专题8.5椭圆十大题型（原卷版）_11.4-2_2026-11-04

      
      
专题8.5 椭圆【十大题型】

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³ 1f2024·ĨĩĪī·öĬgĭˆ𝐹1,𝐹2X¿l𝑀:𝑥2
16 + 𝑦2
5 = 1 Į8į12i𝑃²𝑀<Ž2iđ|𝑃𝐹1
      
      
| = 3iă|𝑃𝐹2| = f    g
A–2
B–3
C–5
D–6
Ä} 1-1f2024·½İıIJ·öĬgĭˆ2 P ¢ C
𝑥2
9 + 𝑦2
8 = 1<iC Į12² Fiđ+¾𝑃𝐹 02
¢yÜ2 O ²Ei|𝑂𝐹|²Àij <iă|𝑃𝐹| A²f    g
A–2
B–4
C–6
D–8
Ä} 1-2f2024·ĴIJĵĶ·öĬgĭˆ𝐸:𝑥2
𝑎2 + 𝑦2
𝑏2 = 1(𝑎> 𝑏> 0) Į8į12X¿²𝐹1,𝐹2–2𝑃¢𝐸
<iÐ𝑃𝐹1 ⊥𝑃𝐹2–|𝑃𝐹1| ⋅|𝑃𝐹2| = 2iă𝑏= f    g
A–
1
2
B–1
C– 3
D–2
Ä} 1-3f2024·ķīķĸ·ŽĬgđ𝑃²𝐶∶𝑥2
121 + 𝑦2
96 = 1<Ž2i𝐹1,𝐹2²𝐶 ˜12iÐ|𝑃𝐹2| = 8i
ă|𝑃𝐹1| = fĹĹg
A–10
B–12
C–14
D–16
 2   &'()
³ 2f2024·ĵĺ‹Ļ·ĬļĽľgĭˆ12¢𝑥°<  ?EF² 2
3 i1>²2 2iăĿ ()
²f    g
A–𝑥2
3 + 𝑦2 = 1
B–𝑥2
9 + 𝑦2 = 1
C–𝑥2
9 + 𝑦2
7 = 1
D–𝑥2
36 + 𝑦2
28 = 1
Ä} 2-1f2024·bc·ĬļĽľgĭˆ𝐹1,𝐹2X¿l𝐸:𝑥2
𝑎2 + 𝑦2
𝑏2 = 1(𝑎> 𝑏> 0) Į8į12i𝑀¢𝐸<i
𝑁¢𝑦°<i2𝐹2𝑁= 3𝑀𝐹2iy𝑀𝑁²ûij ŀ𝐹1iÐ △𝑀𝐹1𝐹2 9:²
20
3 iă𝐸 &'()²f    g
A–𝑥2
25 + 𝑦2
10 = 1
B–𝑥2
25 + 𝑦2
20 = 1
C–𝑥2
20 + 𝑦2
5 = 1
D–𝑥2
15 + 𝑦2
10 = 1
Ä} 2-2f2024·Łĩĵł·ĬļĽľgĭˆ𝐶:𝑥2
𝑎2 + 𝑦2
𝑏2 = 1(𝑎> 𝑏> 0) Į､į12X¿²𝐹1(−2,0)，
𝐹2( 2,0)iT+𝑙,𝑥° ¸2²𝑃(−3 2,0)iŀį12𝐹2œ𝐹2𝑀⊥𝑙™2𝑀，|𝐹2𝑀| = 4iÐ𝐹2𝑀 02𝑄¢
𝐶<iă𝐶 ()²f    g
A–𝑥2
6 + 𝑦2
4 = 1
B–𝑥2
5 + 𝑦2
3 = 1
      
      
C–𝑥2
4 + 𝑦2
2 = 1
D–𝑥2
3 + 𝑦2 = 1
Ä} 2-3f2024·ŃĩńŃ·öĬgĭˆ𝐸:𝑥2
𝑎2 + 𝑦2
𝑏2 = 1(𝑎> 𝑏> 0) <`2²𝑃i?EF²
1
2iŀ$Į1
2Ņņ4² 30° û+𝑙¸𝐸™𝐴i𝐵˜2iđ △𝑃𝐴𝐵 67² 16iă𝐸 ()²f    g
A–𝑥2
4 + 𝑦2
3 = 1
B–𝑥2
12 + 𝑦2
9 = 1
C–𝑥2
16 + 𝑦2
12 = 1
D–𝑥2
36 + 𝑦2
27 = 1
 3  *+(),
³ 3f2024·ķī·öĬgĭˆ() 𝑥2
𝑘−4 + 𝑦2
8−𝑘= 1 ¡ *+liăQ
 k HAIJlf    g
A–(4,6)
B–(6,8)
C–(4,8)
D–(4,6) ∪(6,8)
Ä} 3-1f2024·bc·ĬļĽľgq“Q
𝑝∈(1,3)”lq“*+(3−𝑝)𝑥2 + (𝑝−1)𝑦2 = (3−𝑝)(𝑝−1) 
¡” Žf    g
A–ŇSéê
B–ŇX‚ňSéê
C–ňS‚ŇXéê
D–ʼn‚ŇX‚ňSéê
Ä} 3-2f2024·Łĩĩĵ·ĬļĽľgĭˆÚ[²ĝ
①𝑦= 𝑥𝑐„②𝑦= 𝑎𝑥„③𝑦= 𝑏𝑥 [Ŋiă()
(𝑎−1)𝑥2 + (𝑏−1)𝑦2 +𝑐= 0 ¡f    g
A–12¢𝑥°< ŋ*+
B–12¢𝑦°< ŋ*+
C–12¢𝑥°< 
D–12¢𝑦°< 
Ä} 3-3f2024·ŌĪ·ĬļĽľgđ()(𝑚+ 1)𝑥2 + (1−𝑚)𝑦2 = 1−𝑚2 ¡12¢𝑥°< iă
f    g
A–−1 < 𝑚< 1
B–0 < 𝑚< 1
C–−1 < 𝑚< 0
D–−1 < 𝑚< 0G0 < 𝑚< 1
 4  -./——
³ 4f23-24 3î·nō·Ŏ0g7² 2 +¾𝐴𝐵 ˜ŏ2𝐴B𝐵X¿¢𝑥°B𝑦°<Őőiă2𝐴™
2𝐵 ^_2𝑀 -.()²f   g
A–𝑥2
4 + 𝑦2
2 = 1
B–𝑦2
4 + 𝑥2
2 = 1
C–𝑥2
16 + 𝑦2
4 = 1
D–𝑦2
16 + 𝑥2
4 = 1
      
      
Ä} 4-1f2024·bc·ŽĬg—9û4£&¤0i‡Œ △𝐴𝐵𝐶 Œ7² 2iM ²𝐵𝐶02iBiC X¿¢
œ+𝑦=
3𝑥i𝑦= −3𝑥(𝑦≥0)<Ŕőiŕ M -.²𝐶1iăf    
g
A–𝐶1²ŖX
B–𝐶1²ŖX+¾
C–𝐶1²ŖXŗ+
D–𝐶1²ŖX
Ä} 4-2f2024·ŘřŚś·ĬļĽľgŜŝŞæŎ 
şŠšŢ¢; ţ
ŤťŦ0ŧòÇm*
+ 12B'+ OP
—9o=Ž!2B!û+ >?þŽ!Á³ º¼2 -.lŽm*+–ß
lm*+ hö!"G_²UŽ!"–đ—9oŽő2𝑃(𝑥,𝑦)=!2𝐴(1,0)B=!û+𝑥= 4 >?ŨÁl
1
2iă2𝑃 -.²f    g
A–
B–
C–ŋ*+
D–ŗ+
Ä} 4-3f2023·ŃũĪŪ·ĬļĽľgĭˆ𝐶 ()²𝑥2 + 𝑦2 = 16iû+𝑙²𝐶 ū+iŕ𝐴(−2,0),𝐵
(2,0)˜2=û+𝑙 >?X¿²𝑑1,𝑑2iő2𝑃Ŭŭ|𝑃𝐴| = 𝑑1i|𝑃𝐵| = 𝑑2iăő2𝑃 -.()²f   g
A–𝑥2 + 𝑦2 = 4
B–𝑥2
16 + 𝑦2
12 = 1
C–𝑥2
16−𝑦2
12 = 1
D–𝑦2 = 4𝑥
 5  012345 6789:#$;/
³ 5f2024·ŮůŃŰűŲ·3Ĭgĭˆ𝐹1l𝐶:𝑥2
2 + 𝑦2 = 1 Į12iû+𝑥= 1,𝐶¸™𝐴8𝐵˜2i
ă △𝐹1𝐴𝐵67²f    g
A– 2
B– 3
C–2 2
D–4 2
Ä} 5-1f2024·nō·ĬļĽľgĭˆ𝐹1,𝐹2l𝐶:𝑥2
9 + 𝑦2
5 = 1 Į8į12i2 P ¢ C <iÐ+¾𝑃𝐹1
 02¢y𝐹1𝐹2²ûij <iă345𝑃𝐹2𝐹1 9:²f   g
A–1
B– 15
2
C– 15
D–8
Ä} 5-2f2024·ŌĪųŴŵ·öĬgĭˆ𝐶:𝑥2
9 + 𝑦2
𝑚= 1(0 < 𝑚< 9,𝑚∈𝑍) Į､į12X¿²𝐹1,𝐹2i
2𝑃¢𝐶<Ĥ‚¢£&°<iÐ △𝑃𝐹1𝐹2l‡Ŷ345i$0Žo4 ĀāA²
7
8iă𝑚= f    g
A–4
B–5
C–6
D–8
Ä} 5-3f2024·ĨŷŸIJ·3Ĭgĭˆ𝐶:𝑥2
9 + 𝑦2
5 = 1 Į8į12X¿²𝐹1i𝐹2iŀ2𝐹2 û+𝑙,
𝐶 Ž¸2²𝐴iđ|𝐴𝐹2| = 4iă △𝐴𝐹1𝐹2 9:²f    g
A–2 3
B– 13
C–4
D– 15
 6  <2=$;2>? @A/
³ 6f23-24 ö<·ŮůŃŰűŲ·ŎŹgĭˆ𝐶:𝑥2
25 + 𝑦2
9 = 1 Į12²𝐹,𝑃²𝐶<źÃŽ2iă|𝑃𝐹|
      
      
 @ƒA²f    g
A–5
B–9
C–10
D–18
Ä} 6-1f23-24 ö<·Řř7Ż·ŎŹgĭˆ𝐵l𝑥2
3 + 𝑦2 = 1 <`2i2𝑀l< źÃŽ2i
ă|𝑀𝐵| @ƒA²f    g
A–2
B–2 2
C–3 2
2
D–
9
2
Ä} 6-2f23-24 ö<·żŃ·Ŏ0gĭˆ2𝐹²𝐶:𝑥2
25 + 𝑦2
16 = 1 į12i2𝑃l𝐶< ő2i2𝑄
l𝑀:(𝑥+ 3)2 + 𝑦2 = 1< ő2iă
|𝑃𝐹|
|𝑃𝑄| @ÎAlf    g
A–
1
2
B–
2
9
C–
2
3
D–
8
3
Ä} 6-3f2024·Łĩĩĵ·ŽĬgĭˆ2𝑀¢𝑥2
18 + 𝑦2
9 = 1<Ŕői2𝑁¢𝑥2 + (𝑦−1)2 = 1<Ŕői
ă|𝑀𝑁| @ƒA²f    g
A–1 +
19
B–1 + 2 5
C–5
D–6
 7  <2=12B!2>? B8C@A
³ 7f2024·Ž·öĬgè𝐹1,𝐹2X¿l𝑥2
9 + 𝑦2
4 = 1 Įiį12iŀ𝐹1 û+𝑙¸™𝐴,𝐵˜2i
ă|𝐴𝐹2| + |𝐵𝐹2| @ƒA²(    )
A–
29
3
B–
28
3
C–
8
3
D–6
Ä} 7-1f2024·žſ!ĩ·ĬļĽľgĭˆ C
𝑥2
9 + 𝑦2
5 = 1 Į8į12X¿²𝐹1i𝐹2iA l C <Ž
2i𝐵(2,1)iă|𝐴𝐵| + |𝐴𝐹1| @ƒA²f    g
A–7
B–8
C–9
D–11
Ä} 7-2f24-25 3<·bc·ƀ¾ƁƂgè𝐶：
𝑥2
4 + 𝑦2
3 = 1 į12²𝐹iő2𝑃¢𝐶<i2𝐴l
û+4𝑥−5𝑦−12 = 0< ő2iă|𝑃𝐴|−|𝑃𝐹| @ÎA²f    g
A–−16 41
41
B–16 41
41
C–16 41
41 −4
D–4−16 41
41
Ä} 7-3f2024·ŃũƃIJ·ĬļĽľgĭˆ F ²𝐶:𝑥2
4 + 𝑦2 = 1 į12iP ² C <Ž2iQ ²𝑀:𝑥2
+ (𝑦−3)2 = 1<Ž2iă|𝑃𝑄| + |𝑃𝐹| @ƒA²f    g
A–5
B–5 + 2 3
C–3 + 2 3
D–6
      
      
 8  D ?EFG$HAIJ
³ 8f2024·Ĩŷ·ŽĬgĭˆ2 FiA X¿l𝑥2
𝑎2 + 𝑦2
𝑏2 = 1(𝑎> 𝑏> 0) Į128į`2i𝐵(0,𝑏)Ŭŭ
𝐹𝐵⋅𝐴𝐵= 0iă ?EF‡™f    g
A– 3+1
2
B– 5−1
2
C– 3−1
2
D– 5+1
2
Ä} 8-1f2024·ŁĩƄİ·3ĬgĭˆÜ2²𝑂i𝐶:𝑥2
𝑎2 + 𝑦2
𝑏2 = 1(𝑎> 𝑏> 0),û+𝑙:𝑥−𝑦+ 1 = 0¸™𝐴,𝐵
˜2i+¾𝐴𝐵 02²𝑀iđû+𝑂𝑀 ņF²−
1
4iă𝐶 ?EF²f    g
A–
1
2
B– 3
2
C– 5−1
2
D– 6
3
Ä} 8-2f2024·ŌĪƅĸ·ĬļĽľg2𝑀l𝑥2
𝑎2 + 𝑦2
𝑏2 = 1(𝑎> 𝑏> 0)< 2iy𝑀²E ,𝑥°ě
ū™ 12𝐹i𝑀,𝑦°ě¸™𝑃,𝑄˜2iđ △𝑃𝑄𝑀lƆ4345iă?EF HAIJlf    g
A–(2−3,1)
B–
5−1
2
,1
C–
6−2
2
,1
D–
6−2
2
,
5−1
2
Ä} 8-3f2024·śŷƇƈ·3Ĭgĭˆ 𝑂 ²£&Ü2i𝐸:𝑥2
𝑎2 + 𝑦2
𝑏2 = 1(𝑎> 𝑏> 0) Įiį12X¿
²𝐹1，𝐹2iĮ8į`2X ¿²𝐴，𝐵i1>²2𝑐iy 𝐹1𝐹2 ²ûij ,𝐸 ¢hŽBh3ŊƉX¿¸
™ 𝑀，𝑁 ˜2.Ð𝑁𝑀⋅𝐴𝐵= 2 3𝑎𝑐iă𝐸 ?EF²f     g
A– 2
2
B– 2
C– 3
3
D– 6
3
 9   KLMNOP/
³ 9f2024·Ńĩ·ĬļĽľg𝐶:𝑥2
80 + 𝑦2
35 = 1 7°7,1>ŨC‡™f    g
A– 5
B–2 5
C–2 6
D–3 6
Ä} 9-1f2024·żŃ·ĬļĽľgĭˆ𝐸
𝑥2
𝑎2 + 𝑦2
𝑏2 = 1(𝑎> 𝑏> 0) Įį12=û+𝑙
𝑥−𝑦−4 = 0 
>?ŨC² 2iă𝐸 1>lf    g
A– 2
B–2
C–2 2
D–4
Ä} 9-2f2024·bc·ĬļĽľgĭˆ𝐸:𝑥2
𝑎2 + 𝑦2
𝑏2 = 1(𝑎> 𝑏> 0) Į8į12X¿²𝐹1,𝐹2i?EF²
2 2
3 i2𝐴¢𝐸<iÐ|𝐴𝐹1| = 2|𝐴𝐹2|i △𝐴𝐹1𝐹2 9:²4 7iă𝐸 1>²f    g
A–4 2
B–8 2
C–6
D–12
      
      
Ä} 9-3f2024·śŷƊƋ·3Ĭgĭˆ𝐹1i𝐹2X¿²𝐶
𝑥2
6 + 𝑦2
2 = 1 Į8į12i2𝑃(𝑥0,𝑦0) ¢𝐶
<iđ∠𝐹1𝑃𝐹2ƒ™
π
3iă𝑥0 HAIJlf      g
A–(−∞,−3) ∪( 3, + ∞)
B–(−3, 3)
C– −∞,−5 ∪
5, + ∞
D– −5, 5
 10   QR%/
³ 10f2024·Ĩŷƌ·ĬļĽľgƌIJƒƍlŽƎƏƐŋƑ9ƒƓƔ‹ƕņƖƍič¼ƗƘiƐ8ƕ
n8ƙƒ Ĕ2i»ƚŪÇ*Ńƛ8ƜŃƛ8ƝŃƛ¸ŪƞĐ ƟƠiþ²ơË*Ńƛ,ƢƣƤ nS
¸ŪƍƕiƒƍƥƦƧQƌ“3ƛƨ‹” nSƩqiƌIJƒƍ ƍƐƪ5sƫiđƍƐº¢—9Ƭ
ƍ9²+¾𝐴𝐵iÐ𝐴𝐵ŀ î12i𝐴𝐵= 44ƭiƍƐ@2𝑃>ƍ9110ƭiăÏ ?EF²
f    g
A–
1
3
B–
2
5
C–
2
3
D–
4
5
Ä} 10-1f2024·oƮŜƯư·ŽĬgÚ[º¡i¼Ʊ ƲOP
r Ž12~Ƴ Ʋ+i
ƴƵœæiƵœƲ+ƴŀ ƶŽ12.ÓÔ ƲOPZðî9 Ē
ĭˆ*+ C ()
²𝑥2
25 + 𝑦2
16 = 1i$Į8į12X¿l𝐹1i𝐹2iû+ l , C ū™2 PiÐ|𝑃𝐹1| = 2iŀ2 P Ð,û+ l Ʒ
û û+𝑙′,7°¸™2 Miă|𝐹1𝑀|:|𝐹2𝑀| = f    g
A–1: 3
B–1:2
C–1:3
D–1:4
Ä} 10-2f2024·nō·3ĬgÚ[º¡i“ƸƹŽ	”ŧƺƻƼƽƾƺĉƿ-ƞǀ…ƺǁi¢ƺǁǂsŽ
2𝑃Ä-ǃDŽyƺǁǁE𝐹²Ž12 -ƞ I DžƺǀdžiŨæƻƼ¢𝑃2höČÄ-ǃDŽLJy𝐹²Ž
12 -ƞ II Džƺǀdži@LjƻƼ¢𝑃2h3ČÄ-ǃDŽy𝐹²E 5-ƞ III Džƺǀdžiđ2𝑐1
      
      
B2𝑐2X¿ ¡-ƞ I B II 1>i2𝑎1B2𝑎2X¿ ¡-ƞ I B II 7° 7iăf    g
  
A–𝑎1−𝑐1 < 𝑎2−𝑐2
B–𝑎1 + 𝑐1 < 𝑎2 + 𝑐2
C–
𝑐2
𝑎1 >
𝑐1
𝑎2
D–
𝑐1
𝑎1 >
𝑐2
𝑎2
Ä} 10-3f2023·ĴIJljNJ·ĬļĽľgNjnjǍl0cǎUǏǐǑiǒǓĭ¼ 1000 ğa ǔǕi²ǖǎ
BǗĨŽǎUǏǐijkǘǙ™ŻXÌNJǚǛNjnjǍǐǜNJ.•ő0ĞNjnjǍǝǚæǞǟ¢Ǡƪ
ǛǡǢƾ<iÚ[º¡iĿǍ ǍƽlŽÀij² 2 iE=ǍǣǤŏ>?² 2iĸƲǥœNjnjǍ¢
ƾ95þǎ5ĦǦfŻXÌijk ĸƲ,ƾ9ǧ4²60°giđǍǣǤŏǨǩ¥™Ŀ 12
¥¦iăĿ ?EF²f    g
A–2−3
B– 2−1
C– 3−1
D– 2
2
Ž8L
1–f2024·śŷǪĪ·öĬg𝑥2
5 + 𝑦2 = 1 12£&lf    g
A–( ±
6,0)
B–( ± 2,0)
C–(0, ±
6)
D–(0, ± 2)
2–f2024·Ōjǫ!·3Ĭgĭˆ𝑃l𝐸
𝑥2
16 + 𝑦2
9 = 1<Ž2i𝐹1i𝐹2X¿²𝐸 Į8į12iă|𝑃𝐹1| + |𝑃𝐹2|
= f    g
A–8
B–6
C–4
D–3
3–f2024·ǬjǭIJ·3Ĭgĭˆ𝐶:(𝑥−1)2 + 𝑦2 = 2iû+𝑙:𝑎𝑥+ 𝑦−3 = 0i()𝑚:𝑥2
𝑎2 + 𝑦2 = 1iă“𝐶,
û+𝑙ěū”l“()𝑚 ¡ *+²” f    g
      
      
A–ŇXǮňSéê
B–ňSǮŇXéê
C–ŇSéê
D–ʼnǮŇXǮňSéê
4–f23-24 ö<·jkĩƤ·ŎŹgÚ[lŽ5ǯƍiÉǰ9¢𝑙DZÌi¢Ú[º¡ Ƭ9Dziƍdz
,$ǴĦǵǩýþŽ.ÏÌǯ`?ǰ92miǰ9Ƕ6miÝÞÉǰ¥<Ƿ1mÌiǰ9Ƕ²f    g
  
A–3 3m
B–3 3
2 m
C–4 2m
D–4 2
3 m
5–f2024·ŃĩĀ·ĬļĽľgĭˆ12¢𝑥°< 𝐶 Įį12X¿²𝐹1、𝐹2iƴŀ𝐹2 û+𝑙,𝐶¸
™𝐴、𝐵˜2iđ𝐹1𝐴⋅𝐹1𝐵= 16i𝐴𝐹1 ⋅𝐴𝐵= 9i𝐵𝐹1 ⋅𝐵𝐴= 0iă𝐶 ()²
f    g.
A–𝑥2
9 + 𝑦2
4 = 1
B–𝑥2
3 + 𝑦2
2 = 1
C–𝑥2
9 + 𝑦2
8 = 1
D–𝑥2
3 + 𝑦2 = 1
6–f2024·½İoŃ·3Ĭgè𝐹1、𝐹2l𝐶:𝑥2
9 + 𝑦2 = 1 ˜12i2 P ¢ C <iđ △𝑃𝐹1𝐹2²û
4345iă △𝑃𝐹1𝐹2 9:²f    g
A–2 2
3
B–1 G 2
C– 2
D–1 G2 2
3
7–f2024·ǬĪǸĸ·ĬļĽľgĭˆ𝐶:𝑥2
4 + 𝑦2
3 = 1 Į､į12²𝐹1,𝐹2,𝑀l𝐶<Žő2iû+
𝑙:𝑦= 𝑘(𝑥−1) +2ƴŀ !2²𝑁iă|𝑀𝐹1|−|𝑀𝑁| @ƒA²f    g
A– 2
B–2
C–2 2
D–6
8–f2024·ǬĪ·3Ĭgĭˆ𝐹1,𝐹2l𝐶:𝑥2
𝑎2 + 𝑦2
𝑏2 = 1(𝑎> 𝑏> 0) Į8į12iO l£&Ü2iŀ𝐹1œû+
, C ¸™ AiB ˜2iđ|𝐴𝐹2| = |𝐴𝐵|iÐ △𝑂𝐴𝐹2 9:² 3
6 𝑏2iă C ?EF²f    g
A– 3
12
B– 3
6
C– 3
3
D– 3
2
ö8ğ
9–f2024·ŃĩǹŻ·3Ĭgè C
𝑥2
8 + 𝑦2
4 = 1 Į8į12X¿²𝐹1i𝐹2i£&Ü2² O–đ C <
Ǻ¢Ž2 PiƩµ|𝑂𝑃| =
7iăîǻ¶ÒǨÕ ¼f    g
      
      
A–cos∠𝐹1𝑃𝐹2 =
3
5
B–𝑃𝐹1 ⋅𝑃𝐹2 = 5
C– △𝐹1𝑃𝐹2 9:² 2
D– △𝐹1𝑃𝐹2 oūÀij² 2−1
10–f2024·Ōj·3ĬgĭˆŽǼ¼ÀƟǰ ǽ5ǾǿȀi$Ǥ9Àij²3 cmiǾǿȀ²16 cmfǾ
ǿȁȂą‚Vgi$ŅņȃȄ Ǩȅ[Ú[º¡i𝑃𝑄 ¡ǰ—Ȇ9–ÉǾǿȀŅņÌiƟoǰ9²
5iȇĦŖX𝐴𝐵𝑁𝑀²Ɵoǰ Ǩȅ[–è∠𝐶𝐵𝑄= 𝜃iăîǻŠóǨÕ lf    g
A–É𝜃= 30°Ìi ?EF² 3
2
B–É ?EF@ƒÌitan𝜃=
1
2
C–É 1>² 4 Ìitan𝜃=
3
4
D–É𝜃= 45°Ìi 1>² 6
11–f2024·Ńĩ·ĬļĽľgĭˆ𝐴(−2,0)i𝐵(2,0)i𝐶(1,0)iő2𝑀Ŭŭ𝑀𝐴,𝑀𝐵 ņFŨ:²−
3
4iő2𝑀
 -.ŕ²Γiŀ2𝐶 û+¸Γ™𝑃i𝑄˜2iÐ𝑃i𝑄 02²𝑅iăf   g
A–𝑀 -.()²𝑥2
4 + 𝑦2
3 = 1
B–|𝑀𝐶| @ÎA² 1
C–đ𝑂²£&Ü2iă △𝑂𝑃𝑄9: @ƒA²
3
2
D–đ+¾𝑃𝑄 Ʒû—X+¸𝑥°™2𝐷iă𝑅2 Ȉ£&l𝐷2 Ȉ£& 4ȉ
38{|
12–f2024·<Ȋ·ĬļĽľgĭˆ() 𝑥2
𝑘−4 + 𝑦2
8−𝑘= 1 ¡ *+liăQ
𝑘 HAIJl      .
13–f2024·ĵĺŴȋś·ĬļĽľg0E¥™£&Ü2i^_°²£&° 𝐸 <8î12X¿²𝐹1,
𝐹2iį`2²𝐴iđ𝐸 7°7²4i∠𝐹1𝐴𝐹2 =
2π
3 iă𝐸 &'()²      
–
14–f2024·śŷǪĪ·3Ĭgĭˆ𝐹1、𝐹2l𝑥2
𝑎2 + 𝑦2
𝑏2 = 1(𝑎> 𝑏> 0) Įiį12i2𝑃²<Ž2i𝑂
²£&Ü2i △𝑃𝑂𝐹2²Ǩ345iăĿ ?EF²      .
½8Zõ
      
      
15–f24-25 ö<·bc·ȌȍĽƂgDȎ‹îǻéê  &'()–
(1)12¢𝑦°<iÐƴŀ˜2(0,2)B(1,0)„
(2)ĭˆ ˜12£&X¿l(−2,0)i(2,0)iĚÐƴŀ2 5
2 ,−3
2 –
16–f2024·jkƒȏ·3Ĭgĭˆ𝐶:𝑥2
𝑎2 + 𝑦2 = 1 Į8į12X¿²𝐹1i𝐹2iÐ2𝐴1,
2
2 ¢<iő
2 CiD X¿¢û+𝐴𝐹2B<.
(1)D &'()#$12£&„
(2)đ<Ǻ¢Ž2 EiƩµ½Œ5𝐹1𝐷𝐸𝐶lȐ5iD2 D Ȉ£&.
17–f23-24 öî·ĵĺĵō·ǚȑgĵōǘĎȒȓ ȔȕȖȗÿ˜ƒÎ‚ú ŋȘǁȗěșÍþi
$ȔȕȖȗ7°ȚțȜȝ 97 ƭi±°ȚțȜȝ 77 ƭi±°7,7°7 —(ÁËsȞȟÁ 0.618–Ƞüȡ
±°7,7°7 —(Á² 5−1
2 _²Ȟȟ–¼ŽȞȟ𝐶:𝑥2
𝑎2 + 𝑦2
𝑏2 = 1(𝑎> 𝑏> 0)$0 AiF X
¿²$Į`2Bį12iB ²<`2–
(1)DȞȟ C ?EF„
(2)Ȣú¢ȣȤȞȟ OPÌȥľ △𝐴𝐵𝐹Öí²û4345iȑġȦĿú ȥľlȧǨÕi̶·
ÿ–
      
      
18–f2024 3î·bc·ƁƂgĭˆ𝐶
𝑥2
𝑎2 + 𝑦2
𝑏2 = 1(𝑎> 𝑏> 0) Į12²𝐹iį`2²𝐴iÐ𝐶
<Ǻ¢2𝑃Ʃµ𝑃𝐹⊥𝑃𝐴iD𝐶 ?EF𝑒 HAIJ–
19–f23-24 ö·bc·ȌæœȨgĭˆ𝐶
𝑥2
25 + 𝑦2
16 = 1o¼Ž2𝑀(2,3)i𝐹1i𝐹2X¿² Į8į1
2i𝑃²𝐶< Ž2iD
(1)|𝑃𝑀|−|𝑃𝐹1| @ƒA,@ÎA„
(2)|𝑃𝑀| + |𝑃𝐹1| @ƒA,@ÎA–