专题8.6双曲线十一大题型（原卷版）_11.4-2_2026-11-04

      
      
专题8.6 双曲线【十一大题型】

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 4  , !-.)* ............................................................................................................................6
 5   /012345 ....................................................................................................................7
 6   61701!89$:;5 ................................................................................................7
 7   / <=>?!:;5 ........................................................................................................8
 8  , !9@AB%C;DE ........................................................................................................8
 9   !FGHIJK5 ................................................................................................................9
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(tİ0).
 1   !"#$%&
ı 1j2024·IJlijĴ·ĵĶkí1 P p C
𝑥2
16−𝑦2
9 = 16·1f𝐹1f𝐹2Wð½ C !ñ>ò01fÉ
“|𝑃𝐹1| = 8”p“|𝑃𝐹2| = 16”!j    k
AšķàĸWĹàĺRÙÚ
BšĺRàĸWÙÚ
CšĸRÙÚ
DšĸWàĺRÙÚ
Ę÷ 1-1j2024·Ļļ·ĶĽľĿkŀ}𝐹1f𝐹2Wðp C
𝑥2
𝑎2−𝑦2
𝑏2 = 1(𝑎> 0,𝑏> 0)!ñ>ò01f
|𝐹1𝐹2| = 2𝑐f1 P © C !òĨ6fĮ △𝑃𝐹1𝐹2!Ł¼½6𝑐fÉ|𝑃𝐹1| = j   k
Aš3𝑐−𝑎
Bš3𝑐+ 𝑎
Cš2𝑐−𝑎
Dš2𝑐+ 𝑎
Ę÷ 1-2j23-24 ĵĄ·lmļł·Ńńkŀ} 𝐶:𝑥2
𝑎2−𝑦2
16 = 1!ñò01Ņû½𝐹1f𝐹2fĮ|𝐹1𝐹2|
= 10fí1𝑃© !òĨ6fÉ|𝑃𝐹1|−|𝑃𝐹2| = j    k
Aš−6
Bš6
Cš8
Dš10
Ę÷ 1-3j2024·ņŇňʼn·ĵĶkÖ𝐹1f𝐹2p C
𝑥2
4 −𝑦2
3 = 1!ñ>ò01fĭ𝐹2!Í + C !ò
ĨŊ  PfQ 1fÉ|𝐹1𝑃| + |𝐹1𝑄|−|𝑃𝑄| = j    k
Aš5
Bš6
Cš8
Dš12
      
      
 2   !'()*
ı 2j2024·lmŋŌō··Ķkŀ} 𝐶Ŏĭ1(0,1)f 9@A½ 2fÉ𝐶!'()*½j    k
Aš𝑥2−𝑦2
3 = 1
Bš𝑥2
3 −𝑦2 = 1
Cš𝑦2−𝑥2
3 = 1
Dš𝑦2
3 −𝑥2 = 1
Ę÷ 2-1j2024·lmļł··Ķkí 𝑥2
𝑎2−𝑦2
𝑏2 = 1(𝑎> 0,𝑏> 0)6!·1701(−5,0)!8970
1( 5,0)!89𝑏fÉŏ !)*½j    k
Aš𝑥2
4 −𝑦2 = 1
Bš𝑥2
2 −𝑦2 = 1
Cš𝑥2−𝑦2
2 = 1
Dš𝑥2−𝑦2
4 = 1
Ę÷ 2-2j2024·ŐőŒœ··ĶkÞ[fŔŕŖŗŘřŚśŕŜŠŝ Şşf§ÚŕŜŠċšޑţ
ŤťŦŧŨfĸWċ‹ŔĔŕũŪū¦!ŬĥŭfpŔĔŕũŮů!ŰDŧ¦.ŏŜzċűW!¹Ųœˆ
Çbųy¦ 𝐶!·űWfí𝐶!/@©º1f01©𝑥¹6f9@A𝑒= 2fĮ1𝑃( 6,3)© 𝐶6f
É 𝐶!'()*½j    k
Aš𝑥2−𝑦2
3 = 1
Bš𝑥2
2 −𝑦2
6 = 1
Cš𝑥2
3 −𝑦2
9 = 1
Dš𝑥2
4 −𝑦2
12 = 1
Ę÷ 2-3j2024·ņŇŴş··Ķkŀ}𝐹1,𝐹2½ 𝐶:𝑥2
𝑎2−𝑦2
𝑏2 = 1(𝑎> 0,𝑏> 0)!ñ>ò01f1𝐴©𝐶6f
í|𝐹1𝐴| = 2|𝐹2𝐴|f∠𝐴𝐹1𝐹2 = 30°, △𝐴𝐹1𝐹2!œŵ½6 3fÉ𝐶!)*½j    k
Aš𝑥2
9 −𝑦2
6 = 1
Bš𝑥2
3 −𝑦2
6 = 1
Cš𝑥2
6 −𝑦2
9 = 1
Dš𝑥2
6 −𝑦2
3 = 1
 3   )*+ 
ı 3j2024·ņŇőĸ·ĵĶkŀ}𝑚f𝑛pL
fÉ“𝑚𝑛< 0”p“ 𝑚𝑥2 +𝑛𝑦2 = 1p01©𝑥¹!
 ”!j    k
AšĸWàĺRÙÚ
BšĺRàĸWÙÚ
      
      
CšĸRÙÚ
DšķàĸWĹàĺRÙÚ
Ę÷ 3-1j23-24 ĵ6·6ļ·Ńńkç𝑎𝑏< 0’f)*𝑎𝑥2−𝑎𝑦2 = 𝑏Æęì! pj    k
Aš01©𝑥¹!NO
Bš01©𝑥¹! 
Cš01©𝑦¹!NO
Dš01©𝑦¹! 
Ę÷ 3-2j2024·şŶŷŸ·ĶĽľĿkŀ} 𝐶:𝑥2
4 + 𝑦2
𝑚= 1(𝑚≠0)fÉ“𝑚∈(0,4)”p“ C !01©
x ¹6”!j    k
AšĸWàĺRÙÚ
BšĺRàĸWÙÚ
CšĸRÙÚ
DšķàĸWĹàĺRÙÚ
Ę÷ 3-3j23-24 ĵĄ·Źź·Ń/k“𝑚> 1”p“)* 𝑥2
𝑚−1−𝑦2
𝑚+3 = 1ęì! p ”!j    k
AšĸWàĺRÙÚ
BšĺRàĸWÙÚ
CšĸRÙÚ
DšķàĸWĹàĺRÙÚ
 4  , !-.)*
ı 4j23-24 ĵ6·Żż·Ńńkŀ}ŽO+O𝐹1:(𝑥+ 4)2 + 𝑦2 = 1$O𝐹2:(𝑥−4)2 + 𝑦2 = 9‡žſfƀ
ƁŽOO@-.)*pj    k
Aš𝑥2−𝑦2
15 = 1
Bš𝑥2−𝑦2
15 = 1(𝑥≤−1)
Cš𝑥2
15−𝑦2 = 1
Dš𝑥2
15−𝑦2 = 1(𝑥≤−15)
Ę÷ 4-1j23-24 ĵ6·ŻżżƂ·Ń/kÖ𝐹1>𝐹2p"1f|𝐹1𝐹2| = 6fŽ1 P ƃƄ|𝑃𝐹1|−|𝑃𝐹2|
= 4fÉŽ1 P !-.pj    k
Aš 
Bš !·Ĩ
Cš·Ùƅ 
Dš-.àƆ©
Ę÷ 4-2j24-25 ĵ6·gh·ƇĜ¦ƈkË8 1600m !žƉÆ𝐴,𝐵fƊ7Ƌƌƍx!ƎƏƐƑƒfŀ
}ç’!ƒƓƔp320m/sf©𝐴ƉÆƊ7!ƐƑƒ!’¨©𝐵ƉÆƊ7’ƕ4sšíÇ𝐴𝐵Æ©Í ½𝑥¹f
Ç <𝐴𝐵!/Ì ½𝑦¹fÉƐƑ1Æ© !)*ˆÇpj    k
Aš
𝑥2
435600−
𝑦2
564400 = 1(𝑥> 0)
Bš 𝑥2
6402−𝑦2
4802 = 1(𝑥> 0)
Cš
𝑥2
435600 +
𝑦2
564400 = 1
Dš 𝑥2
6402 +
𝑦2
4802 = 1
Ę÷ 4-3j24-25 ĵ6·6ļ·ƇƖıkŀ}ŽO P +O M
(𝑥+ 3)2 + 𝑦2 = 1fO N
(𝑥−3)2 + 𝑦2 = 9
ƗžſfƘO@ P !ƙŽ-.½ CfÉ C !)*½j   k
Aš𝑥2−𝑦2
8 = 1
Bš𝑥2−𝑦2
8 = 1(𝑥≤−1)
      
      
Cš𝑥2−𝑦2
8 = 1(𝑥≥1)
Dš𝑦2−𝑥2
8 = 1
 5   /012345
ı 5j2024·ņŇ뇷2ĶkÖ𝐹1f𝐹2p 𝐶:𝑥2−𝑦2
3 = 1!ñfò01f1 P © C !òĨ6f
ç|𝑃𝐹1| = 6’f △𝑃𝐹1𝐹2œŵ½j    k
Aš4 3
Bš3 7
Cš 455
2
Dš6 7
Ę÷ 5-1j2023·gh·ĶĽľĿkŀ}1𝐴(−2,0)f𝐴′(2,0)fŽ1 P ƃƄ4𝑘𝐴𝑃⋅𝑘𝐴′𝑃= 1fO E
𝑥2 + 𝑦2 = 5
+1 P !-.!·žŊ1½ MfO E + x ¹!Ŋ1½ BfCfÉ △𝑀𝐵𝐶!Ł¼½j    k
Aš 5 +
6
Bš2 5 +
6
Cš 5 +2 6
Dš2 5 +2 6
Ę÷ 5-2j2024·ƚŞƛƜ·ĶĽľĿkÖ𝐹1f𝐹2p C
𝑥2
4 −𝑦2
8 = 1!ñfò01fĭ𝐹1!Í + y
¹= C !òĨWðŊ 1 PfQfí △𝑃𝑄𝐹2pƝ234fÉ|𝑄𝐹1| = j    k
Aš2
Bš4
Cš8
Dš16
Ę÷ 5-3j2024·ŻŞőƞ··ĶkÖ𝐹1、𝐹2p 𝐶:𝑥2
8 −𝑦2
10 = 1!ñ>òž01fO ½ª'º1f1
P © C 6Į|𝑂𝑃| =
1
2|𝑃𝐹1−𝑃𝐹2|fÉ △𝑃𝐹1𝑂!œŵ½j    k
Aš5
Bš8
Cš10
Dš12
 6   61701!89$:;5
ı 6j2024·gh·ĶĽľĿkŀ} 𝐶:𝑥2
3 −𝑦2 = 1!ò01½𝐹fŽ1𝑀©Í 𝑙:𝑥=
3
26f <𝐹𝑀Ŋ𝐶
 𝑃1fĭ𝑃¦𝑙!Ì fÌƄ½𝑅fÉ
|𝑃𝑅|
|𝑃𝐹|!;½j    k
Aš 6
2
Bš 3
3
Cš 6
3
Dš 3
2
Ę÷ 6-1j2024·ĻļƟƠ·ĶĽľĿkŀ}𝐹1f𝐹2½ 𝐶:𝑥2
4 −𝑦2
2 = 1!ñ>ò01f1 P p C !òĨ
6!·1fÉ
|𝑃𝐹1|2
|𝑃𝐹2| !:¤;½j    k
Aš16
Bš18
Cš8 + 4 2
Dš9 + 15 2
2
Ę÷ 6-2j2024·IJőơʼn··ĶkÖ𝐹1f𝐹2½ C
𝑥2
3 −𝑦2 = 1!ñ>ò01fQ ½ òĨ6·
1f1 Pj0f2kšç|𝑄𝐹1| + |𝑃𝑄|C:¤;’f|𝑄𝐹2|!;½j    k
Aš 3−2
Bš 3 +
2
Cš 6−2
Dš 6 +2
      
      
Ę÷ 6-3j2024·ƢżƣƤ··Ķkĭ 𝑥2
4 −𝑦2
12 = 1!òĨ6·1 PfWð” ⊙𝐶1:(𝑥+ 4)2 + 𝑦2 = 3=
⊙𝐶2:(𝑥−4)2 + 𝑦2 = 1¦ſ fſ1Wð½ MfNfÉ 𝑃𝑀+ 𝑃𝑁⋅𝑁𝑀!:¤;½j    k
Aš28
Bš29
Cš30
Dš32
 7   / <=>?!:;5
ı 7j2024·IJőơʼn··Ķkŀ} C
𝑥2
𝑎2−𝑦2
𝑏2 = 1j𝑎> 0f𝑏> 0k!ñò01Wð½𝐹1f𝐹2fL
¹¼½ 6fab )*½𝑦=±
1
3𝑥fŽ1𝑀© ñĨ6f1𝑁½O𝐸:𝑥2 + 𝑦+
6
2 = 16·1fÉ|𝑀𝑁| +
|𝑀𝐹2|!:¤;½j    k
Aš8
Bš9
Cš10
Dš11
Ę÷ 7-1j2024·gh·ĶĽľĿkÖ 𝐶
𝑥2−𝑦2
24 = 1!ñ01=ò01Wðp𝐹1f𝐹2f1𝐴p𝐶òĨ
6!·1fÉ|𝐴𝐹1| +
4
|𝐴𝐹2|!:¤;½j    k
Aš5
Bš6
Cš7
Dš8
Ę÷ 7-2j23-24 ĵ6·gh·GƥĿƦkŀ}|¹ !/@©ª'º1f01©𝑥¹6fñ01½
𝐹1f08½ 4f1𝐴!ª'½(2,1)f𝑃½ òĨ6·Ž1fÉ|𝑃𝐹1|−|𝑃𝐴|!:;½jƧƧk
Aš2 2
Bš 17
Cš2 2 +1
Dš2 2 +
5
Ę÷ 7-3j23-24 ĵ6·şŶƨʼn·Ńńkŀ}1𝑀(1,2)f1𝑃p 𝐶
𝑥2
4 −𝑦2
12 = 1ñĨ6!Ž1f𝑁p
O𝐷
(𝑥+ 4)2 + 𝑦2 = 16!Ž1fÉ|𝑃𝑀|−|𝑃𝑁|!:¤;½j    k
Aš5−10
Bš 10−5
Cš 13−3
Dš3−13
 8  , !9@AB%C;DE
ı 8j2024·şŶ·ĶĽľĿk 𝑥2
𝑎2−𝑦2
𝑏2 = 1(𝑎> 0,𝑏> 0)!·Ùab ĭ1(−1, 2)fÉ !9@
A½j    k
Aš 3
Bš 2
Cš2 3
Dš2 2
Ę÷ 8-1j2024·ņŇ·ĶĽľĿkŀ} 𝐸:𝑥2
𝑎2−𝑦2
𝑏2 = 1(𝑎> 0,𝑏> 0),𝐹,𝐴Wð½𝐸!ò01=ñ`1f
      
      
1𝑀(−2,3)p 𝐸6!1fí △𝐴𝑀𝐹!œŵ½
9
2fÉ 𝐸!9@A½j    k
Aš 3
Bš2
Cš 6
2
Dš 6
Ę÷ 8-2j2024·ņŇŴş·2ĶkÖ𝐹1,𝐹2Wð½ 𝐶:𝑥2
𝑎2−𝑦2
𝑏2 = 1(𝑎> 0,𝑏> 0)!ñò01fĭ1𝐹2!
Í Ŋ òĨ 1𝑀fŊ𝑦¹ 1𝑁fĮ𝐹2½ <𝑀𝑁!/1fēƃƄ𝐹1𝑀⊥𝐹1𝑁fÉ 𝐶!9@A
½j    k
Aš 3+1
2
Bš 3 +1
Cš2
Dš 5 +1
Ę÷ 8-3j2024·ŹźƩʼn·2Ķkŀ} 𝑥2
𝑎2−𝑦2
𝑏2 = 1(𝑎,𝑏> 0)6Ɔ© º1/@^_!1 AfBf
Ç$ 6!ƪ·1 CfƫƬ △𝐴𝐵𝐶½Ɲ234fÉŏ 9@A!C;DEpj    k
Aš( 2, + ∞)
Bš( 3, + ∞)
Cš(2, + ∞)
Dš 2 3
3 , + ∞
 9   !FGHIJK5
ı 9j2024·ƭĖƭʼn·ĶĽľĿkÇ𝑦=± 3𝑥½ab ! ˆÇpj    k
Aš𝑥2
3 −𝑦2 = 1
Bš𝑥2−𝑦2
9 = 1
Cš𝑦2
3 −𝑥2 = 1
Dš𝑦2−𝑥2
9 = 1
Ę÷ 9-1j2024·Őő·2Ķk 𝐶:𝑦2
𝑎2−𝑥2 = 1(𝑎> 0)!601𝐹27 ·Ùab !89½
𝑎
2fÉ
 Ùab !ÁAŧŵ½j    k
Aš−4
Bš4
Cš−2
Dš2
Ę÷ 9-2j2024·ƮƯưƱ·2Ķkŀ} )*½2𝑥2−𝑦2 = 𝜆j𝜆≠0kfÉàÅ𝜆!;ĘÂĘ!
pj    k
Aš`1ª'
Bš08
Cš9@A
Dšab )*
Ę÷ 9-3j2024·IJl·ĶĽľĿk 𝛤:𝑦2
𝑎2−𝑥2
𝑏2 = 1(𝑎> 0,𝑏> 0)!01Wð½𝐹1,𝐹2fĭ𝐹2!Í +
%·ĨŊ 𝐴f𝐵1f1𝐵©eņƲƳ.Ç𝐹1½O@f𝛤!L¹¼½»Ĭ!O+ <𝐴𝐹1,𝐵𝐹1WðŊ  MfN
1fĮ|𝐴𝑀| = 3|𝐵𝑁|,𝐹1𝐵⊥𝐹2𝐵fÉ𝛤!ab )*pj   k
Aš𝑦=±
6𝑥
Bš𝑦=±
6
2 𝑥
Cš𝑦=±
6
3 𝑥
Dš𝑦=±
6
4 𝑥
      
      
 10   !LM&5
ı 10j2024·gh·ĶĽľĿkOq !ƴJK©LMƵ™/‘ƶŻƷ!&šƸhƹÔƺūëƻ!
“ Ƽƽƾƿ”ďpöY !ƴJKfÛv !·ž01ƅŠ!ƴ fŎĭ ǀƅĜf
ǀƅƴ !ǀ”ǁ¼ ‡Ŏĭ !ƪ·ž01šÞ[fŀ} C
𝑥2
𝑎2−𝑦2
𝑏2 = 1(𝑎> 0,𝑏> 0)!ñ>ò
01Wð½𝐹1f𝐹2fçǂƅƴ 𝐹2𝑃=ǀƅƴ PE ÊËÌ͒j%/ P ½ǂƅ1kfcos∠𝐹1𝐹2𝑃=
7−1
4 f
Éŏ !9@A½j    k
  
Aš 2
Bš2
Cš 7
2
Dš 7
Ę÷ 10-1j23-24 ĵĄ·Źź·ǃ<DŽDžkźődžLJĚLjljNJf‹‘|¹ 4!LjljNJjÞ[kf
lj`9džœ 10 Njfdžœnj𝐴𝐵= 20 5Njfídžœ6Ǎ 5 NjfÉdžœnj½j    k
Aš10 2Nj
Bš15 2Nj
Cš12 3Nj
Dš30 Nj
Ę÷ 10-2j2024·rǎǏǐǑ·ĶĽľĿkǒ/@õ7%Ɲż>ƝŞ>Ɲl)”2žǓĿ1!ǔǕǖƝŞ>
ƝlžǓĿ1ã’Ɗ7Y·ƒǗĤfƝżǓĿ1Ɗ7!’¨%èǓĿ1ǘ 2sfŀ}ǙǓĿ17ŏ/
@!89p 680mfÉŏǗĤǚƵ©õǔ/@!j    kƌ(Ǜ"ç’ƒƓƍǜ!Ɣ½ 340m/sfËǙ1Ɨ
©ã·›œ6)
AšŞŽl 45°)”f89 340 3m
BšżŽő 45°)”f89 340 3m
CšŞŽl 45°)”f89 170 3m
DšżŽő 45°)”f89 170 3m
Ę÷ 10-3j23-24 ĵ6·IJő·ǃ<DŽDžkGǝœp:ǞÖUǟĻǠ!–êŧ·fèˆÇÍ!ǡ
ǢĖšfķ‰ǣǤǥ!ǦǧfǨ‰:Ǥ!ǩǪxǫǬ–ê!ǭŠ.Þ[ 1fǮ_¤ǯǰ!ŻʼnDZ¬ /h
ŻʼnDzfè!ž4ďpGǝœfˆyëp !·űWdz%¾¹ǴĀÆ4ë!œ.ǒDzUǵĖšĂ
ų ŻʼnDZ!Ƕ'ĖǷfāǶ'ĖǷ!›œ[4p fÞ[ 2f:Ůƌ!ÍĬ½100 mfǸǹ!ÍĬ½
      
      
50 22 mfǸ`ÍĬ½50 6 mf:Ůƌ8Ǹǹ300 mfÉŏǶ'ĖǷ!½j    k
Aš350 m
Bš375 m
Cš400 m
Dš450 m
 11  NO+ PQ
ı 11j2024·ņŇǺƢ·2ĶkÖ 𝐶1:𝑥2
𝑎2−𝑦2 = 1(𝑎> 0)fNO𝐶2:𝑥2
4 + 𝑦2 = 1!9@AWð½𝑒1,𝑒2f
í𝑒1 = 2 3𝑒2fÉ𝑎= j    k
Aš 2
8
Bš 2
4
Cš 2
2
Dš 6
3
Ę÷ 11-1j2024·ƢŞǻº··ĶkÖ 𝑥2
𝑎2−𝑦2
𝑏2 = 1j𝑎>𝑏Ɨ½Ɲ;k!ab !ǼÁ3½𝛼fĮŏ
 +NO𝑥2
4 + 𝑦2
3 = 1!9@Aŧŵ½ 1fĮ‘Ëã!08fÉsin𝛼= j    k
Aš 3
7
Bš 7
13
Cš 3
2
Dš 2
13
Ę÷ 11-2j2024·ƢżǽǾ·ĵĶkŀ}𝑒1,𝑒2Wð½NO𝑥2
𝑎2 + 𝑦2
𝑏2 = 1(𝑎> 𝑏> 0)= 𝑥2
𝑎2−𝑦2
𝑏2 = 1!9@
Af ab !ÁAàŬĭ2 5
5 fÉ
𝑒2
𝑒1!:;pj    k
Aš2
Bš3
Cš4
Dš5
Ę÷ 11-3j2024·gh·ĶĽľĿkŀ}NO𝐶1: 𝑥2
𝑚2 + 𝑦2
𝑛2 = 1(𝑚> 𝑛> 0)+ 𝐶2:𝑥2
𝑎2−𝑦2
𝑏2 = 1(𝑎> 0,𝑏> 0)
‘įã!01𝐹1,𝐹2f1𝑃½ !·žį1fĮ∠𝐹1𝑃𝐹2 = 60∘fNO!9@A½𝑒1f !9@A½
𝑒2fƀƁ𝑒2
1 + 𝑒2
2:¤½j    k
Aš2+ 3
4
Bš2+ 3
2
Cš3+2 2
4
Dš3+2 2
2
·>G
1šj2024·Ƣżǿş·ĶĽľĿkŀ} 𝐶:𝑥2
8 + 𝑦2
𝑚= 1(𝑚≠0,𝑚≠8)fÉ“𝑚∈(0,8)”p“ 𝐶!01©𝑥¹
6!j    k
      
      
AšĸWàĺRÙÚ
BšĺRàĸWÙÚ
CšĸRÙÚ
DšķàĸWĹàĺRÙÚ
2šj2024·ƚŞƛƜ·ĶĽľĿkÖ𝐹1f𝐹2p 𝐶: 𝑥2
4 −𝑦2
8 = 1!ñfò01fĭ𝐹1!Í +𝑦¹=𝐶!òĨ
WðŊ 1𝑃f𝑄fí △𝑃𝑄𝐹2pƝ234fÉ|𝑃𝐹1| = j    k
Aš2
Bš4
Cš8
Dš16
3šj2024·IJőȀœ·ĶĽľĿk©›œÍ3ª'«𝑥𝑂𝑦/f1 F !ª'½(2,0)fÇ < FP ½ÍĬ!O+O
𝑂:𝑥2 + 𝑦2 = 3ËſfÉŽ1 P !-.)*½j    k
Aš𝑥2
4 −𝑦2
3 = 1
Bš𝑥2
3 −𝑦2 = 1
Cš𝑥2
12−𝑦2
9 = 1
Dš𝑥2
16−𝑦3
3 = 1
4šj2024·noő÷ĵĶkŀ} 𝑥2
𝑎2−𝑦2
𝑏2 = 1j𝑎> 0f𝑏> 0k!ñ>ò01Wð½𝐹1f𝐹2fĭ𝐹2ĮÁA
½
24
7 !Í + ©e·ƲƳ!Ŋ1½ Afí|𝐹1𝐹2| = |𝐴𝐹2|fÉā !'()*ˆ‰½j    k
Aš𝑥2
3 −𝑦2
4 = 1
Bš𝑥2
4 −
𝑦
3 = 1
Cš𝑥2
9 −𝑦2
16 = 1
Dš𝑥2
16−𝑦2
9 = 1
5šj2024·rǎȁ/·ĶĽľĿkŀ} 𝐶:𝑥2
𝑎2−𝑦2
𝑏2 = 1(𝑎> 0,𝑏> 0)!ñ01½𝐹fĭª'º1𝑂!Í +
 𝐶Ŋ 𝑀,𝑁1fĮ1𝑀©e·ƲƳfƃƄ𝑂𝑀= 𝑂𝐹.í1𝑃© 𝐶6fĮ𝑁𝑃= 4𝑁𝐹fÉ 𝐶!
9@A½j    k
Aš 5
2
Bš 10
2
Cš2 2
Dš 5
6šj2024·ŐőȂœ·2Ķkŀ} 𝐶
𝑥2
𝑎2−𝑦2
𝑏2 = 1j𝑎> 0f𝑏> 0k!ò01½𝐹fñ>ò`1Wð½𝐴1f
𝐴2f1𝑀©𝐶6Į𝑀𝐹⊥𝑥¹fÍ 𝑀𝐴1f𝑀𝐴2+𝑦¹WðŊ 1𝑃f𝑄fí3|𝑂𝑄| = 4|𝑂𝑃|j𝑂½ª'º1kf
É𝐶!ab )*½j    k
Aš𝑦=± 2 6𝑥
Bš𝑦=± 2 10𝑥
Cš𝑦=± 4 3𝑥
Dš𝑦=± 2 15𝑥
7šj2024·ƢŞǻº·ĶĽľĿk©›œÍ3ª'«/fŀ}1 A ª'½(0,−6)fíŽ1 P ¬  y ¹òȃf
Į7"1𝐹1(−3,0)f𝐹2(3,0)!89ŧ?½"; 4fÉ △𝐴𝑃𝐹1Ł¼!:¤;½j    k
Aš3 + 4 5
Bš3 + 6 5
Cš4 + 4 5
Dš4 + 6 5
8šj2024·ƚŞȄŇ·ĶĽľĿkŀ} C
𝑥2
2 −𝑦2
2 = 1!ñ>ò01Wð½𝐹1f𝐹2f1 P p C !òĨ6
!·1fC ©1 P ƌ!ſ + C !ab Ŋ  MfN 1fO ½ª'º1fȅŠĄøņž–ä
①Í 𝐹1𝑃!ÁA!C;DEp(−1,1)ǖ
      
      
②1 P 7 C !Ùab !89ŧŵ½
1
2ǖ
③|𝑃𝑂|2 = |𝑃𝐹1| ⋅|𝑃𝐹2|ǖ
④|𝑃𝑀| = |𝑃𝑁|š
%/ƑƝіä!ž
pj    k
Aš1
Bš2
Cš3
Dš4
ĵ>Ě
9šj2024·ŻżȆǎ·ĶĽľĿkŀ} 𝐶!)*½𝑥2
𝑎+ 𝑦2
3 = 1fÉj    k
Ašç𝑎<0’f 𝐶ęì 
Bšç0 < 𝑎< 3’f 𝐶ęì01©𝑥¹6!NO
Cšç𝑎= 3’f 𝐶ęìO
Dšç𝑎> 3’f 𝐶ęì01©𝑦¹6!NO
10šj2024·rǎ·2Ķkŀ} 𝐶:𝑥2
𝑎2−𝑦2
16 = 1(𝑎> 0)!ñfò01Wð½𝐹1,𝐹2,𝑃½ 𝐶61fĮ △𝑃𝐹1
𝐹2!sſOO@½𝐼(3,1)fÉĄøȇÎƝÑ!pj    k
Aš𝑎= 3
BšÍ PF1 !ÁA½
1
4
Cš △𝑃𝐹1𝐹z!Ł¼½
64
3
Dš △𝑃𝐹1𝐹2!žõO»Ĭ½
65
12
11šj2024·Ȉȉźǎ·2Ķkŀ}1𝑃(1, 2)p 𝐶:3𝑥2−𝑦2 = 16·1fĭ𝑃” !Ùab ¦
Ì fÌƄWð½𝐴,𝐵fÉĄøȇÎƝÑ!pj    k
Aš !Źb )*½𝑦=±
3𝑥
Bš !017ab !89½ 1
Cš|𝑃𝐴| ⋅|𝑃𝐵| =
1
3
Dš △𝑃𝐴𝐵!œŵ½ 3
16
2>„…
12šj2024·lmȊ·2Ķk 𝑦2−𝑥2 = 1!01ª'p      .
13šj2024·ƞȋũŇ··Ķkŀ} 𝐶:𝑥2
𝑎2−𝑦2
𝑏2 = 1(𝑎> 0,𝑏> 0)!ñ>ò01Wð½𝐹1,𝐹2fÇ <𝐹1𝐹2½
ÍĬ!O+ 𝐶©e·ƲƳ!Ŋ1½𝑃fí∠𝐹1𝑃𝐹2!s3›W +𝑥¹!Ŋ1𝑀›W <𝑂𝐹2fÉ 
𝐶!9@A½      š
      
      
14šj2024·źȌőī·ĶĽľĿk©›œÍ3ª'«𝑥𝑂𝑦/f𝐹1f𝐹2Wðp 𝐸
𝑥2
4 −𝑦2
5 = 1!ñfò0
1fÖ1𝑃p𝐸!òĨ6·1fÉ
5
|𝑃𝐹1|−
1
|𝑃𝐹2|!:;½      .
ņ>Z†
15šj23-24 ĵ6·gh·GƥĿƦkWð,ȍQĄøÙÚ! !'()*
(1)Ŏĭ𝐴(−7,−6 2),𝐵(2 7,3)1ǖ
(2)+ 𝑥2
2 −𝑦2 = 1‘į!ab ,Įĭ1( 2, 2)š
16šj23-24 ĵ6·no·ǃ<DŽDžkŀ} C
𝑥2
𝑎2−𝑦2
𝑏2 = 1j𝑎> 0f𝑏> 0k+ 𝑦2
4 −𝑥2
2 = 1‘Ëã!
ab fĮŎĭ1𝑀( 2,−2)š
(1), 𝐶!)*ǖ
(2), 𝐶!L¹¼f01ª'f9@Aš
17šj23-24 ĵ6·IJlLjȎȏ·ǃ<DŽDžkŽ1𝑀(𝑥,𝑦)+"1𝐹(5,0)!89=è7"Í 𝑙:𝑥=
9
5!89!
p
5
3fƘŽ1 M !-.½ Cš
(1), C !)*ǖ
(2)íŽ1 M © y ¹òȃf"1𝐴(5,2),,|𝑀𝐴| +
3
5|𝑀𝐹|!:¤;š
      
      
18šj23-24 ĵ6·ƮƯȐũ·Ńńkŀ} 𝐶:𝑥2
2 −𝑦2 = 1,𝑃p𝐶6!ï¿·1.
(1)Ö1𝐴!ª'½(4,0)f,|𝑃𝐴|!:¤;ǖ
(2)í𝐹1,𝐹2Wð½ !ñ､ò01f∠𝐹1𝑃𝐹2 = 60∘f, △𝑃𝐹1𝐹2!œŵ.
19šj2024·şŶȑŐ·ĶĽľĿkÖ 𝐸
𝑥2
𝑎2−𝑦2
𝑏2 = 1j𝑎> 0f𝑏> 0kĭ𝑃1( 3,0)f𝑃2(3,4)f𝑃3(3, 2)f
𝑃4(−3, 2)ņž1/!2ž1š
(1), 𝐸!)*ǖ
(2)ĭ1𝐹¦ÙÊËÌÍ!Í 𝑚f𝑛f%/𝑚+𝐸!òĨŊ 𝐴f𝐵1f+Í 𝑥=
3
2Ŋ 1𝑀f𝑛+𝐸!ò
ĨËŊ 𝐶f𝐷1f+Í 𝑥=
3
2Ŋ 1𝑁f,
1
|𝑀𝐴| +
1
|𝑀𝐵| +
1
|𝑁𝐶| +
1
|𝑁𝐷|!:;š