Question

19£®ÉèÔ²${F_1}£º{x^2}+{y^2}+4x=0$µÄÔ²ÐÄΪF₁£¬Ö±Ïßl¹ýµãF₂£¨2£¬0£©ÇÒ²»ÓëxÖá¡¢yÖá´¹Ö±£¬ÇÒÓëÔ²F₁ÓÚC£¬DÁ½µã£¬¹ýF₂×÷F₁CµÄƽÐÐÏß½»Ö±ÏßF₁DÓÚµãE£¬
£¨1£©Ö¤Ã÷||EF₁|-|EF₂||Ϊ¶¨Öµ£¬²¢Ð´³öµãEµÄ¹ì¼£·½³Ì£»
£¨2£©ÉèµãEµÄ¹ì¼£ÎªÇúÏߦ££¬Ö±Ïßl½»¦£ÓÚM£¬NÁ½µã£¬¹ýF₂ÇÒÓël´¹Ö±µÄÖ±ÏßÓëÔ²F₁½»ÓÚP£¬QÁ½µã£¬Çó¡÷PQMÓë¡÷PQNµÄÃæ»ýÖ®ºÍµÄÈ¡Öµ·¶Î§£®

Answer 1

·ÖÎö £¨1£©ÇóµÃÔ²F₁µÄÔ²ÐĺͰ뾶£¬ÔËÓÃƽÐÐÏßµÄÐÔÖʺ͵ÈÑüÈý½ÇÐεÄÐÔÖÊ£¬¿ÉµÃED=EF₂£¬ÔÙÓÉË«ÇúÏߵĶ¨Ò壬¼´¿ÉµÃµ½ËùÇó¶¨ÖµºÍË«ÇúÏߵķ½³Ì£»
£¨2£©Éè³öl£ºx=my+2£¨m¡Ù0£©£¬l_PQ£ºy=-m£¨x-2£©£¬ÉèM£¨x₁£¬y₁£©£¬N£¨x₂£¬y₂£©£¬Çó³öÔ²Ðĵ½Ö±ÏßPQµÄ¾àÀ룬ÔËÓÃÏÒ³¤¹«Ê½¿ÉµÃ|PQ|£»ÔÙÓÉÖ±ÏßlµÄ·½³ÌºÍË«ÇúÏߵķ½³ÌÁªÁ¢£¬ÔËÓÃΤ´ï¶¨ÀíºÍÏÒ³¤¹«Ê½£¬¿ÉµÃ|MN|£¬ÔÙÓÉÈý½ÇÐεÄÃæ»ý¹«Ê½¿ÉµÃ¡÷PQMÓë¡÷PQNµÄÃæ»ýÖ®ºÍΪ$\frac{1}{2}$|MN|•|PQ|£¬»¯¼òÕûÀí£¬½áºÏ²»µÈʽµÄÐÔÖÊ£¬¼´¿ÉµÃµ½ËùÇó·¶Î§£®

½â´ð ½â£º£¨1£©Ö¤Ã÷£ºÔ²${F_1}£º{£¨{x+2}£©^2}+{y^2}=4$£¬Ô²ÐÄF₁£¨-2£¬0£©£¬°ë¾¶r=2£¬ÈçͼËùʾ£®

ÒòΪF₁C¡ÎEF₂£¬ËùÒÔ¡ÏF₁CD=¡ÏEF₂D£®
ÓÖÒòΪF₁D=F₁C£¬ËùÒÔ¡ÏF₁CD=¡ÏF₁DC£¬
ËùÒÔ¡ÏEF₂D=¡ÏF₁DC£¬
ÓÖÒòΪ¡ÏF₁DC=¡ÏEDF₂£¬ËùÒÔ¡ÏEF₂D=¡ÏEDF₂£¬
¹ÊED=EF₂£¬¿ÉµÃ||EF₁|-|EF₂||=||EF₁|-|ED||=|F₁D|=2£¼|F₁F₂|£¬
¸ù¾ÝË«ÇúÏߵĶ¨Ò壬¿ÉÖªµãEµÄ¹ì¼£ÊÇÒÔF₁£¬F₂Ϊ½¹µãµÄË«ÇúÏߣ¨¶¥µã³ýÍ⣩£¬
ÇÒa=1£¬c=2£¬b=$\sqrt{{c}^{2}-{a}^{2}}$=$\sqrt{3}$£¬
¹ÊµãEµÄ¹ì¼£·½³ÌΪ${x^2}-\frac{y^2}{3}=1£¨{y¡Ù0}£©$£®
£¨2£©$¦££º{x^2}-\frac{y^2}{3}=1£¨{y¡Ù0}£©$£®
ÒÀÌâÒâ¿ÉÉèl£ºx=my+2£¨m¡Ù0£©£¬M£¨x₁£¬y₁£©£¬N£¨x₂£¬y₂£©£¬
ÓÉÓÚPQ¡Íl£¬Éèl_PQ£ºy=-m£¨x-2£©£®
Ô²ÐÄF₁£¨-2£¬0£©µ½Ö±ÏßPQµÄ¾àÀë$d=\frac{{|{-m£¨{-2-2}£©}|}}{{\sqrt{1+{m^2}}}}=\frac{{|{4m}|}}{{\sqrt{1+{m^2}}}}$£¬
ËùÒÔ$|{PQ}|=2\sqrt{{r^2}-{d^2}}=\frac{{4\sqrt{1-3{m^2}}}}{{\sqrt{1+{m^2}}}}$£¬
ÓÖÒòΪd£¼2£¬½âµÃ$0£¼{m^2}£¼\frac{1}{3}$£®
ÁªÁ¢Ö±ÏßlÓëË«ÇúÏߦ£µÄ·½³Ì$\left\{{\begin{array}{l}{{x^2}-\frac{y^2}{3}=1}\\{x=my+2}\end{array}}\right.$£¬ÏûÈ¥xµÃ£¨3m²-1£©y²+12my+9=0£¬
Ôò${y_1}+{y_2}=-\frac{12m}{{3{m^2}-1}}£¬{y_1}{y_2}=\frac{9}{{3{m^2}-1}}$£¬
ËùÒÔ$|{MN}|=\sqrt{1+{m^2}}|{{y_2}-{y_1}}|=\sqrt{1+{m^2}}\sqrt{{{£¨{{y_1}+{y_2}}£©}^2}-4{y_1}{y_2}}=\frac{{6£¨{{m^2}+1}£©}}{{1-3{m^2}}}$£¬
¼Ç¡÷PQM£¬¡÷PQNµÄÃæ»ý·Ö±ðΪS₁£¬S₂£¬
Ôò${S_1}+{S_2}=\frac{1}{2}|{MN}|•|{PQ}|=\frac{{12\sqrt{{m^2}+1}}}{{\sqrt{1-3{m^2}}}}=12\sqrt{\frac{1}{{-3+\frac{4}{{{m^2}+1}}}}}$£¬
ÓÖÒòΪ$0£¼{m^2}£¼\frac{1}{3}$£¬ËùÒÔS₁+S₂¡Ê£¨12£¬+¡Þ£©£¬
ËùÒÔS₁+S₂µÄÈ¡Öµ·¶Î§Îª£¨12£¬+¡Þ£©£®

µãÆÀ ±¾Ì⿼²é¹ì¼£·½³ÌµÄÇ󷨣¬×¢ÒâÔËÓÃË«ÇúÏߵĶ¨Ò壬¿¼²éÖ±ÏߺÍÔ²¡¢Ö±ÏߺÍÍÖÔ²µÄλÖùØϵ£¬×¢ÒâÁªÁ¢·½³Ì×飬ÔËÓÃΤ´ï¶¨ÀíºÍÏÒ³¤¹«Ê½£¬¿¼²éÈý½ÇÐÎÃæ»ýµÄÇ󷨣¬ÒÔ¼°»¯¼òÕûÀíµÄÔËËãÄÜÁ¦£¬ÊôÓÚÖеµÌ⣮