11£®ÈçͼËùʾ£¬ÒÑÖªÍÖÔ²CµÄ·½³ÌΪ$\frac{x^2}{2}+{y^2}$=1£¬F1£¬F2·Ö±ðÊÇÍÖÔ²CµÄ×ó¡¢ÓÒ½¹µã£¬Ö±ÏßAB£ºy=kx+m£¨k£¼0£©ÓëÍÖÔ²C½»ÓÚ²»Í¬µÄA£¬BÁ½µã£®
£¨¢ñ£© Èôk=-1£¬m=$\sqrt{2}$£¬µãPÔÚÖ±ÏßABÉÏÇó|PF1|+|PF2|µÄ×îСֵ£»
£¨¢ò£© ÈôÒÔÏß¶ÎABΪֱ¾¶µÄÔ²¾­¹ýµãF2£¬ÇÒÔ­µãOµ½Ö±ÏßABµÄ¾àÀëΪ$\frac{{2\sqrt{5}}}{5}$£®
£¨1£©ÇóÖ±ÏßABµÄ·½³Ì£»
£¨2£©ÔÚÍÖÔ²CÉÏÇóµãQµÄ×ø±ê£¬Ê¹µÃ¡÷ABQµÄÃæ»ý×î´ó£®

·ÖÎö £¨¢ñ£©Çó³öÍÖÔ²µÄ½¹µã×ø±ê£¬Ö±ÏßABµÄ·½³Ì£¬Çó³öF2¹ØÓÚÖ±ÏßABµÄ¶Ô³Æ${F}_{2}¡ä£¨\sqrt{2}£¬\sqrt{2}-1£©$£¬È»ºóÇó½â|PF1|+|PF2|µÄ×îСֵ£®
£¨¢ò£©£¨1£©ÉèµãA£¬BµÄ×ø±ê·Ö±ðΪA£¨x1£¬y1£©£¬B£¨x2£¬y2£©£®ÀûÓÃÔ­µãOµ½Ö±ÏßABµÄ¾àÀëµÃµ½m¡¢kµÄ¹ØÏµ£¬ÁªÁ¢y=kx+mÓë$\frac{x^2}{2}+{y^2}=1$£¬Í¨¹ýΤ´ï¶¨ÀíÒÔ¼°$\overrightarrow{A{F_2}}•\overrightarrow{B{F_2}}=0$£¬Çó³öm¡¢kµÄÖµ£¬È»ºóÇó³öABµÄ·½³Ì£®
£¨2£©ÓÉ£¨1£©¿ÉÖª£¬|AB|ÊǶ¨Öµ£¬µ±ÍÖÔ²CÉϵĵãQʹµÃ¡÷ABQµÄÃæ»ý×î´óʱ£¬µãQµ½Ö±ÏßABµÄ¾àÀëΪ×î´ó£¬¼´µãQΪÔÚÖ±ÏßABµÄÏ·½Æ½ÐÐÓÚABÇÒÓëÍÖÔ²CÏàÇеÄÇе㣮ÉèÆ½ÐÐÓÚABÇÒÓëÍÖÔ²CÏàÇеÄÇÐÏß·½³Ì£¬ÓëÍÖÔ²ÁªÁ¢£¬ÀûÓÃÅбðʽΪ0£¬Çó½â¼´¿É£®

½â´ð ½â£º£¨¢ñ£© ÓÉÍÖÔ²·½³Ì¿ÉµÃ£¬½¹µã×ø±êΪF1£¨-1£¬0£©£¬F2£¨1£¬0£©£®        ¡­£¨1·Ö£©
µ±k=-1£¬$m=\sqrt{2}$ʱ£¬Ö±ÏßABµÄ·½³ÌΪ$y=-x+\sqrt{2}$£®        ¡­£¨2·Ö£©
Ôò¿ÉµÃF2£¨1£¬0£©¹ØÓÚÖ±ÏßABµÄ¶Ô³ÆµãΪ${F}_{2}¡ä£¨\sqrt{2}£¬\sqrt{2}-1£©$£®         ¡­£¨3·Ö£©
¡à|PF1|+|PF2|µÄ×îСֵΪ£º$|{F}_{1}{F}_{2}¡ä|=\sqrt{{£¨\sqrt{2}+1£©}^{2}+{£¨\sqrt{2}-1£©}^{2}}=\sqrt{6}$£®  ¡­£¨4·Ö£©
£¨¢ò£©£º£¨1£©ÉèµãA£¬BµÄ×ø±ê·Ö±ðΪA£¨x1£¬y1£©£¬B£¨x2£¬y2£©£®
ÓÉÔ­µãOµ½Ö±ÏßABµÄ¾àÀëΪ$\frac{{2\sqrt{5}}}{5}$£¬µÃ$\frac{|m|}{{\sqrt{1+{k^2}}}}=\frac{{2\sqrt{5}}}{5}$£¬¼´${m^2}=\frac{4}{5}£¨1+{k^2}£©$£®¢Ù¡­£¨5·Ö£©
½«y=kx+m´úÈë$\frac{x^2}{2}+{y^2}=1$£¬µÃ£¨1+2k2£©x2+4kmx+2m2-2=0£¬
¡à¡÷=16k2m2-4£¨1+2k2£©£¨2m2-2£©=8£¨2k2-m2+1£©£¾0£¬¡à${x_1}+{x_2}=-\frac{4km}{{1+2{k^2}}}£¬{x_1}{x_2}=\frac{{2{m^2}-2}}{{1+2{k^2}}}$£®                   ¡­£¨6·Ö£©
ÓÉÒÑÖª£¬µÃ$\overrightarrow{A{F_2}}•\overrightarrow{B{F_2}}=0$£¬¼´£¨x1-1£©£¨x2-1£©+y1y2=0£®         ¡­£¨7·Ö£©
¡à£¨x1-1£©£¨x2-1£©+£¨kx1+m£©£¨kx2+m£©=0£¬
¼´$£¨1+{k^2}£©{x_1}{x_2}+£¨km-1£©£¨{x_1}+{x_2}£©+{m^2}+1=0$£¬
¡à$£¨1+{k^2}£©•\frac{{2{m^2}-2}}{{1+2{k^2}}}+£¨km-1£©•\frac{-4km}{{1+2{k^2}}}+{m^2}+1=0$£¬
»¯¼ò£¬µÃ3m2+4km-1=0£®¢Ú¡­£¨8·Ö£©
ÓÉ¢Ù¢Ú£¬µÃ${m^2}=\frac{4}{5}[1+{£¨\frac{{1-3{m^2}}}{4m}£©^2}]$£¬¼´11m4-10m2-1=0£¬¡àm2=1£®
¡ßk£¼0£¬¡à$\left\{\begin{array}{l}m=1\\ k=-\frac{1}{2}\end{array}\right.$£¬Âú×ã¡÷=8£¨2k2-m2+1£©£¾0£®
¡àABµÄ·½³ÌΪ$y=-\frac{1}{2}x+1$£® ¡­£¨9·Ö£©
£¨2£©ÓÉ£¨1£©¿ÉÖª£¬|AB|ÊǶ¨Öµ£¬µ±ÍÖÔ²CÉϵĵãQʹµÃ¡÷ABQµÄÃæ»ý×î´óʱ£¬µãQµ½Ö±ÏßABµÄ¾àÀëΪ×î´ó£¬¼´µãQΪÔÚÖ±ÏßABµÄÏ·½Æ½ÐÐÓÚABÇÒÓëÍÖÔ²CÏàÇеÄÇе㣮ÉèÆ½ÐÐÓÚABÇÒÓëÍÖÔ²CÏàÇеÄÇÐÏß·½³ÌΪ$y=-\frac{1}{2}x+n£¨n£¼0£©$£¬ÓÉ$\left\{\begin{array}{l}y=-\frac{1}{2}x+n\\ \frac{x^2}{2}+{y^2}=1\end{array}\right.$µÃ$\frac{3}{2}{x^2}-2nx+2{n^2}-2=0$£¬¡à¡÷=-8n2+12=0£¬
¡à$n=-\frac{{\sqrt{6}}}{2}$£¬£¨$n=\frac{{\sqrt{6}}}{2}$ÉáÈ¥£©£¬¡­£¨11·Ö£©
´Ó¶ø£¬¿ÉµÃQµÄ×ø±êΪ$Q£¨-\frac{{\sqrt{6}}}{3}£¬-\frac{{\sqrt{6}}}{3}£©$£®                 ¡­£¨12·Ö£©

µãÆÀ ±¾Ì⿼²éÖ±ÏßÓëÍÖÔ²µÄλÖùØÏµµÄÓ¦Ó㬵㵽ֱÏߵľàÀ빫ʽµÄÓ¦Ó㬿¼²é·ÖÎöÎÊÌâ½â¾öÎÊÌâµÄÄÜÁ¦£®

Á·Ï°²áϵÁдð°¸
Ïà¹ØÏ°Ìâ

¿ÆÄ¿£º¸ßÖÐÊýѧ À´Ô´£º ÌâÐÍ£º½â´ðÌâ

17£®ÒÑÖªÏòÁ¿$\overrightarrow{m}$=£¨cosA£¬-sinA£©£¬$\overrightarrow{n}$=£¨cosB£¬sinB£©£¬$\overrightarrow{m}$•$\overrightarrow{n}$=cos2C£¬ÆäÖÐA£¬B£¬CÊÇ¡÷ABCµÄÄÚ½Ç
£¨1£©Çó½ÇCµÄ´óС£»
£¨2£©ÇósinA+2sinBµÄȡֵ·¶Î§£®

²é¿´´ð°¸ºÍ½âÎö>>

¿ÆÄ¿£º¸ßÖÐÊýѧ À´Ô´£º ÌâÐÍ£ºÌî¿ÕÌâ

18£®º¯Êýy=sin3xµÄͼÏó¿ÉÒÔÓɺ¯Êýy=cos3xµÄͼÏóÏò×óÆ½ÒÆa¸öµ¥Î»µÃµ½µÄ£¬ÔòaµÄ×îСֵΪ-$\frac{¦Ð}{6}$£®

²é¿´´ð°¸ºÍ½âÎö>>

¿ÆÄ¿£º¸ßÖÐÊýѧ À´Ô´£º ÌâÐÍ£ºÌî¿ÕÌâ

15£®Ô²£¨x-3£©2+£¨y+1£©2=3¹ØÓÚÖ±Ïßx+2y-3=0¶Ô³ÆµÄÔ²µÄ·½³ÌΪ£¨x-$\frac{19}{5}$£©2+£¨y-$\frac{3}{5}$£©2=3£®

²é¿´´ð°¸ºÍ½âÎö>>

¿ÆÄ¿£º¸ßÖÐÊýѧ À´Ô´£º ÌâÐÍ£ºÑ¡ÔñÌâ

6£®ÉèÈ«¼¯UÊÇʵÊý¼¯R£¬M={x|x£¾2»òx£¼-2}£¬N={x|x¡Ý3»òx£¼1}£¬Ôò£¨∁UM£©¡ÉNÊÇ£¨¡¡¡¡£©
A£®{x|-2¡Üx£¼1}B£®{x|-2¡Üx¡Ü2}C£®{x|1£¼x¡Ü2}D£®{x|x£¼2}

²é¿´´ð°¸ºÍ½âÎö>>

¿ÆÄ¿£º¸ßÖÐÊýѧ À´Ô´£º ÌâÐÍ£º½â´ðÌâ

16£®Èçͼ£¬ÔÚÈýÀâ×¶P-ABCÖУ¬AC¡ÍBC£¬Æ½ÃæPAC¡ÍÆ½ÃæABC£¬PA=PC=AC=2£¬BC=4£¬E¡¢F·Ö±ðÊÇPC£¬PBµÄÖе㣬¼ÇÆ½ÃæAEFÓëÆ½ÃæABCµÄ½»ÏßΪֱÏßl£®
£¨¢ñ£©ÇóÖ¤£ºÖ±Ïßl¡ÍÆ½ÃæPAC£»
£¨¢ò£©Ö±ÏßlÉÏÊÇ·ñ´æÔÚµãQ£¬Ê¹Ö±ÏßPQ·Ö±ðÓëÆ½ÃæAEF¡¢Ö±ÏßEFËù³ÉµÄ½Ç»¥ÓࣿÈô´æÔÚ£¬Çó³ö|AQ|µÄÖµ£»Èô²»´æÔÚ£¬Çë˵Ã÷ÀíÓÉ£®

²é¿´´ð°¸ºÍ½âÎö>>

¿ÆÄ¿£º¸ßÖÐÊýѧ À´Ô´£º ÌâÐÍ£ºÌî¿ÕÌâ

3£®Èô{an}ΪµÈ²îÊýÁУ¬SnÊÇǰnÏîµÄºÍ£¬ÇÒS11=$\frac{22}{3}$¦Ð£¬{bn}ΪµÈ±ÈÊýÁУ¬b5¡Áb7=$\frac{{¦Ð}^{2}}{4}$£¬Ôòtan£¨a6+b6£©=$\frac{\sqrt{3}}{3}$£®

²é¿´´ð°¸ºÍ½âÎö>>

¿ÆÄ¿£º¸ßÖÐÊýѧ À´Ô´£º ÌâÐÍ£ºÑ¡ÔñÌâ

20£®ÈçͼËùʾ£¬ÓÉÖ±Ïßx=a£¬x=a+1£¨a£¾0£©£¬y=x2¼°xÖáΧ³ÉµÄÇú±ßÌÝÐεÄÃæ»ý½éÓÚÏàӦС¾ØÐÎÓë´ó¾ØÐεÄÃæ»ýÖ®¼ä£¬¼´a2£¼${¡Ò}_{a}^{a+1}$x2dx£¼£¨a+1£©2£®Àà±ÈÖ®£¬?n¡ÊN*£¬$\frac{1}{n+1}$+$\frac{1}{n+2}$+¡­+$\frac{1}{2n}$£¼A£¼$\frac{1}{n}$+$\frac{1}{n+1}$+¡­+$\frac{1}{2n-1}$ºã³ÉÁ¢£¬ÔòʵÊýAµÈÓÚ£¨¡¡¡¡£©
A£®$\frac{1}{2}$B£®$\frac{3}{5}$C£®ln2D£®ln$\frac{5}{2}$

²é¿´´ð°¸ºÍ½âÎö>>

¿ÆÄ¿£º¸ßÖÐÊýѧ À´Ô´£º ÌâÐÍ£ºÑ¡ÔñÌâ

1£®ÒÑÖª m¡¢n ÊÇÁ½Ìõ²»ÖغϵÄÖ±Ïߣ¬¦Á¡¢¦Â¡¢¦ÃÊÇÈý¸ö»¥²»ÖØºÏµÄÆ½Ã棬ÔòÏÂÁÐÃüÌâÖРÕýÈ·µÄÊÇ£¨¡¡¡¡£©
A£®Èô m¡Î¦Á£¬n¡Î¦Á£¬Ôò  m¡ÎnB£®Èô¦Á¡Í¦Ã£¬¦Â¡Í¦Ã£¬Ôò ¦Á¡Î¦Â
C£®Èôm¡Í¦Á£¬n¡Í¦Á£¬Ôò m¡ÎnD£®Èô m¡Î¦Á£¬m¡Î¦Â£¬Ôò ¦Á¡Î¦Â

²é¿´´ð°¸ºÍ½âÎö>>

ͬ²½Á·Ï°²á´ð°¸