,¦Â)ÉÏΪÔöº¯Êý.
(3)¡ß¦Ë,¦Ì¡ÊR+,ÇÒ¦Á£¼¦Â,¡à
.¡à¦Á£¼
£¼¦Â.
ÓÉ(2)¿ÉÖªf(¦Á)£¼f()£¼f(¦Â),ͬÀí,¿ÉµÃf(¦Á)£¼f(
)£¼f(¦Â).
¡àf(¦Á)-f(¦Â)£¼f()-f(
)£¼f(¦Â)-f(¦Á).
¡à|f()-f(
)|£¼|f(¦Á)-f(¦Â)|.
ÓÖÓÉ(1)Öªf(¦Á)=,f(¦Â)=
,¦Á¦Â=-1,
¡à|f(¦Á)-f(¦Â)|=|-
|=|
|=|¦Á-¦Â|.¡à|f(
)-f(
)|£¼|¦Á-¦Â|.
(ÎÄ)½â£º(1)ÓÉÒÑÖªM(0,y),N(x,-y).
Ôò=(x,y)¡¤(x,-2y)=x2-2y2=4,¼´
=1.
(2)ÉèA(x1,y1),B(x2,y2),Èçͼ,ÓÉQA¡ÍQB¿ÉµÃ,
=(x1-2,y1)¡¤(x2-2,y2)=(x1-2)(x2-2)+y1y2=0.
¢ÙÈôÖ±ÏßAB¡ÍxÖá,Ôòx1=x2,|y1|=|y2|=,ÇÒy1¡¢y2ÒìºÅ,´Ëʱ(x1-2)(x2-2)+y1y2=(x1-2)2
=0Ôòx12-8x1+12=0,
½âÖ®,µÃx1=6»òx1=2.Èôx1=2,ÔòÖ±ÏßAB¹ýQµã,²»¿ÉÄÜÓÐQA¡ÍQB.
Èôx1=6,ÔòÖ±ÏßABµÄ·½³ÌΪx=6,´ËʱQµãµ½Ö±ÏßABµÄ¾àÀëΪ4.
¢ÚÈôÖ±ÏßABбÂÊ´æÔÚ,ÉèÖ±ÏßABµÄ·½³ÌΪy=kx+m,Ôò
(2k2-1)x2+4kmx+2m2+4=0.
Ôò¼´
ÓÖx1+x2=,x1x2=
.
¡ày1y2=(kx1+m)(kx2+m)=k2x1x2+km(x1+x2)+m2
=.
¡à=(x1-2,y1)¡¤(x2-2,y2)=(x1-2)(x2-2)+y1y2=x1x2-2(x1+x2)+4+y1y2
=
Ôòm2+8km+12k2=0,¿ÉµÃm=-6k»òm=-2k.Èôm=-2k,ÔòÖ±ÏßABµÄ·½³ÌΪy=k(x-2),´ËÖ±Ïß¹ýµãQ,ÕâÓëQA¡ÍQBì¶Ü,¹ÊÉáÈ¥.Èôm=-6k,ÔòÖ±ÏßABµÄ·½³ÌΪy=kx-6k,¼´kx-y-6k=0.
´ËʱÈôk=0,ÔòÖ±ÏßABµÄ·½³ÌΪy=0,ÏÔÈ»ÓëQA¡ÍQBì¶Ü,¹Êk¡Ù0.
¡àd=.
Óɢ٢ڿɵÃ,dmax=4.
˵Ã÷:ÆäËûÕýÈ·½â·¨°´ÏàÓ¦²½Öè¸ø·Ö.
